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ons as ratios of the sides of a right triangle. 45° 1 2 45° 1 FIGURE 4.28 In Example 1, you were given the lengths of two sides of the right triangle, but not the angle Often, you will be asked to find the trigonometric functions of a given acute angle To do this, construct a right triangle having as one of its angles. . . Example 2 Evaluating Trigonometric Functions of 45 Find the values of sin 45, cos 45, and tan 45. Solution Construct a right triangle having as one of its acute angles, as shown in Figure 4.28. Choose the length of the adjacent side to be 1. From geometry, you know that the other acute angle is also So, the triangle is isosceles and the length of the opposite side is also 1. Using the Pythagorean Theorem, you find the length of the hypotenuse to be 2. 45. 45 sin 45 opp hyp 1 2 cos 45 adj hyp 1 2 2 2 2 2 tan 45 opp adj 1 1 1 Now try Exercise 17. 333202_0403.qxd 12/7/05 11:03 AM Page 303 Section 4.3 Right Triangle Trigonometry 303 Example 3 Evaluating Trigonometric Functions of 30 and 60 and 30, 45, Because the angles 6, 4, and 3 60 occur frequently in trigonometry, you should learn to construct the triangles shown in Figures 4.28 and 4.29. Use the equilateral triangle shown in Figure 4.29 to find the values of cos 60, sin 30, cos 30. and sin 60, 30° 2 3 2 60° 1 1 FIGURE 4.29 Solution Use the Pythagorean Theorem and the equilateral triangle in Figure 4.29 to verify adj 1, the lengths of the sides shown in the figure. For opp 3, hyp 2. 60, you have and So, Te c h n o l o g y You can use a calculator to convert the answers in Example 3 to decimals. However, the radical form is the exact value and in most cases, the exact value is preferred. For 3 sin 60 opp 2 hyp adj 3, 30, sin 30 opp hyp 1 2 and opp 1, and cos 60 adj hyp 1 2 . and hyp 2. So, cos 30 adj hyp 3 2 . Now try Exercise 19. Sines, Cosines, and Tangents of Special Angles sin 30 sin sin 45 sin sin 60 sin 6 4 3 1 2 2 2 3 2 cos 30 cos cos 45 cos cos 60 cos 6 4 3 2 2 2 3 1 2 tan 30 tan tan 45 tan tan 60 tan 3 3 1 3 6 4 3 sin 30 1 2 In the box, note that 60 are complementary angles. In general, it can be shown from the right triangle definitions that cofunctions of complementary angles are equal. That is, if is an acute angle, the following relationships are true. This occurs because cos 60. and 30 sin90 cos tan90 cot sec90 csc cos90 sin cot90 tan csc90 sec 333202_0403.qxd 12/7/05 11:03 AM Page 304 304 Chapter 4 Trigonometry Trigonometric Identities In trigonometry, a great deal of time is spent studying relationships between trigonometric functions (identities). Fundamental Trigonometric Identities Reciprocal Identities sin 1 cos 1 csc sec tan 1 cot csc 1 sin sec 1 cos cot 1 tan Quotient Identities tan sin cos Pythagorean Identities sin2 cos2 1 cot cos sin 1 tan2 sec2 1 cot2 csc2 Note that sin2 represents sin 2, cos2 represents cos 2, and so on. Example 4 Applying Trigonometric Identities Let be an acute angle such that (b) using trigonometric identities. tan sin 0.6. Find the values of (a) cos and Solution a. To find the value of cos , use the Pythagorean identity sin2 cos2 1. So, you have 0.62 cos2 1 cos2 1 0.6 2 0.64 cos 0.64 0.8. Substitute 0.6 for sin . Subtract 0.62 from each side. Extract the positive square root. b. Now, knowing the sine and cosine of , you can find the tangent of to be tan sin cos 0.6 0.8 0.75. 0.6 Use the definitions of cos and tan check these results. , Now try Exercise 29. and the triangle shown in Figure 4.30, to 1 0.8 θ FIGURE 4.30 333202_0403.qxd 12/7/05 11:03 AM Page 305 Section 4.3 Right Triangle Trigonometry 305 Example 5 Applying Trigonometric Identities be an acute angle such that tan 3. Find the values of (a) cot and Let (b) sec using trigonometric identities. 10 3 θ 1 FIGURE 4.31 You can also use the reciprocal identities for sine, cosine, and tangent to evaluate the cosecant, secant, and cotangent functions with a calculator. For instance, you could use the following keystroke sequence to evaluate sec 28. 1 COS 28 ENTER The calculator should display 1.1325701. Solution a. cot 1 tan b. cot 1 3 sec2 1 tan2 sec2 1 32 sec2 10 sec 10 Reciprocal identity Pythagorean identity Use the definitions of check these results. cot and sec , and the triangle shown in Figure 4.31, to Now try Exercise 31. Evaluating Trigonometric Functions with a Calculator To use a calculator to evaluate trigonometric functions of angles measured in degrees, first set the calculator to degree mode and then proceed as demonstrated as follows. in Section 4.2. For instance, you can find values of cos and sec 28 28 Function cos 28 sec 28 a. b. Mode Degree Degree Calculator Keystrokes Display COS 28 ENTER 0.8829476 COS 28 x 1 ENTER 1.1325701 Throughout this text, angles are assumed to be measured in radians unless means noted otherwise. For example, sin 1 means the sine of 1 radian and the sine of 1 degree. sin 1 Example 6 Using a Calculator Use a calculator to evaluate sec5 40 12. Solution Begin by converting to decimal degree form. [Recall that 1 1 3600 1. 1 1 60 1 and 5 40 12 5 40 60 5.67 12 3600 sec 5.67. Then, use a calculator to evaluate Function sec5 40 12 sec 5.67 Calculator Keystrokes Display COS 5.67 x 1 ENTER 1.0049166 Now try Exercise 47. 333202_0403.qxd 12/7/05 11:03 AM Page 306 306 Chapter 4 Trigonometry Object Angle of elevation Horizontal Observer Observer Horizontal Angle of depression Object FIGURE 4.32 y Angle of elevation 78.3° x = 115 ft Not drawn to scale FIGURE 4.33 Applications Involving Right Triangles Many applications of trigonometry involve a process called solving right triangles. In this type of application, you are usually given one side of a right triangle and one of the acute angles and are asked to find one of the other sides, or you are given two sides and are asked to find one of the acute angles. In Example 7, the angle you are given is the angle of elevation, which represents the angle from the horizontal upward to an object. For objects that lie below the horizontal, it is common to use the term angle of depression, as shown in Figure 4.32. Example 7 Using Trigonometry to Solve a Right Triangle A surveyor is standing 115 feet from the base of the Washington Monument, as shown in Figure 4.33. The surveyor measures the angle of elevation to the top of the monument as How tall is the Washington Monument? 78.3. Solution From Figure 4.33, you can see that tan 78.3 opp adj y x and x 115 where Washington Monument is y x tan 78.3 y 1154.82882 555 feet. is the height of the monument. So, the height of the Now try Exercise 63. Example 8 Using Trigonometry to Solve a Right Triangle An historic lighthouse is 200 yards from a bike path along the edge of a lake. A walkway to the lighthouse is 400 yards long. Find the acute angle between the bike path and the walkway, as illustrated in Figure 4.34. 200 yd θ 400 yd FIGURE 4.34 Solution From Figure 4.34, you can see that the sine of the angle 1 2 sin opp hyp 200 400 . is Now you should recognize that 30. Now try Exercise 65. 333202_0403.qxd 12/7/05 11:03 AM Page 307 Section 4.3 Right Triangle Trigonometry 307 By now you are able to recognize that is the acute angle that Suppose, however, that you were given the 30 satisfies the equation equation sin 0.6 sin 1 2. and were asked to find the acute angle Because . and sin 30 1 2 0.5000 sin 45 1 2 0.7071 you might guess that In a later section, you will study a method by which a more precise value of can be determined. lies somewhere between and 45. 30 Example 9 Solving a Right Triangle Find the length of the skateboard ramp shown in Figure 4.35. c c 18.4° 4 ft FIGURE 4.35 Solution From Figure 4.35, you can see that sin 18.4 opp hyp 4 c . So, the length of the skateboard ramp is c 4 sin 18.4 4 0.3156 12.7 feet. Now try Exercise 67. 333202_0403.qxd 12/7/05 11:03 AM Page 308 308 Chapter 4 Trigonometry 4.3 Exercises VOCABULARY CHECK: 1. Match the trigonometric function with its right triangle definition. (a) Sine (b) Cosine (c) Tangent (d) Cosecant (e) Secant (f) Cotangent (i) hypotenuse adjacent (ii) adjacent opposite (iii) hypotenuse opposite (iv) adjacent hypotenuse (v) opposite hypotenuse (vi) opposite adjacent In Exercises 2 and 3, fill in the blanks. 2. Relative to the angle , the three sides of a right triangle are the ________ side, the ________ side, and the ________. 3. An angle that measures from the horizontal upward to an object is called the angle of ________, whereas an angle that measures from the horizontal downward to an object is called the angle of ________. PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. In Exercises 1–4, find the exact values of the six trigonometric functions of the angle shown in the figure. (Use the Pythagorean Theorem to find the third side of the triangle.) In Exercises 9 –16, sketch a right triangle corresponding to the trigonometric function of the acute angle Use the Pythagorean Theorem to determine the third side and then find the other five trigonometric functions of . . 1. 6 θ 8 3. θ 41 9 2. 4. 13 θ 5 9. 11. 13. 15. sin 3 4 sec 2 tan 3 cot 3 2 10. 12. 14. 16. cos 5 7 cot 5 sec 6 csc 17 4 4 Function θ 4 In Exercises 5–8, find the exact values of the six trigonometric functions of the angle for each of the two triangles. Explain why the function values are the same. 5. 3 θ 1 2 6 θ 7. 1.25 1θ 5 4 θ 6. 8. θ 15 8 4 1 3 θ 7.5 θ 2 θ 6 17. sin 18. cos 19. tan 20. sec 21. cot 22. csc 23. cos 24. sin 25. cot 26. tan In Exercises 17–26, construct an appropriate triangle to complete the table. 0 ≤ ≤ 90, 0 ≤ ≤ /2 (deg) (rad) 30 45 Function Value 333202_0403.qxd 12/7/05 11:03 AM Page 309 In Exercises 27–32, use the given function value(s), and trigonometric identities (including the cofunction identities), to find the indicated trigonometric functions. 27. sin 60 3 2 , cos 60 1 2 28. (a) tan 60 cos 30 (c) sin 30 1 2 csc 30 cos 30 (a) (c) , tan 30 29. csc 13 2 , sec (b) (d) sin 30 cot 60 3 3 (b) (d) 13 3 cot 60 cot 30 30. 31. 32. t
an 26 (a) sin tan (c) sec 5, cos (a) cot90 (c) cos 1 3 sec (a) cot (c) tan 5 cot (a) tan90 (c) (b) (d) cos sec90 (b) (d) cot sin (b) (d) sin sin90 (b) (d) cos csc In Exercises 33–42, use trigonometric identities to transform the left side of the equation into the right side 0 < < /2. 33. tan cot 1 cos sec 1 tan cos sin cot sin cos 1 cos 1 cos sin2 1 sin 1 sin cos2 sec tan sec tan 1 sin2 cos2 2 sin2 1 sin cos sin cos tan cot tan csc sec csc2 34. 35. 36. 37. 38. 39. 40. 41. 42. In Exercises 43–52, use a calculator to evaluate each function. Round your answers to four decimal places. (Be sure the calculator is in the correct angle mode.) 43. (a) 44. (a) sin 10 tan 23.5 (b) (b) cos 80 cot 66.5 Section 4.3 Right Triangle Trigonometry 309 45. (a) 46. (a) 47. (a) 48. (a) 49. (a) 50. (a) 51. (a) 52. (a) sin 16.35 cos 16 18 sec 42 12 cos 4 50 15 cot 11 15 sec 56 8 10 csc 32 40 3 sec9 5 20 32 (b) (b) (b) (b) (b) (b) (b) (b) csc 16.35 sin 73 56 csc 48 7 sec 4 50 15 tan 11 15 cos 56 8 10 tan 44 28 16 cot9 5 30 32 In Exercises 53–58, find the values of 0 < < 90 0 < < /2 and radians of a calculator. in degrees without the aid 53. (a) 54. (a) sin 1 2 2 2 cos 55. (a) sec 2 56. (a) tan 3 57. (a) 58. (a) csc 23 3 3 3 cot (b) csc 2 (b) tan 1 (b) (b) (b) cot 1 cos 1 2 2 2 sin (b) sec 2 In Exercises 59– 62, solve for x, y, or as indicated. r 59. Solve for x. 60. Solve for y. 30 30° x 18 y 60° 61. Solve for x. 62. Solve for r. 32 60° x r 20 45° 63. Empire State Building You are standing 45 meters from the base of the Empire State Building. You estimate that the angle of elevation to the top of the 86th floor (the observatory) is If the total height of the building is another 123 meters above the 86th floor, what is the approximate height of the building? One of your friends is on the 86th floor. What is the distance between you and your friend? 82. 333202_0403.qxd 12/7/05 11:03 AM Page 310 310 Chapter 4 Trigonometry 64. Height A six-foot person walks from the base of a broadcasting tower directly toward the tip of the shadow cast by the tower. When the person is 132 feet from the tower and 3 feet from the tip of the shadow, the person’s shadow starts to appear beyond the tower’s shadow. 68. Height of a Mountain In traveling across flat land, you notice a mountain directly in front of you. Its angle of 3.5. After you drive 13 miles elevation (to the peak) is 9. closer to the mountain, the angle of elevation is Approximate the height of the mountain. (a) Draw a right triangle that gives a visual representation of the problem. Show the known quantities of the triangle and use a variable to indicate the height of the tower. (b) Use a trigonometric function to write an equation involving the unknown quantity. (c) What is the height of the tower? 65. Angle of Elevation You are skiing down a mountain with a vertical height of 1500 feet. The distance from the top of the mountain to the base is 3000 feet. What is the angle of elevation from the base to the top of the mountain? 66. Width of a River A biologist wants to know the width w of a river so in order to properly set instruments for the studying the pollutants in the water. From point C biologist walks downstream 100 feet and sights to point (see figure). From this sighting, it is determined that 54. How wide is the river? A, C w θ = 54° 100 ft A 67. Length A steel cable zip-line is being constructed for a competition on a reality television show. One end of the zip-line is attached to a platform on top of a 150-foot pole. The other end of the zip-line is attached to the top of a 23 5-foot stake. The angle of elevation to the platform is (see figure). θ = 23° 5 ft 150 ft (a) How long is the zip-line? (b) How far is the stake from the pole? (c) Contestants take an average of 6 seconds to reach the ground from the top of the zip-line. At what rate are contestants moving down the line? At what rate are they dropping vertically? 3.5° 13 mi 9° Not drawn to scale 69. Machine Shop Calculations A steel plate has the form of one-fourth of a circle with a radius of 60 centimeters. Two two-centimeter holes are to be drilled in the plate positioned as shown in the figure. Find the coordinates of the center of each hole. y 60 56 ( x y ) , 2 2 30° 30° 30° ( x y ) , 1 1 x 56 60 70. Machine Shop Calculations A tapered shaft has a diameter of 5 centimeters at the small end and is 15 centimeters long (see figure). The taper is Find the diameter of the large end of the shaft. 3. d 3° 5 cm d 15 cm 333202_0403.qxd 12/7/05 11:03 AM Page 311 Model It Synthesis Section 4.3 Right Triangle Trigonometry 311 71. Height A 20-meter line is used to tether a heliumfilled balloon. Because of a breeze, the line makes an angle of approximately with the ground. 85 (a) Draw a right triangle that gives a visual representation of the problem. Show the known quantities of the triangle and use a variable to indicate the height of the balloon. (b) Use a trigonometric function to write an equation involving the unknown quantity. (c) What is the height of the balloon? (d) The breeze becomes stronger and the angle the balloon makes with the ground decreases. How does this affect the triangle you drew in part (a)? (e) Complete the table, which shows the heights (in meters) of the balloon for decreasing angle measures . Angle, 80 70 60 50 Height Angle, 40 30 20 10 Height (f) As the angle the balloon makes with the ground approaches how does this affect the height of the balloon? Draw a right triangle to explain your reasoning. 0, 20 72. Geometry Use a compass to sketch a quarter of a circle of radius 10 centimeters. Using a protractor, construct an angle of in standard position (see figure). Drop a perpendicular line from the point of intersection of the terminal side of the angle and the arc of the circle. By actual measurement, calculate the coordinates of the point of intersection and use these measurements to angle. approximate the six trigonometric functions of a x, y 20 y 10 (x, y) x 10 1 c m 0 20° True or False? statement is true or false. Justify your answer. In Exercises 73–78, determine whether the 73. 75. 77. sin 60 csc 60 1 sin 45 cos 45 1 sin 60 sin 30 sin 2 74. 76. sec 30 csc 60 cot2 10 csc2 10 1 78. tan52 tan25 79. Writing In right triangle trigonometry, explain why sin 30 1 2 regardless of the size of the triangle. 80. Think About It You are given only the value Is without finding the tan . it possible to find the value of measure of Explain. ? sec 81. Exploration (a) Complete the table. 0.1 0.2 0.3 0.4 0.5 sin (b) Is or (c) As Explain. greater for sin approaches 0, how do in the interval and 0, 0.5? sin compare? 82. Exploration (a) Complete the table. 0 18 36 54 72 90 sin cos (b) Discuss the behavior of the sine function for in the range from 0 to 90. (c) Discuss the behavior of the cosine function for in the range from 0 to 90. (d) Use the definitions of the sine and cosine functions to explain the results of parts (b) and (c). Skills Review 83. In Exercises 83–86, perform the operations and simplify. x 2 12x 36 x 2 36 t 2 16 4t 2 12t 9 x 2 6x x 2 4x 12 2t 2 5t 12 9 4t 2 84. 85 4x 4 86. 1 4 1 3 x 12 x 333202_0404.qxd 12/7/05 11:05 AM Page 312 312 Chapter 4 Trigonometry 4.4 Trigonometric Functions of Any Angle What you should learn • Evaluate trigonometric functions of any angle. • Use reference angles to evaluate trigonometric functions. • Evaluate trigonometric functions of real numbers. Why you should learn it You can use trigonometric functions to model and solve real-life problems. For instance, in Exercise 87 on page 319, you can use trigonometric functions to model the monthly normal temperatures in New York City and Fairbanks, Alaska. Introduction In Section 4.3, the definitions of trigonometric functions were restricted to acute angles. In this section, the definitions are extended to cover any angle. If is an acute angle, these definitions coincide with those given in the preceding section. Definitions of Trigonometric Functions of Any Angle Let be an angle in standard position with of and r x2 y2 0. x, y a point on the terminal side sin y r tan y x sec r x , x 0 , x 0 cos x r cot x y csc Because . functions are defined for any real value of However, if are undefined. For example, secant of y 0, Similarly, if x 0, 90 the cotangent and cosecant of are undefined. cannot be zero, it follows that the sine and cosine the tangent and is undefined. the tangent of Example 1 Evaluating Trigonometric Functions be a point on the terminal side of . Find the sine, cosine, and 3, 4 Let . tangent of Solution Referring to Figure 4.36, you can see that x 3, r x 2 y 2 32 42 25 5. y 4, and James Urbach/SuperStock So, you have the following. − ( 3, 4) r y 4 3 2 1 −3 −2 −1 FIGURE 4.36 θ 1 x sin y r cos x r tan y x 4 5 3 5 4 3 Now try Exercise 1. 333202_0404.qxd 12/7/05 11:05 AM Page 313 Section 4.4 Trigonometric Functions of Any Angle 313 π < <θ 3 2 0 < < < < 2θ π y y Quadrant II θ sin : + − θ cos : − θ tan : Quadrant I θ sin : + θ cos : + θ tan : + Quadrant III θ − sin : θ − cos : θ tan : + Quadrant IV θ − sin : θ cos : + θ − tan : x x FIGURE 4.37 y π 2 (0, 1) (−1, 0) π (1, 0) 0 x π 3 2 (0, −1) FIGURE 4.38 The signs of the trigonometric functions in the four quadrants can be determined easily from the definitions of the functions. For instance, because cos xr, which is in Quadrants I and IV. (Remember, is always positive.) In a similar manner, you can verify the results shown in Figure 4.37. is positive wherever it follows that cos r x > 0, Example 2 Evaluating Trigonometric Functions Given tan 5 4 and cos > 0, find sin and sec . Solution Note that tangent is negative and the cosine is positive. Moreover, using lies in Quadrant IV because that is the only quadrant in which the tan y x 5 4 and the fact that r 16 25 41 y and you have is negative in Quadrant IV, you can let y 5 and x 4. So, sin y r 5 41 0.7809 sec r x 41 4 1.6008. Now try Exercise 17. Example 3 Trigonometric Functions of Quadrant Angles Evaluate the cosine an
d tangent functions at the four quadrant angles 0, 3 . 2 2 , , and Solution To begin, choose a point on the terminal side of each angle, as shown in Figure 4.38. For each of the four points, and you have the following. cos 0 x r 1 1 1 0 cos x r 2 cos x r 0 1 1 1 1 r 1, tan 0 y x tan y x 2 tan y x cos 3 2 x r 0 1 0 tan 3 2 y x Now try Exercise 29. 0 1 1 0 0 x, y 1, 0 ⇒ undefined x, y 0, 1 0 0 1 1 0 x, y 1, 0 ⇒ undefined x, y 0, 1 333202_0404.qxd 12/7/05 11:05 AM Page 314 314 Chapter 4 Trigonometry Reference Angles (or less than ) can be determined from their values at corresponding acute angles called The values of the trigonometric functions of angles greater than 0 reference angles. 90 Definition of Reference Angle Let be an angle in standard position. Its reference angle is the acute angle formed by the terminal side of and the horizontal axis. Figure 4.39 shows the reference angles for in Quadrants II, III, and IV. Quadrant II θ Reference ′θ angle: θ Reference ′θ angle: θ Reference ′ θ angle: πθ θ ′ = − (radians) θ θ ′ = 180° − (degrees) FIGURE 4.39 Quadrant III ′ = − (radians) θ π θ ′ = − 180° (degrees) θ θ Quadrant IV θ θ ′ = 2 − (radians) θ θ ′ = 360° − (degrees) π Example 4 Finding Reference Angles Find the reference angle . 2.3 a. 300 b. c. 135 Solution a. Because 300 lies in Quadrant IV, the angle it makes with the -axis is x 360 300 60. Degrees Figure 4.40 shows the angle 300 2 1.5708 in Quadrant II and its reference angle is b. Because 2.3 lies between and its reference angle 3.1416, and 60. it follows that it is 2.3 0.8416. Radians c. First, determine that Figure 4.41 shows the angle 135 III. So, the reference angle is 225 180 2.3 and its reference angle 225, is coterminal with which lies in Quadrant 2.3. 45. Degrees Figure 4.42 shows the angle 135 and its reference angle 45. y θ = 300° x ′ = 60° θ FIGURE 4.40 y ′ = − 2.3 π θ θ = 2.3 x FIGURE 4.41 y 225° and −135° are coterminal. 225° ′ = 45° θ x = −135° θ FIGURE 4.42 Now try Exercise 37. 333202_0404.qxd 12/7/05 11:05 AM Page 315 y (x, y) r = h y p opp θ ′ θ adj opp y, adj x FIGURE 4.43 Learning the table of values at the right is worth the effort because doing so will increase both your efficiency and your confidence. Here is a pattern for the sine function that may help you remember the values. 0 30 45 60 90 sin Reverse the order to get cosine values of the same angles. Section 4.4 Trigonometric Functions of Any Angle 315 Trigonometric Functions of Real Numbers To see how a reference angle is used to evaluate a trigonometric function, consider the point as shown in Figure 4.43. By definition, you know that on the terminal side of x, y , sin y r and tan y x . For the right triangle with acute angle have x sin opp hyp y r and sides of lengths x and y, you and tan opp adj y x. sin tan and tan are equal, except possibly in sign. The same is So, it follows that and for the other four trigonometric functions. In all true for and cases, the sign of the function value can be determined by the quadrant in which sin lies. Evaluating Trigonometric Functions of Any Angle To find the value of a trigonometric function of any angle : 1. Determine the function value for the associated reference angle . 2. Depending on the quadrant in which lies, affix the appropriate sign to the function value. By using reference angles and the special angles discussed in the preceding section, you can greatly extend the scope of exact trigonometric values. For means that you know the function instance, knowing the function values of values of all angles for which is a reference angle. For convenience, the table below shows the exact values of the trigonometric functions of special angles and quadrant angles. 30 30 Trigonometric Values of Common Angles (degrees) 0 30 45 60 (radians) sin cos tan 90 2 1 0 180 0 270 3 2 1 1 0 1 3 Undef. 0 Undef. 333202_0404.qxd 12/7/05 11:05 AM Page 316 316 Chapter 4 Trigonometry Example 5 Using Reference Angles Evaluate each trigonometric function. a. cos 4 3 b. tan210 c. csc 11 4 Solution a. Because 43 43 3, negative in Quadrant III, so lies in Quadrant III, as shown in Figure 4.44. Moreover, the reference angle is the cosine is cos 4 3 cos 3 1 2 . b. Because 210 360 150, 150. is coterminal with the as shown in Figure 4.45. Finally, because the tangent is 210 reference angle it follows that the So, is second-quadrant angle 180 150 30, negative in Quadrant II, you have tan210 tan 30 3 3 114 2 34, . c. Because with the second-quadrant angle 34 4, positive in Quadrant II, you have it follows that is coterminal the reference angle is as shown in Figure 4.46. Because the cosecant is 34. So, 114 csc 11 4 csc 4 1 sin4 2. y y y ′ = 30° θ θ = π 4 3 x ′θ = π 4 x θ = π 11 210° FIGURE 4.44 FIGURE 4.45 FIGURE 4.46 Now try Exercise 51. 333202_0404.qxd 12/7/05 11:05 AM Page 317 Section 4.4 Trigonometric Functions of Any Angle 317 Example 6 Using Trigonometric Identities Let be an angle in Quadrant II such that by using trigonometric identities. sin 1 3. Find (a) cos and (b) tan Solution a. Using the Pythagorean identity sin2 cos2 1, you obtain 2 1 3 cos2 1 Substitute 1 3 for sin . cos 2 1 1 9 8 9 . Because cos < 0 in Quadrant II, you can use the negative root to obtain cos 8 9 22 3 . b. Using the trigonometric identity tan sin cos , you obtain Substitute for sin and cos . tan 13 223 1 22 2 4 . Now try Exercise 59. You can use a calculator to evaluate trigonometric functions, as shown in the next example. Example 7 Using a Calculator Use a calculator to evaluate each trigonometric function. a. cot 410 b. sin7 c. sec 9 Solution Function Mode Calculator Keystrokes Display a. b. c. cot 410 sin7 sec 9 Degree Radian TAN SIN 410 7 x 1 ENTER ENTER 0.8390996 0.6569866 Radian COS 9 x 1 ENTER 1.0641778 Now try Exercise 69. 333202_0404.qxd 12/7/05 11:05 AM Page 318 318 Chapter 4 Trigonometry 4.4 Exercises VOCABULARY CHECK: In Exercises 1– 6, let be an angle in standard position, with x, y a point on the terminal side of and rx2 y2 0. 1. sin ________ 3. 5. tan ________ x r ________ 2. 4. 6. ________ r y sec ________ x y ________ 7. The acute positive angle that is formed by the terminal side of the angle and the horizontal axis is called the ________ angle of and is denoted by . PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. In Exercises 1–4, determine the exact values of the six trigonometric functions of the angle . 9. 3.5, 6.8 10. 31 2, 73 4 1. (a) 2. (a) (4, 3) θ y y θ (−12, −5) 3. (a) y θ − 3, −1 ) ( 4. (a) y (3, 1b) θ (8, 15)− (b) − ( 1, 1) (b) − ( 4, 1) (b) θ θ θ x x x x (4, 4)− In Exercises 5–10, the point is on the terminal side of an angle in standard position. Determine the exact values of the six trigonometric functions of the angle. 5. 7. 7, 24 4, 10 6. 8. 8, 15 5, 2 In Exercises 11–14, state the quadrant in which lies. 11. 12. 13. 14. sin < 0 sin > 0 sin > 0 sec > 0 and and and and cos < 0 cos > 0 tan < 0 cot < 0 In Exercises 15–24, find the values of the six trigonometric functions of with the given constraint. Function Value Constraint 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. sin 3 5 cos 4 5 tan 15 8 cos 8 17 cot 3 csc 4 sec 2 sin 0 cot tan is undefined. is undefined. lies in Quadrant II. lies in Quadrant III. sin < 0 tan < 0 cos > 0 cot < 0 sin > 0 sec 1 2 ≤ ≤ 32 ≤ ≤ 2 lies on the given In Exercises 25–28, the terminal side of line in the specified quadrant. Find the values of the six trigonometric functions of by finding a point on the line. Line y x y 1 3x 2x y 0 4x 3y 0 25. 26. 27. 28. Quadrant II III III IV 333202_0404.qxd 12/7/05 11:05 AM Page 319 In Exercises 29–36, evaluate the trigonometric function of the quadrant angle. Section 4.4 Trigonometric Functions of Any Angle 319 In Exercises 65–80, use a calculator to evaluate the trigonometric function. Round your answer to four decimal places. (Be sure the calculator is set in the correct angle mode.) 29. sin 31. sec 33. sin 3 2 2 35. csc 30. csc 3 2 32. sec 34. cot 36. cot 2 In Exercises 37–44, find the reference angle in standard position. and 65. 67. 69. 71. 73. 75. 77. 79. , and sketch sin 10 cos110 tan 304 sec 72 tan 4.5 tan 9 sin0.65 cot11 8 66. 68. 70. 72. 74. 76. 78. 80. sec 225 csc330 cot 178 tan188 cot 1.35 tan 9 sec 0.29 csc15 14 37. 39. 41. 203 245 2 3 43. 3.5 38. 40. 42. 44. 309 145 7 4 11 3 In Exercises 45–58, evaluate the sine, cosine, and tangent of the angle without using a calculator. 46. 48. 50. 52. 54. 56. 300 405 840 4 2 10 3 45. 47. 49. 51. 53. 55. 57. 58. 225 750 150 4 3 6 11 4 3 2 25 4 In Exercises 59–64, find the indicated trigonometric value in the specified quadrant. Function sin 3 5 cot 3 tan 3 2 csc 2 cos 5 8 sec 9 4 59. 60. 61. 62. 63. 64. Quadrant IV II III IV I III Trigonometric Value cos sin sec cot sec tan In Exercises 81–86, find two solutions of the equation. Give and in radians your answers in degrees 0 ≤ < 2. Do not use a calculator. 0 ≤ < 360 81. (a) 82. (a) 83. (a) sin 1 2 2 cos 2 csc 23 3 84. (a) 85. (a) sec 2 tan 1 86. (a) sin 3 2 (b) (b) sin 1 2 2 2 cos (b) cot 1 (b) (b) sec 2 cot 3 3 2 (b) sin Model It 87. Data Analysis: Meteorology The table shows the monthly normal temperatures (in degrees Fahrenheit) and for selected months for New York City F. Fairbanks, Alaska (Source: National Climatic Data Center) N Month January April July October December New York City, N Fairbanks, F 33 52 77 58 38 10 32 62 24 6 (a) Use the regression feature of a graphing utility to find a model of the form y a sinbt c d for each city. Let represent the month, with corresponding to January. t t 1 333202_0404.qxd 12/7/05 11:05 AM Page 320 320 Chapter 4 Trigonometry Model It (co n t i n u e d ) (b) Use the models from part (a) to find the monthly normal temperatures for the two cities in February, March, May, June, August, September, and November. (c) Compare the models for the two cities. 88. Sales A company that produces snowboards, which are seasonal products, forecasts monthly sales over the next 2 years to be FIGURE FO
R 92 Synthesis d θ 6 mi Not drawn to scale S 23.1 0.442t 4.3 cos t 6 S is measured in thousands of units and where the time in months, with Predict sales for each of the following months. is representing January 2006. t 1 t (a) February 2006 (b) February 2007 (c) June 2006 (d) June 2007 89. Harmonic Motion The displacement from equilibrium of an oscillating weight suspended by a spring is given by yt 2 cos 6t True or False? the statement is true or false. Justify your answer. In Exercises 93 and 94, determine whether 93. In each of the four quadrants, the signs of the secant function and sine function will be the same. 94. To find the reference angle for an angle in degrees), find 0 ≤ 360n ≤ 360. reference angle. integer the The difference n such 360n (given that is the 95. Writing Consider an angle in standard position with centimeters, as shown in the figure. Write a short x, y, 0 r 12 paragraph describing the changes in the values of sin , 90. to increases continuously from cos , tan and as y is the displacement (in centimeters) and t where time (in seconds). Find the displacement when (a) (b) and (c) t 1 2. t 1 4, is the t 0, y 90. Harmonic Motion The displacement from equilibrium of an oscillating weight suspended by a spring and subject to the damping effect of friction is given by y t 2et cos 6t y is the displacement (in centimeters) and t where time (in seconds). Find the displacement when (a) (b) and (c) t 1 2. t 1 4, is the t 0, 12 cm θ (x, y) x 96. Writing Explain how reference angles are used to find 91. Electric Circuits The current I (in amperes) when the trigonometric functions of obtuse angles. 100 volts is applied to a circuit is given by I 5e2t sin t Skills Review t where is the time (in seconds) after the voltage is applied. second after the voltage Approximate the current at is applied. t 0.7 92. Distance An airplane, flying at an altitude of 6 miles, is on a flight path that passes directly over an observer (see is the angle of elevation from the observer to figure). If d the plane, find the distance from the observer to the plane 30, 90, 120. when (a) and (c) (b) In Exercises 97–106, graph the function. Identify the domain and any intercepts and asymptotes of the function. 97. 99. 101. y x2 3x 4 f x x3 8 f x x 7 x2 4x 4 103. 105. y 2x1 y ln x4 98. 100. 102. y 2x2 5x gx x4 2x2 3 hx x2 1 x 5 y 3 x1 2 104. 106. y log10 x 2 333202_0405.qxd 12/7/05 11:06 AM Page 321 Section 4.5 Graphs of Sine and Cosine Functions 321 4.5 Graphs of Sine and Cosine Functions What you should learn • Sketch the graphs of basic sine and cosine functions. • Use amplitude and period to help sketch the graphs of sine and cosine functions. • Sketch translations of the graphs of sine and cosine functions. • Use sine and cosine functions to model real-life data. Why you should learn it Sine and cosine functions are often used in scientific calculations. For instance, in Exercise 73 on page 330, you can use a trigonometric function to model the airflow of your respiratory cycle. Basic Sine and Cosine Curves In this section, you will study techniques for sketching the graphs of the sine and cosine functions. The graph of the sine function is a sine curve. In Figure 4.47, the black portion of the graph represents one period of the function and is called one cycle of the sine curve. The gray portion of the graph indicates that the basic sine curve repeats indefinitely in the positive and negative directions. The graph of the cosine function is shown in Figure 4.48. Recall from Section 4.2 that the domain of the sine and cosine functions is the set of all real numbers. Moreover, the range of each function is the interval 1, 1, Do you see how this information is consistent with the basic graphs shown in Figures 4.47 and 4.48? and each function has a period of 2. Range: − ≤ ≤y 1 1 FIGURE 4.47 y = sin 1 y 1 −1 Period: 2π y = cos x π π 2 π 3 2 2π π 5 2 Period: 2π Note in Figures 4.47 and 4.48 that the sine curve is symmetric with respect to the origin, whereas the cosine curve is symmetric with respect to the -axis. These properties of symmetry follow from the fact that the sine function is odd and the cosine function is even. y © Karl Weatherly/Corbis Range: − ≤ ≤y 1 1 − π 3 2 −π FIGURE 4.48 333202_0405.qxd 12/7/05 11:06 AM Page 322 322 Chapter 4 Trigonometry To sketch the graphs of the basic sine and cosine functions by hand, it helps to note five key points in one period of each graph: the intercepts, maximum points, and minimum points (see Figure 4.49). y y Intercept Maximum Intercept ) , 1 ( y π 2 Minimum Intercept Intercept Minimum Intercept = sin x (0, 1) Maximum y = cos x π ( , 0) ( π 3 2 ) , 1− (0, 0) Quarter period Half period π Period: 2 Three-quarter period x π(2 , 0) Full period Quarter period π Period)− π Half period Three-quarter period Maximum π (2 , 1) x Full period FIGURE 4.49 Example 1 Using Key Points to Sketch a Sine Curve Sketch the graph of y 2 sin x on the interval , 4. Solution Note that y 2 sin x 2sin x indicates that the -values for the key points will have twice the magnitude of those on the graph of into four equal parts to get Divide the period the key points for y sin x. y 2 sin x. 2 y Intercept Maximum Intercept , 2, 2 , 0, 0, 0, Minimum , 2, 3 2 Intercept and 2, 0 By connecting these key points with a smooth curve and extending the curve in you obtain the graph shown in Figure both directions over the interval 4.50. , 4, Te c h n o l o g y When using a graphing utility to graph trigonometric functions, pay special attention to the viewing window you use. For instance, y [sin10x]/10 try graphing the standard viewing window in radian mode. What do you observe? Use the zoom feature to find a viewing window that displays a good view of the graph. in y 3 2 1 − π 2 −2 FIGURE 4.50 y = 2 sin x y = sin x 3π 2 5π 2 7π 2 x Now try Exercise 35. 333202_0405.qxd 12/7/05 11:06 AM Page 323 Section 4.5 Graphs of Sine and Cosine Functions 323 Amplitude and Period In the remainder of this section you will study the graphic effect of each of the constants in equations of the forms d a, and c,b, y d a sinbx c and y d a cosbx c. A quick review of the transformations you studied in Section 1.7 should help in this investigation. y a sin x in The constant factor a stretch or vertical shrink of the basic sine curve. If is stretched, and if y a sin x graph of a absolute value of function acts as a scaling factor—a vertical a > 1, the basic sine curve the basic sine curve is shrunk. The result is that the and 1. The The range of the a < 1, ranges between instead of between y a sin x. is the amplitude of the function a ≤ y ≤ a. y a sin x a and a > 0 1 a for is Definition of Amplitude of Sine and Cosine Curves The amplitude of and between the maximum and minimum values of the function and is given by represents half the distance y a cos x y a sin x Amplitude a. y = 3 cos x Example 2 Scaling: Vertical Shrinking and Stretching y = cos x On the same coordinate axes, sketch the graph of each function. a. y 1 2 cos x b. y 3 cos x x 2π y 1 = cos 2 x Solution a. Because the amplitude of minimum value is get the key points 1 2. y 1 is Divide one cycle, 2 cos x 1 2, 0 ≤ x ≤ 2, the maximum value is and the into four equal parts to 1 2 Maximum Intercept 0, , 0, , 2 1 2 Minimum , , 1 2 Intercept , 0, 3 2 Maximum . 2, 1 2 and y 3 −1 −2 −3 FIGURE 4.51 Exploration y cos bx Sketch the graph of b 1 2, and 3. How does for 2, b the value of affect the graph? How many complete cycles occur between 0 and for each value of b? 2 b. A similar analysis shows that the amplitude of y 3 cos x is 3, and the key points are Maximum Intercept Minimum , 0, , 3, 0, 3, 2 Intercept , 0, 3 2 Maximum and 2, 3. The graphs of these two functions are shown in Figure 4.51. Notice that the graph y 1 of and the graph of y 3 cos x is a vertical shrink of the graph of is a vertical stretch of the graph of y cos x. y cos x 2 cos x Now try Exercise 37. 333202_0405.qxd 12/7/05 11:06 AM Page 324 324 Chapter 4 Trigonometry y y = 3 cos x y = −3 cos x 3 1 You know from Section 1.7 that the graph of y f x. y f x For instance, the graph of is a reflection in the y 3 cos x is a x -axis of the graph of reflection of the graph of y a sin x Because y a sin bx that y 3 cos x, completes one cycle from to x 0 completes one cycle from as shown in Figure 4.52. x 0 to x 2b. x 2, it follows π− π 2π x Period of Sine and Cosine Functions Let be a positive real number. The period of b is given by y a sin bx and y a cos bx −3 FIGURE 4.52 Period 2 . b Note that if 0 < b < 1, Exploration Sketch the graph of y sinx c c 4, 0, 4. where How does the value of affect the graph? and c In general, to divide a period-interval into four equal parts, successively add “period/4,” starting with the left endpoint of the interval. For instance, for the period-interval 6, 2 you would successively add of length 23, 6 23 4 6, 0, 6, 3, and as the -values for the key to get 2 points on the graph. x the period of represents a horizontal stretching of the graph of the period of of the graph of cosx cos x y a sin bx y a sin x. are used to rewrite the function. is less than If 2 b is negative, the identities y a sin x. y a sin bx is greater than Similarly, if 2 and b > 1, and represents a horizontal shrinking and sinx sin x Example 3 Scaling: Horizontal Stretching Sketch the graph of y sin x 2 . Solution The amplitude is 1. Moreover, because b 1 2, the period is 2 b 2 1 2 4. Substitute for b. Now, divide the period-interval 2, and 3 to obtain the key points on the graph. 0, 4 into four equal parts with the values , Intercept Maximum Intercept 0, 0, , 1, 2, 0, Minimum 3, 1, Intercept 4, 0 and The graph is shown in Figure 4.53. y = sin x y 1 y = sin x 2 − π π −1 Period: 4 π FIGURE 4.53 Now try Exercise 39. x 333202_0405.qxd 12/7/05 11:06 AM Page 325 Section 4.5 Graphs of Sine and Cosine Functions 325 Translations of Sine and Cosine Curves The constant c y a sinbx c in the general equations and y a cosbx c creates a horizo
ntal translation (shift) of the basic sine and cosine curves. y a sinbx c, Comparing you find that the graph of y a sinbx c By solving for bx c 0 you can find the interval for one cycle to be completes one cycle from bx c 2. y a sin bx with to x, Left endpoint Right endpoint Period This implies that the period of y a sin bx is shifted by an amount y a sinbx c is The number cb. 2b, cb and the graph of is the phase shift. Graphs of Sine and Cosine Functions y a sinbx c The graphs of b > 0. ) characteristics. (Assume and y a cosbx c have the following Amplitude a Period 2 b The left and right endpoints of a one-cycle interval can be determined by solving the equations and bx c 2. bx c 0 Example 4 Horizontal Translation Sketch the graph of y 1 2 sinx . 3 Solution 1 The amplitude is and the period is 2 2. and By solving the equations 3, 73 you see that the interval Dividing this interval into four equal parts produces the key points corresponds to one cycle of the graph. Intercept Maximum , 3 5 , 6 , 0, 1 2 Intercept , 0, 4 3 Minimum , 11 , 1 2 6 Intercept , 0. 7 3 and y 1 2 y = sin 2π π 5 3 π 8 3 x Period: 2 π The graph is shown in Figure 4.54. FIGURE 4.54 Now try Exercise 45. 333202_0405.qxd 12/7/05 11:06 AM Page 326 326 Chapter 4 Trigonometry y = −3 cos(2 x + 4 ) π π Example 5 Horizontal Translation y 3 2 −3 −2 Period 1 FIGURE 4.55 Sketch the graph of y 3 cos2x 4. Solution The amplitude is 3 and the period is x 1 2x 4 0 22 1. By solving the equations 2x 4 x 2 and 2x 4 2 2x 2 x 1 2, 1 you see that the interval Dividing this interval into four equal parts produces the key points corresponds to one cycle of the graph. Minimum 2, 3, Intercept Maximum Intercept 7 3 , 0, 4 2 5 4 , 0, , 3, Minimum and 1, 3. The graph is shown in Figure 4.55. Now try Exercise 47. The final type of transformation is the vertical translation caused by the constant d in the equations y d a sinbx c and y d a cosbx c. d The shift is units upward for words, the graph oscillates about the horizontal line x -axis. d > 0 d and units downward for d < 0. In other instead of about the y d y = 2 + 3 cos 2x Example 6 Vertical Translation y 5 Sketch the graph of y 2 3 cos 2x. − π 1 −1 π x Period π FIGURE 4.56 Solution . The amplitude is 3 and the period is The key points over the interval , 1, , 2, , 2, 0, 5, , 5. and 4 2 3 4 0, are The graph is shown in Figure 4.56. Compared with the graph of the graph of is shifted upward two units. y 2 3 cos 2x f x 3 cos 2x, Now try Exercise 53. 333202_0405.qxd 12/7/05 11:06 AM Page 327 Section 4.5 Graphs of Sine and Cosine Functions 327 Mathematical Modeling Sine and cosine functions can be used to model many real-life situations, including electric currents, musical tones, radio waves, tides, and weather patterns. Time, t Depth, y Example 7 Finding a Trigonometric Model Midnight 2 A.M. 4 A.M. 6 A.M. 8 A.M. 10 A.M. Noon 3.4 8.7 11.3 9.1 3.8 0.1 1.2 Changing Tides y 12 10 FIGURE 4.57 4 A.M. 8 A.M. Time Noon t Throughout the day, the depth of water at the end of a dock in Bar Harbor, Maine varies with the tides. The table shows the depths (in feet) at various times during the morning. (Source: Nautical Software, Inc.) a. Use a trigonometric function to model the data. b. Find the depths at 9 A.M. and 3 P.M. c. A boat needs at least 10 feet of water to moor at the dock. During what times in the afternoon can it safely dock? Solution a. Begin by graphing the data, as shown in Figure 4.57. You can use either a sine or cosine model. Suppose you use a cosine model of the form y a cosbt c d. The difference between the maximum height and the minimum height of the graph is twice the amplitude of the function. So, the amplitude is a 1 2 maximum depth minimum depth 1 2 11.3 0.1 5.6. The cosine function completes one half of a cycle between the times at which the maximum and minimum depths occur. So, the period is p 2time of min. depth time of max. depth b 2p 0.524. 210 4 which implies that midnight, consider the left endpoint to be 1 because the average depth is 2 you can model the depth with the function given by 11.3 0.1 5.7, Because high tide occurs 4 hours after Moreover, So, c 2.094. it follows that cb 4, d 5.7. 12 so 12 (14.7, 10) (17.3, 10) b. The depths at 9 A.M. and 3 P.M. are as follows. y 5.6 cos0.524t 2.094 5.7. y = 10 0 24 0 y = 5.6 cos(0.524t − 2.094) + 5.7 FIGURE 4.58 y 5.6 cos0.524 9 2.094 5.7 0.84 foot y 5.6 cos0.524 15 2.094 5.7 10.57 feet 9 A.M. 3 P.M. c. To find out when the depth y 10 is at least 10 feet, you can graph the model with the line using a graphing utility, as shown in Figure 4.58. Using the intersect feature, you can determine that the depth is at least 10 feet between t 17.3. 2:42 P.M. and 5:18 P.M. t 14.7 y Now try Exercise 77. 333202_0405.qxd 12/7/05 11:06 AM Page 328 328 Chapter 4 Trigonometry 4.5 Exercises VOCABULARY CHECK: Fill in the blanks. 1. One period of a sine or cosine function function is called one ________ of the sine curve or cosine curve. 2. The ________ of a sine or cosine curve represents half the distance between the maximum and minimum values of the function. 3. The period of a sine or cosine function is given by ________. 4. For the function given by y a sinbx c, c b represents the ________ ________ of the graph of the function. 5. For the function given by y d a cosbx c, d represents a ________ ________ of the graph of the function. PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. In Exercises 1–14, find the period and amplitude. 1. y 3 sin 2x 2. y 2 cos 3x 3 π x 3. y 5 2 cos x 2 4. y 3 sin x 3 y 3 −2 −3 x π2 y 4 π π− −2 −4 x 5. y 1 2 sin x 3 6. y 3 2 cos x 2 y 1 π 2 −1 x y 2 − π −2 x π 7. y 2 sin x 9. 11. y 3 sin 10x 2x y 1 2 3 cos 8. 10. 12. y cos 2x 3 3 sin 8x x 4 cos y 1 y 5 2 In Exercises 15–22, describe the relationship between the graphs of and Consider amplitude, period, and shifts. g. 13. 14. y 1 4 y 2 3 sin 2x cos x 10 15. 17. 19. 21. f f x sin x gx sinx f x cos 2x gx cos 2x f x cos x gx cos 2x f x sin 2x gx 3 sin 2x 16. 18. 20. 22. f x cos x gx cosx f x sin 3x gx sin3x f x sin x gx sin 3x f x cos 4x gx 2 cos 4x In Exercises 23–26, describe the relationship between the graphs of and Consider amplitude, period, and shifts. g. f 23. 252 −3 3 2 1 −2 −3 −2 π 24. y x x 26. 3 2 g −2 −2 π π 2 −2 x x 333202_0405.qxd 12/7/05 11:06 AM Page 329 In Exercises 27–34, graph and on the same set of coordinate axes. (Include two full periods.) g f 27. f x 2 sin x gx 4 sin x 29. 31. 33. sin f x cos x gx 1 cos x x f x 1 2 2 gx 3 1 2 f x 2 cos x gx 2 cosx sin x 2 28. f x sin x x 3 gx sin 30. f x 2 cos 2x gx cos 4x 32. f x 4 sin x gx 4 sin x 3 34. f x cos x gx cosx In Exercises 35–56, sketch the graph of the function. (Include two full periods.) 65. 35. 37. 39. y 3 sin x y 1 3 cos x x 2 y cos 41. y cos 2x 43. 45. y sin 2x 3 y sinx 4 47. y 3 cosx 49. 51. 53. 54. 55. y 2 sin 2x 3 y 2 1 10 cos 60x y 3 cosx 3 4 y 4 cosx 4 cosx 2 y 2 3 4 36. 38. y 1 4 sin x y 4 cos x 40. y sin 4x 42. y sin x 4 44. y 10 cos x 6 46. y sinx y 4 cosx 48. 4 t 12 50. y 3 5 cos 52. y 2 cos x 3 56. y 3 cos6x 57. In Exercises 57– 62, use a graphing utility to graph the function. Include two full periods. Be sure to choose an appropriate viewing window. y 2 sin4x y cos2x y 3 cosx 2 y 4 sin2 3 2 2 1 x 60. 58. 59. 3 2 Section 4.5 Graphs of Sine and Cosine Functions 329 y 0.1 sinx 10 y 1 100 sin 120t 61. 62. Graphical Reasoning for the function matches the figure. f x a cos x d In Exercises 63– 66, find d and f such that the graph of a 63. y y 4 1 −1 −2 10 8 6 4 − π −2 64 66. − π f π f y −3 −4 1 −1 −2 −5 x x Graphical Reasoning for the function f matches the figure. f x a sinbx c In Exercises 67–70, find c such that the graph of and a, b, 67. 69. y y f 1 −3 3 2 1 −2 −3 f 68. x π − π 70. x π y 3 2 1 −3 y 3 2 −2 −3 f π f 2 4 x x In Exercises 71 and 72, use a graphing utility to graph [2, 2]. and Use the graphs to find y2. y1 real numbers such that in the interval x y2 y1 71. y1 y2 sin x 1 2 72. cos x 1 y1 y2 333202_0405.qxd 12/7/05 11:06 AM Page 330 330 Chapter 4 Trigonometry 73. Respiratory Cycle For a person at rest, the velocity (in liters per second) of air flow during a respiratory cycle (the time from the beginning of one breath to the beginning of v the next) is given by t 3 seconds). (Inhalation occurs when occurs when v 0.85 sin v < 0. ) , where is the time (in t v > 0, and exhalation (a) Find the time for one full respiratory cycle. (b) Find the number of cycles per minute. (c) Sketch the graph of the velocity function. 74. Respiratory Cycle After exercising for a few minutes, a person has a respiratory cycle for which the velocity of air flow is approximated by v 1.75 sin t 2 , where t is the time (in seconds). (Inhalation occurs when exhalation occurs when v < 0. ) v > 0, and (a) Find the time for one full respiratory cycle. (b) Find the number of cycles per minute. (c) Sketch the graph of the velocity function. 75. Data Analysis: Meteorology The table shows the maxiand t 1 (Source: National Climatic mum daily high temperatures for Tallahassee Chicago corresponding to January. Data Center) T (in degrees Fahrenheit) for month with C t, (c) Use a graphing utility to graph the data points and the model for the temperatures in Chicago. How well does the model fit the data? (d) Use the models to estimate the average maximum temperature in each city. Which term of the models did you use? Explain. (e) What is the period of each model? Are the periods what you expected? Explain. (f) Which city has the greater variability in temperature throughout the year? Which factor of the models determines this variability? Explain. 76. Health The function given by approximates the blood pressure t mercury at time (in seconds) for a person at rest. P 100 20 cos P (in millimeters) of 5t 3 (a) Find the period of the function. (b) Find the number of heartbeats per minute. 77. Piano Tuning When tuning a piano, a technician strikes a tuning fork for the A above middle C an
d sets up a wave y 0.001 sin 880t, motion that can be approximated by where is the time (in seconds). t (a) What is the period of the function? (b) The frequency f is given by f 1p. What is the frequency of the note? Month, t Tallahassee, T Chicago, C Model It 1 2 3 4 5 6 7 8 9 10 11 12 63.8 67.4 74.0 80.0 86.5 90.9 92.0 91.5 88.5 81.2 72.9 65.8 29.6 34.7 46.1 58.0 69.9 79.2 83.5 81.2 73.9 62.1 47.1 34.4 78. Data Analysis: Astronomy The percent moon’s face that is illuminated on day 2007, where the table. of the of the year represents January 1, is shown in (Source: U.S. Naval Observatory) x 1 y x x 3 11 19 26 32 40 y 1.0 0.5 0.0 0.5 1.0 0.5 (a) A model for the temperature in Tallahassee is given by Tt 77.90 14.10 cost 6 3.67. Find a trigonometric model for Chicago. (b) Use a graphing utility to graph the data points and the model for the temperatures in Tallahassee. How well does the model fit the data? (a) Create a scatter plot of the data. (b) Find a trigonometric model that fits the data. (c) Add the graph of your model in part (b) to the scatter plot. How well does the model fit the data? (d) What is the period of the model? (e) Estimate the moon’s percent illumination for March 12, 2007. 333202_0405.qxd 12/7/05 11:06 AM Page 331 79. Fuel Consumption The daily consumption lons) of diesel fuel on a farm is modeled by C 30.3 21.6 sin2t 365 10.9 C (in gal- is the time (in days), with t where January 1. t 1 corresponding to (a) What is the period of the model? Is it what you expected? Explain. (b) What is the average daily fuel consumption? Which term of the model did you use? Explain. (c) Use a graphing utility to graph the model. Use the graph to approximate the time of the year when consumption exceeds 40 gallons per day. 80. Ferris Wheel A Ferris wheel is built such that the height (in feet) above ground of a seat on the wheel at time (in t h seconds) can be modeled by ht 53 50 sin 10 t . 2 (a) Find the period of the model. What does the period tell you about the ride? (b) Find the amplitude of the model. What does the ampli- tude tell you about the ride? (c) Use a graphing utility to graph one cycle of the model. Section 4.5 Graphs of Sine and Cosine Functions 331 87. Exploration Using calculus, it can be shown that the sine and cosine functions can be approximated by the polynomials sin x x x3 3! cos x 1 x 2 2! x5 5! x4 4! and where x is in radians. (a) Use a graphing utility to graph the sine function and its polynomial approximation in the same viewing window. How do the graphs compare? (b) Use a graphing utility to graph the cosine function and its polynomial approximation in the same viewing window. How do the graphs compare? (c) Study the patterns in the polynomial approximations of the sine and cosine functions and predict the next term in each. Then repeat parts (a) and (b). How did the accuracy of the approximations change when an additional term was added? 88. Exploration Use the polynomial approximations for the sine and cosine functions in Exercise 87 to approximate the following function values. Compare the results with those given by a calculator. Is the error in the approximation the same in each case? Explain. (a) sin 1 2 (b) sin 1 (c) sin 6 4 cos Synthesis (d) cos0.5 (e) cos 1 (f) True or False? statement is true or false. Justify your answer. In Exercises 81– 83, determine whether the Skills Review 81. The graph of the function given by translates the graph of the right so that the two graphs look identical. f x sin x f x sinx 2 exactly one period to 82. The function given by y 1 is twice that of the function given by 83. The graph of y sinx 2 y cos x x in the -axis. 2 cos 2x has an amplitude that y cos x. is a reflection of the graph of y d a sinbx c, 84. Writing Use a graphing utility to graph the function given a, and Write a paragraph describing the changes in the by c,b, graph corresponding to changes in each constant. for several different values of d. Conjecture In Exercises 85 and 86, graph and on the same set of coordinate axes. Include two full periods. Make a conjecture about the functions. f x sin x, gx cosx 85. g f 2 f x sin x, gx cosx 86. 2 In Exercises 89–92, use the properties of logarithms to write the expression as a sum, difference, and/or constant multiple of a logarithm. log10 x 2 t 3 t 1 log2 ln z x2x 3 z2 1 92. 89. 91. 90. ln In Exercises 93–96, write the expression as the logarithm of a single quantity. 93. 95. 96. log10 x log10 y 1 2 ln 3x 4 ln y ln 2x 2 ln x 3 ln x 1 2 94. 2 log2 x log2 xy 97. Make a Decision To work an extended application analyzing the normal daily maximum temperature and normal precipitation in Honolulu, Hawaii, visit this text’s website at college.hmco.com. (Data Source: NOAA) 333202_0406.qxd 12/8/05 8:43 AM Page 332 332 Chapter 4 Trigonometry 4.6 Graphs of Other Trigonometric Functions What you should learn • Sketch the graphs of tangent functions. • Sketch the graphs of cotangent functions. • Sketch the graphs of secant and cosecant functions. • Sketch the graphs of damped trigonometric functions. Graph of the Tangent Function Recall that the tangent function is odd. That is, the graph of from the identity cos x 0. which Consequently, is symmetric with respect to the origin. You also know that the tangent is undefined for values at x ± 2 ±1.5708. tan x sin xcos x Two such values are y tan x tanx tan x. Why you should learn it tan x Undef. 1255.8 14.1 x 2 1.57 1.5 4 1 0 0 4 1 1.5 1.57 2 14.1 1255.8 Undef. Trigonometric functions can be used to model real-life situations such as the distance from a television camera to a unit in a parade as in Exercise 76 on page 341. y tan x increases without bound as approaches x As indicated in the table, tan from from the right. So, the left, and decreases without bound as approaches x 2, as the graph of and , shown in Figure 4.59. Moreover, because the period of the tangent function is is an integer. The vertical asymptotes also occur when domain of the tangent function is the set of all real numbers other than x 2 n, and the range is the set of all real numbers. x 2 x 2 has vertical asymptotes at x 2 n, where n x 2 PERIOD: x DOMAIN: ALL 2 , RANGE: VERTICAL ASYMPTOTES: x 2 n n y y = tan x 3 2 1 Photodisc/Getty Images π 33 FIGURE 4.59 y a tanbx c Sketching the graph of is similar to sketching the graph of y a sinbx c in that you locate key points that identify the intercepts and asymptotes. Two consecutive vertical asymptotes can be found by solving the equations bx c 2 and bx c . 2 The midpoint between two consecutive vertical asymptotes is an -intercept of the graph. The period of the function is the distance between two consecutive vertical asymptotes. The amplitude of a tangent function is not defined. After plotting the asymptotes and the -intercept, plot a few additional points between the two asymptotes and sketch one cycle. Finally, sketch one or two additional cycles to the left and right. y a tanbx c x x 333202_0406.qxd 12/8/05 8:43 AM Page 333 Section 4.6 Graphs of Other Trigonometric Functions 333 Example 1 Sketching the Graph of a Tangent Function Sketch the graph of y tan x 2 . y y = tan x 2 3 2 1 −π π x π3 −3 FIGURE 4.60 y y = −3 tan 2x x π 4 π 2 π3 4 6 −2 −4 −6 − π3 4 − π 2 − π 4 FIGURE 4.61 Solution By solving the equations and . and you can see that two consecutive vertical asymptotes occur at x Between these two asymptotes, plot a few points, including the -intercept, as shown in the table. Three cycles of the graph are shown in Figure 4.60. x 2 tan x 2 Undef. 1 2 1 Undef. 0 0 Now try Exercise 7. Example 2 Sketching the Graph of a Tangent Function Sketch the graph of y 3 tan 2x. Solution By solving the equations 2x x 2 4 and 2x x 2 4 you can see that two consecutive vertical asymptotes occur at and x 4. Between these two asymptotes, plot a few points, including the -intercept, as shown in the table. Three cycles of the graph are shown in Figure 4.61. x x 4 x 4 8 3 tan 2x Undef. 3 8 4 3 Undef. 0 0 Now try Exercise 9. By comparing the graphs in Examples 1 and 2, you can see that the graph of increases between consecutive vertical asymptotes when In is a reflection in the -axis of the graph for a > 0. y a tanbx c a > 0, other words, the graph for and decreases between consecutive vertical asymptotes when a < 0. a < 0 x 333202_0406.qxd 12/8/05 8:43 AM Page 334 334 Chapter 4 Trigonometry Graph of the Cotangent Function The graph of the cotangent function is similar to the graph of the tangent function. It also has a period of However, from the identity . y cot x cos x sin x Te c h n o l o g y Some graphing utilities have difficulty graphing trigonometric functions that have vertical asymptotes. Your graphing utility may connect parts of the graphs of tangent, cotangent, secant, and cosecant functions that are not supposed to be connected. To eliminate this problem, change the mode of the graphing utility to dot mode. y y = 2 cot x 3 3 2 1 −2 π π 3π π 4 π 6 x FIGURE 4.63 x n, is zero, you can see that the cotangent function has vertical asymptotes when is an integer. The graph of the cotangent funcwhich occurs at tion is shown in Figure 4.62. Note that two consecutive vertical asymptotes of the bx c 0 graph of and can be found by solving the equations y a cotbx c bx c . where sin x n y y = cot x 3 2 1 −π − π 2 π 2 π π3 2 π2 x PERIOD: DOMAIN: ALL RANGE: VERTICAL ASYMPTOTES: x n , x n FIGURE 4.62 Example 3 Sketching the Graph of a Cotangent Function Sketch the graph of y 2 cot x 3 . Solution By solving the equations and . you can see that two consecutive vertical asymptotes occur at Between these two asymptotes, plot a few points, including the -intercept, as shown in the table. Three cycles of the graph are shown in Figure 4.63. Note that the distance between consecutive asymptotes. the period is x 0 x 3, and cot x 3 Undef. 2 0 2 Undef. Now try Exercise 19. 333202_0406.qxd 12/8/05 8:43 AM Page 335 Section 4.6 Graphs of Other Trigonometric Functions 335 Graphs of the Reciprocal Functions The
graphs of the two remaining trigonometric functions can be obtained from the graphs of the sine and cosine functions using the reciprocal identities sec x 1 cos x csc x 1 sin x and . y For instance, at a given value of x. the -coordinate of cos Of course, when x, exist. Near such values of of the tangent function. In other words, the graphs of is the reciprocal of the reciprocal does not the behavior of the secant function is similar to that the -coordinate of sec cos x 0, x, x y tan x sin x cos x and sec x 1 cos x x 2 n, have vertical asymptotes at is zero at these -values. Similarly, x cot x cos x sin x and csc x 1 sin x where n is an integer, and the cosine have vertical asymptotes where sin x 0 —that is, at x n. To sketch the graph of a secant or cosecant function, you should first make a y csc x, Then take reciprocals of the -coordinates to This procedure is used to obtain the sketch of its reciprocal function. For instance, to sketch the graph of first sketch the graph of obtain points on the graph of graphs shown in Figure 4.64. y csc x. y sin x. y y 3 y = sin x y = csc x −π −1 −2 −3 y = sec x π π 2 x π 2 y = cos x x n 2 PERIOD: DOMAIN: ALL , 1 RANGE: VERTICAL ASYMPTOTES: SYMMETRY: ORIGIN FIGURE 4.64 1, x n 2 PERIOD: x n DOMAIN: ALL 2 , 1 RANGE: VERTICAL ASYMPTOTES: SYMMETRY: 1, x 2 -AXIS y n In comparing the graphs of the cosecant and secant functions with those of the sine and cosine functions, note that the “hills” and “valleys” are interchanged. For example, a hill (or maximum point) on the sine curve corresponds to a valley (a relative minimum) on the cosecant curve, and a valley (or minimum point) on the sine curve corresponds to a hill (a relative maximum) on the cosecant curve, as shown in Figure 4.65. Additionally, -intercepts of the sine and cosine functions become vertical asymptotes of the cosecant and secant functions, respectively (see Figure 4.65). x Cosecant: relative minimum Sine: minimum x π2 y 4 3 2 1 −1 −2 −3 −4 π Sine: maximum Cosecant: relative maximum FIGURE 4.65 333202_0406.qxd 12/8/05 8:43 AM Page 336 336 Chapter 4 Trigonometry y = 2 csc sin x + ( π 4 ) Example 4 Sketching the Graph of a Cosecant Function 4 3 1 Sketch the graph of y 2 cscx . 4 Solution Begin by sketching the graph of π π 2 x y 2 sinx . 4 For this function, the amplitude is 2 and the period is 2. By solving the equations FIGURE 4.66 and you can see that one cycle of the sine function corresponds to the interval from x 4 The graph of this sine function is represented by the gray curve in Figure 4.66. Because the sine function is zero at the midpoint and endpoints of this interval, the corresponding cosecant function x 74. to y 2 cscx 4 sinx 4 1 2 has vertical asymptotes at the cosecant function is represented by the black curve in Figure 4.66. x 4, x 34, x 74, etc. The graph of Now try Exercise 25. Example 5 Sketching the Graph of a Secant Function Sketch the graph of y sec 2x. y = sec 2x y y = cos 2x 3 −1 −2 −3 −π − π 2 x π π 2 Solution Begin by sketching the graph of Figure 4.67. Then, form the graph of x Note that the -intercepts of , 0, , 0, y cos 2x 3 4 4 4 , 0, . . . y cos 2x, y sec 2x as indicated by the gray curve in as the black curve in the figure. correspond to the vertical asymptotes x 3 4 x x , , 4 4 , . . . of the graph of y sec 2x . is y sec 2x. Moreover, notice that the period of y cos 2x and FIGURE 4.67 Now try Exercise 27. 333202_0406.qxd 12/8/05 8:43 AM Page 337 Section 4.6 Graphs of Other Trigonometric Functions 337 Damped Trigonometric Graphs A product of two functions can be graphed using properties of the individual functions. For instance, consider the function f x x sin x as the product of the functions value and the fact that sin x ≤ 1, y x y sin x. and you have 0 ≤ xsin x ≤ x. Using properties of absolute Consequently, y = x x ≤ x sin x ≤ x which means that the graph of y x. Furthermore, because f x x sin x lies between the lines y x and π x and f x x sin x ±x at x 2 n f x x sin x 0 at x n y = −x y π3 π 2 π −π π−2 π−3 f(x) = x sin x FIGURE 4.68 at x 2 n x Do you see why the graph of f x x sin x touches the lines y ±x and why the graph has -intercepts at Recall that the sine function is equal to 1 at 32, 52, . . . of 2, 2, odd multiples and is equal to 0 at multiples of . 2 3, . . . x n? , f(x) = e−x sin 3x y 6 4 −4 −6 y = e−e−x and x f the graph of x has -intercepts at f x x sin x, tion touches the line x n. the factor x y x A sketch of or the line f and is shown in Figure 4.68. In the func- x 2 n y x at is called the damping factor. Example 6 Damped Sine Wave Sketch the graph of f x ex sin 3x. Solution Consider f x y ex as the product of the two functions y sin 3x and each of which has the set of real numbers as its domain. For any real number you know that that ex sin 3x ≤ ex, x, which means sin 3x ≤ 1. ex ≥ 0 and So, ex ≤ ex sin 3x ≤ ex. Furthermore, because f x ex sin 3x ±ex at x 6 n 3 f x ex sin 3x 0 at x n 3 y ex the graph of and has intercepts at touches the curves x n3. and A sketch is shown in Figure 4.69. at f y ex x 6 n3 FIGURE 4.69 Now try Exercise 65. 333202_0406.qxd 12/8/05 8:43 AM Page 338 338 Chapter 4 Trigonometry Figure 4.70 summarizes the characteristics of the six basic trigonometric functions. y 2 1 y = sin x −π π− 2 π 2 π π 3 2 −2 DOMAIN: ALL REALS RANGE: PERIOD: 2 1, 1 y y = csc x = 1 sin x 3 2 1 y 2 y = cos x x π− π x π 2 −1 −2 DOMAIN: ALL REALS RANGE: PERIOD: 2 1, 1 y y = sec x = 1 cos x 3 y y = tan π5 2 x 2 , DOMAIN: ALL RANGE: PERIOD: n y y = cot x = 1 tan x 3 2 1 −π π π 2 x π2 − π2 x −2 −3 x n DOMAIN: ALL , 1 RANGE: 2 PERIOD: FIGURE 4.70 1, x n 2 , 1 1, DOMAIN: ALL RANGE: PERIOD: 2 x n DOMAIN: ALL RANGE: PERIOD: , W RITING ABOUT MATHEMATICS Combining Trigonometric Functions Recall from Section 1.8 that functions can be combined arithmetically. This also applies to trigonometric functions. For each of the functions hx x sin x and hx cos x sin 3x (a) identify two simpler functions and table to show how to obtain the numerical values of of and (c) use graphs of and gx, f x and g g f f that comprise the combination, (b) use a hx from the numerical values to show how may be formed. h Can you find functions f x d a sinbx c and gx d a cosbx c such that f x gx 0 for all x? 333202_0406.qxd 12/8/05 8:43 AM Page 339 Section 4.6 Graphs of Other Trigonometric Functions 339 4.6 Exercises VOCABULARY CHECK: Fill in the blanks. 1. The graphs of the tangent, cotangent, secant, and cosecant functions all have ________ asymptotes. 2. To sketch the graph of a secant or cosecant function, first make a sketch of its corresponding ________ function. 3. For the functions given by f x gx sin x, gx is called the ________ factor of the function f x. 4. The period of is ________. 5. The domain of is all real numbers such that ________. y tan x y cot x y sec x y csc x 6. The range of 7. The period of is ________. is ________. PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. In Exercises 1–6, match the function with its graph. State the period of the function. [The graphs are labeled (a), (b), (c), (d), (e), and (f).] (b) (d) (f) (a) (c) (e) − 3π 3 −3 y 3 x 2 x π 3 2 In Exercises 7–30, sketch the graph of the function. Include two full periods. 7. 9. 11. 13. 15. 17. 19. 21. 23. 25. 27. 29. 2 sec x y 1 3 tan x y tan 3x y 1 y csc x y sec x 1 x 2 y csc y cot y 1 x 2 2 sec 2x x 4 y tan y csc x y 2 secx cscx y 1 4 4 8. 10. 12. 14. 16. 18. 20. 22. y 1 4 tan x y 3 tan x y 1 4 sec x y 3 csc 4x y 2 sec 4x 2 x 3 y csc y 3 cot y 1 x 2 2 tan x 24. y tanx 26. 28. 30. y csc2x y sec x 1 y 2 cotx 2 In Exercises 31– 40, use a graphing utility to graph the function. Include two full periods. x 1 31. y tan x 3 32. y tan 2x 1. y sec 2x 3. 5. y 1 2 y 1 2 cot x sec x 2 2. y tan x 2 4. y csc x 6. y 2 sec x 2 33. 35. 37. 39. y 2 sec 4x y tanx y csc4x y 0.1 tanx 4 4 4 34. 36. y sec x y 1 4 cotx y 2 sec2x secx 2 38. 40. y 1 3 2 2 333202_0406.qxd 12/8/05 8:43 AM Page 340 340 Chapter 4 Trigonometry In Exercises 41– 48, use a graph to solve the equation on the interval [2, 2]. 41. 42. tan x 1 tan x 3 43. cot x 3 3 44. 45. 46. 47. 48. cot x 1 sec x 2 sec x 2 csc x 2 csc x 23 3 In Exercises 49 and 50, use the graph of the function to determine whether the function is even, odd, or neither. 49. f x sec x 50. f x tan x − π 51. Graphical Reasoning Consider the functions given by f x 2 sin x and gx 1 2 csc x on the interval 0, . g (a) Graph and f in the same coordinate plane. (b) Approximate the interval in which f > g. (c) Describe the behavior of each of the functions as approaches How is the behavior of behavior of as approaches ? x . f g x related to the 52. Graphical Reasoning Consider the functions given by f x tan x 2 and gx 1 2 sec x 2 on the interval 1, 1. (a) Use a graphing utility to graph viewing window. f and g in the same (b) Approximate the interval in which f < g. (c) Approximate the interval in which 2f < 2g. How does the result compare with that of part (b)? Explain. In Exercises 53–56, use a graphing utility to graph the two equations in the same viewing window. Use the graphs to determine whether the expressions are equivalent. Verify the results algebraically. 53. 54. 55. 56. y1 y1 y1 y1 sin x csc x, sin x sec x, cos x sin x y2 , 1 tan x y2 y2 cot x sec2 x 1, tan2 x y2 In Exercises 57– 60, match the function with its graph. Describe the behavior of the function as approaches zero. [The graphs are labeled (a), (b), (c), and (d).] x (a) (c) x π 2 (b) (d) x π − π y y 2 −1 −2 −3 −4 −5 −6 4 2 −2 −4 y 4 2 y −4 4 3 2 1 −1 −2 x π 2 π 3 2 x π 57. 58. 59. 60. f x x cos x f x x sin x gx x sin x gx x cos x In Exercises 61–64, graph the functions and Use the graphs to make a conjecture about the f Conjecture g. relationship between the functions. f x sin x cosx 61. f x sin x cosx f x sin2 x, gx 1 2 62. 63. gx 0 , 2 , 2 1 cos 2x gx 2 sin x 64. f x cos2 x 2 , gx 1 2 1 cos x In Exercises 65–68, use a graphing utility to graph the function and the damping facto
r of the function in the same viewing window. Describe the behavior of the function as increases without bound. x gx ex 22 sin x f x 2x4 cos x 65. 67. 66. 68. f x ex cos x hx 2x24 sin x Exploration In Exercises 69–74, use a graphing utility to graph the function. Describe the behavior of the function as approaches zero. x 69. y 6 x cos x, x > 0 70. y 4 x sin 2x, x > 0 333202_0406.qxd 12/8/05 8:43 AM Page 341 71. gx sin x x 73. f x sin 1 x 72. f x 1 cos x x 74. hx x sin 1 x d 75. Distance A plane flying at an altitude of 7 miles above a radar antenna will pass directly over the radar antenna (see figure). Let be the ground distance from the antenna to x the point directly under the plane and let be the angle of is positive as the elevation to the plane from the antenna. ( d x plane approaches the antenna.) Write and graph the function over the interval d as a function of 0 < x < . x d 7 mi Not drawn to scale 76. Television Coverage A television camera is on a reviewing platform 27 meters from the street on which a parade will be passing from left to right (see figure). Write from the camera to a particular unit in the the distance and graph the function parade as a function of the angle 2 < x < 2. over the interval as (Consider negative when a unit in the parade approaches from the left.) x, d x Not drawn to scale 27 m d x Camera Model It 77. Predator-Prey Model The population of coyotes (a predator) at time (in months) in a region is estimated to be C t C 5000 2000 sin t 12 and the population of rabbits (its prey) is estimated to be R Section 4.6 Graphs of Other Trigonometric Functions 341 Model It (co n t i n u e d ) R 25,000 15,000 cos t 12 . (a) Use a graphing utility to graph both models in the same viewing window. Use the window setting 0 ≤ t ≤ 100. (b) Use the graphs of the models in part (a) to explain the oscillations in the size of each population. (c) The cycles of each population follow a periodic pattern. Find the period of each model and describe several factors that could be contributing to the cyclical patterns. 78. Sales The projected monthly sales (in thousands of units) of lawn mowers (a seasonal product) are modeled by S 74 3t 40 cost6, t is the time (in months), with corresponding to January. Graph the sales function over 1 year. where t 1 S 79. Meterology The normal monthly high temperatures H (in degrees Fahrenheit) for Erie, Pennsylvania are approximated by Ht 54.33 20.38 cos t 6 15.69 sin t 6 and the normal monthly low temperatures mated by L are approxi- Lt 39.36 15.70 cos t 6 14.16 sin t 6 is the time (in months), with t where to January (see figure). Atmospheric Administration) corresponding (Source: National Oceanic and ( 80 60 40 20 H(t) L(t) 1 2 3 4 5 7 8 6 Month of year 9 10 11 12 t (a) What is the period of each function? (b) During what part of the year is the difference between the normal high and normal low temperatures greatest? When is it smallest? (c) The sun is northernmost in the sky around June 21, but the graph shows the warmest temperatures at a later date. Approximate the lag time of the temperatures relative to the position of the sun. 333202_0406.qxd 12/8/05 8:43 AM Page 342 342 Chapter 4 Trigonometry 80. Harmonic Motion An object weighing pounds is suspended from the ceiling by a steel spring (see figure). The weight is pulled downward (positive direction) from its equilibrium position and released. The resulting motion of the weight is described by the function W y 1 2 et4 cos 4t, t > 0 y where seconds). is the distance (in feet) and t is the time (in Equilibrium 86. Approximation Using calculus, it can be shown that the tangent function can be approximated by the polynomial tan x x 2x3 3! 16x 5 5! x is in radians. Use a graphing utility to graph the where tangent function and its polynomial approximation in the same viewing window. How do the graphs compare? 87. Approximation Using calculus, it can be shown that the secant function can be approximated by the polynomial sec x 1 x 2 2! 5x4 4! x where is in radians. Use a graphing utility to graph the secant function and its polynomial approximation in the same viewing window. How do the graphs compare? y 88. Pattern Recognition (a) Use a graphing utility to graph each function. (a) Use a graphing utility to graph the function. (b) Describe the behavior of the displacement function for t. increasing values of time Synthesis True or False? the statement is true or false. Justify your answer. In Exercises 81 and 82, determine whether 81. The graph of y csc x can be obtained on a calculator by graphing the reciprocal of y sin x. 82. The graph of y sec x can be obtained on a calculator by graphing a translation of the reciprocal of y sin x. 83. Writing Describe the behavior of f x tan x as x approaches 2 from the left and from the right. 84. Writing Describe the behavior of f x csc x as x approaches from the left and from the right. 85. Exploration Consider the function given by f x x cos x. (a) Use a graphing utility to graph the function and verify that there exists a zero between 0 and 1. Use the graph to approximate the zero. 1, cosxn1 generate a sequence For example, x1, x2, where x0 xn . (b) Starting with x3, . . . , 1 x0 cosx0 x1 cosx1 x2 cosx2 x3 What value does the sequence approach? y1 y2 sin x 1 4 sin x 1 4 3 3 sin 3x sin 3x 1 5 sin 5x (b) Identify the pattern started in part (a) and find a that continues the pattern one more term. y3. function Use a graphing utility to graph y3 (c) The graphs in parts (a) and (b) approximate the periodic that is a better function in the figure. Find a function approximation. y4 y 1 x 3 Skills Review In Exercises 89–92, solve the exponential equation. Round your answer to three decimal places. 89. 91. e2x 54 300 1 ex 100 90. 92. 83x 98 1 0.15 365 365t 5 In Exercises 93–98, solve the logarithmic equation. Round your answer to three decimal places. 93. 95. ln3x 2 73 lnx2 1 3.2 log8 x log8 97. 98. log6 x log6 x 1 1 3 x2 1 log6 64x 94. 96. ln14 2x 68 ln x 4 5 333202_0407.qxd 12/7/05 11:10 AM Page 343 4.7 Inverse Trigonometric Functions Section 4.7 Inverse Trigonometric Functions 343 What you should learn • Evaluate and graph the inverse sine function. • Evaluate and graph the other inverse trigonometric functions. • Evaluate and graph the compositions of trigonometric functions. Why you should learn it You can use inverse trigonometric functions to model and solve real-life problems. For instance, in Exercise 92 on page 351, an inverse trigonometric function can be used to model the angle of elevation from a television camera to a space shuttle launch. NASA When evaluating the inverse sine function, it helps to remember the phrase “the arcsine of is the angle (or number) whose sine is x.” x Inverse Sine Function Recall from Section 1.9 that, for a function to have an inverse function, it must be one-to-one—that is, it must pass the Horizontal Line Test. From Figure 4.71, x you can see that y yield the same -value. does not pass the test because different values of y sin x y 1 −1 π− y = sin x π x sin x has an inverse function on this interval. FIGURE 4.71 However, if you restrict the domain to the interval 2 ≤ x ≤ 2 (corresponding to the black portion of the graph in Figure 4.71), the following properties hold. 2, 2, 2, 2, the function y sin x y sin x takes on its full range of values, is increasing. 1. On the interval 2. On the interval 1 ≤ sin x ≤ 1. 3. On the interval 2, 2, y sin x 2 ≤ x ≤ 2 , is one-to-one. y sin x So, on the restricted domain function called the inverse sine function. It is denoted by has a unique inverse y arcsin x or y sin1 x. sin1 x is consistent with the inverse function notation x The notation The x arcsin notation (read as “the arcsine of ”) comes from the association of a central angle with its intercepted arc length on a unit circle. So, arcsin means x are the angle (or arc) whose sine is denotes the inverse commonly used in mathematics, so remember that sine function rather than lie in the interval 2 ≤ arcsin x ≤ 2. sin1 x x The values of arcsin y arcsin x 1sin x. The graph of is shown in Example 2. Both notations, arcsin x sin1 x, and x. f 1x. Definition of Inverse Sine Function The inverse sine function is defined by y arcsin x if and only if sin y x 1 ≤ x ≤ 1 and 2 ≤ y ≤ 2. The domain of y arcsin x is where 1, 1, and the range is 2, 2. 333202_0407.qxd 12/7/05 11:10 AM Page 344 344 Chapter 4 Trigonometry Example 1 Evaluating the Inverse Sine Function As with the trigonometric functions, much of the work with the inverse trigonometric functions can be done by exact calculations rather than by calculator approximations. Exact calculations help to increase your understanding of the inverse functions by relating them to the right triangle definitions of the trigonometric functions. y π 2 π ( ) 1, 2 2( 2 , ) π 4 (0, 0) −( 1, − 2 FIGURE 4.72 ( 1 2 , π 6 ) 1 x y = arcsin If possible, find the exact value. arcsin 1 2 sin1 3 2 b. a. c. sin1 2 Solution a. Because sin arcsin 1 2 1 2 6 for 2 ≤ y ≤ , 2 it follows that . 6 Angle whose sine is 1 2 b. Because sin 3 3 2 for 2 ≤ y ≤ , 2 it follows that sin1 3 2 . 3 Angle whose sine is 32 c. It is not possible to evaluate because there is no angle whose sine is 2. Remember that the domain of the inverse sine function is 1, 1. when y sin1 x x 2 Now try Exercise 1. Example 2 Graphing the Arcsine Function Sketch a graph of y arcsin x. Solution By definition, 2 ≤ y ≤ 2. 2, 2, of values. Then plot the points and draw a smooth curve through the points. are equivalent for their graphs are the same. From the interval in the second equation to make a table the equations So, you can assign values to y arcsin x sin y x and y y 2 4 6 x sin arcsin x is shown in Figure 4.72. Note that it is the The resulting graph for reflection (in the line ) of the black portion of the graph in Figure 4.71. Be sure you see that Figure 4.72 shows the entire graph of the inverse sine function. and the Remember that the domain of range is the clo
sed interval y arcsin x 2, 2. is the closed interval 1, 1 y x Now try Exercise 17. 333202_0407.qxd 12/7/05 11:10 AM Page 345 Section 4.7 Inverse Trigonometric Functions 345 Other Inverse Trigonometric Functions The cosine function is decreasing and one-to-one on the interval shown in Figure 4.73. 0 ≤ x ≤ , as y y = cos x π− −1 π π 2 x π 2 cos x has an inverse function on this interval. FIGURE 4.73 Consequently, on this interval the cosine function has an inverse function—the inverse cosine function—denoted by y arccos x or y cos1 x. y tan x Similarly, you can define an inverse tangent function by restricting the domain of The following list summarizes the definitions of the three most common inverse trigonometric functions. The remaining three are defined in Exercises 101–103. 2, 2. to the interval Definitions of the Inverse Trigonometric Functions Function Domain y arcsin x if and only if sin y x 1 ≤ x ≤ 1 Range 2 ≤ y ≤ y arccos x if and only if cos arctan x if and only if tan The graphs of these three inverse trigonometric functions are shown in Figure 4.74 = arcsin x 1 −1 y = arccos x π 2 y = arctan x −2 −1 1 2 x − π 2 1, 1 DOMAIN: RANGE: 2, 2 FIGURE 4.74 −1 x 1 − π 2 1, 1 DOMAIN: RANGE: 0, , DOMAIN: RANGE: 2, 2 333202_0407.qxd 12/8/05 8:25 AM Page 346 346 Chapter 4 Trigonometry Example 3 Evaluating Inverse Trigonometric Functions Find the exact value. a. arccos 2 2 c. arctan 0 Solution a. Because b. cos11 d. tan11 cos4 22, . and 4 lies in 0, , it follows that Angle whose cosine is 22 arccos 2 2 4 b. Because and lies in 0, , it follows that cos 1, cos11 . tan 0 0, arctan 0 0. c. Because and 0 lies in Angle whose cosine is 2, 2, 1 it follows that Angle whose tangent is 0 d. Because tan4 1, and 4 lies in 2, 2, it follows that tan11 . 4 Angle whose tangent is 1 Now try Exercise 11. Example 4 Calculators and Inverse Trigonometric Functions Use a calculator to approximate the value (if possible). a. b. c. arctan8.45 sin1 0.2447 arccos 2 Solution It is important to remember that the domain of the inverse sine function and the inverse cosine 1, 1, function is as indicated in Example 4(c). a. b. c. Mode Function arctan8.45 From the display, it follows that Radian Calculator Keystrokes TAN1 8.45 ENTER arctan8.45 1.453001. sin1 0.2447 From the display, it follows that Radian SIN1 0.2447 ENTER sin1 0.2447 0.2472103. arccos 2 In real number mode, the calculator should display an error message because the domain of the inverse cosine function is 1, 1. Radian ENTER 2 COS1 Now try Exercise 25. In Example 4, if you had set the calculator to degree mode, the displays would have been in degrees rather than radians. This convention is peculiar to calculators. By definition, the values of inverse trigonometric functions are always in radians. 333202_0407.qxd 12/7/05 11:10 AM Page 347 Section 4.7 Inverse Trigonometric Functions 347 Compositions of Functions Recall from Section 1.9 that for all tions have the properties f f 1x x and f 1 f x x. x in the domains of and f f 1, inverse func- Inverse Properties of Trigonometric Functions 2 ≤ y ≤ 2, If 1 ≤ x ≤ 1 and sinarcsin x x then arcsinsin y y. and If 1 ≤ x ≤ 1 and 0 ≤ y ≤ , then cosarccos x x and arccoscos y y. If x is a real number and tanarctan x x 2 < y < 2, then and arctantan y y. Keep in mind that these inverse properties do not apply for arbitrary values of and For instance, x y. arcsinsin arcsin1 3 2 2 3 . 2 In other words, the property arcsinsin y y is not valid for values of outside the interval y 2, 2. Example 5 Using Inverse Properties If possible, find the exact value. a. tanarctan5 arcsinsin b. 5 3 c. coscos1 Solution a. Because 5 applies, and you have lies in the domain of the arctan function, the inverse property tanarctan5 5. 53 2 ≤ y ≤ 2. b. In this case, 5 3 2 3 does not lie within the range of the arcsine function, However, is coterminal with 53 which does lie in the range of the arcsine function, and you have arcsinsin arcsinsin 5 3 coscos1 3 . 3 c. The expression is not defined because cos1 is not defined. Remember that the domain of the inverse cosine function is Now try Exercise 43. 1, 1. 333202_0407.qxd 12/7/05 11:10 AM Page 348 348 Chapter 4 Trigonometry y 3 2 2 3 − 2 5= u = arccos 2 3 2 Angle whose cosine is FIGURE 4.75 2 3 y 52 − −32 = 4 ( ) ( ( u = arcsin − 3 5 −3 5 x x Angle whose sine is FIGURE 4.76 3 5 Example 6 shows how to use right triangles to find exact values of compositions of inverse functions. Then, Example 7 shows how to use right triangles to convert a trigonometric expression into an algebraic expression. This conversion technique is used frequently in calculus. Example 6 Evaluating Compositions of Functions Find the exact value. tanarccos a. 2 3 b. cosarcsin 3 5 Solution a. If you let then cos u 2 3. u arccos 2 3, Because quadrant angle. You can sketch and label angle Consequently, tanarccos 2 3 u arcsin3 tan u opp adj , sin u 3 5. 5 2 then . 5 b. If you let fourth-quadrant angle. You can sketch and label angle 4.76. Consequently, cosarcsin 3 5 cos u adj hyp 4 5 . cos u u is positive, is a firstas shown in Figure 4.75. u Because sin u u is negative, is a as shown in Figure u Now try Exercise 51. Example 7 Some Problems from Calculus Write each of the following as an algebraic expression in x. a. sinarccos 3x, 0 ≤ x ≤ 1 3 b. cotarccos 3x3x)2 Solution If you let u arccos 3x, then cos u 3x, where 1 ≤ 3x ≤ 1. Because u = arccos 3x 3x Angle whose cosine is FIGURE 4.77 3x cos u adj hyp 3x 1 you can sketch a right triangle with acute angle this triangle, you can easily convert each expression to algebraic form. as shown in Figure 4.77. From u, a. b. sinarccos 3x sin u opp hyp cotarccos 3x cot u adj opp 1 9x 2, 0 ≤ x ≤ 3x 1 9x Now try Exercise 59. In Example 7, similar arguments can be made for values lying in the x- interval 1 3, 0. 333202_0407.qxd 12/7/05 11:10 AM Page 349 Section 4.7 Inverse Trigonometric Functions 349 4.7 Exercises VOCABULARY CHECK: Fill in the blanks. Function Alternative Notation Domain 1. y arcsin x 2. __________ y arctan x 3. __________ y cos1 x __________ __________ 1 ≤ x ≤ 1 __________ Range 2 ≤ y ≤ 2 __________ __________ PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. In Exercises 1–16, evaluate the expression without using a calculator. In Exercises 35 and 36, determine the missing coordinates of the points on the graph of the function. 1. 3. 5. 7. 9. 11. 13. 15. arcsin 1 2 arccos 1 2 3 3 arctan cos1 3 2 arctan3 arccos 1 2 sin1 3 2 tan1 0 2. 4. 6. 8. 10. 12. 14. 16. arcsin 0 arccos 0 arctan1 sin1 2 2 arctan 3 2 2 arcsin 3 3 tan1 cos1 1 35. y π 2 π 4 −3 −2 − 3, ( ) y = arctan x ( π ) 4 36. y π π 4 (−1, 1 ( − 2 , ) ) −2 −1 y = arccos x ( 1 π ) 6, x 2 In Exercises 37–42, use an inverse trigonometric function to write as a function of 37. 38. x. In Exercises 17 and 18, use a graphing utility to graph y x and cally that the domain of properly.) f, g, in the same viewing window to verify geometrig (Be sure to restrict is the inverse function of f. f f x sin x, f x tan x, 17. 18. gx arcsin x gx arctan x 39. In Exercises 19–34, use a calculator to evaluate the expression. Round your result to two decimal places. 41. 19. 21. 23. 25. 27. 29. 31. 33. arccos 0.28 arcsin0.75 arctan3 sin1 0.31 arccos0.41 arctan 0.92 arcsin 3 4 tan1 7 2 20. 22. 24. 26. 28. 30. 32. 34. arcsin 0.45 arccos0.7 arctan 15 cos1 0.26 arcsin0.125 arctan 2.8 arccos1 tan195 3 7 x x + 2 40. 42. x θ 4 θ 10 2x θ x + 3 In Exercises 43–48, use the properties of inverse trigonometric functions to evaluate the expression. 43. 45. sinarcsin 0.3 cosarccos0.1 47. arcsinsin 3 44. tanarctan 25 sinarcsin0.2 7 2 46. 48. arccoscos 333202_0407.qxd 12/7/05 11:10 AM Page 350 350 Chapter 4 Trigonometry In Exercises 49–58, find the exact value of the expression. (Hint: Sketch a right triangle.) sinarctan 3 49. 50. 4 51. 53. 55. 57. costan1 2 cosarcsin 5 secarctan3 sinarccos2 13 5 3 52. 54. 56. 58. secarcsin 4 5 sincos1 5 5 cscarctan 5 tanarcsin3 cotarctan 5 12 4 8 In Exercises 59–68, write an algebraic expression that is equivalent to the expression. (Hint: Sketch a right triangle, as demonstrated in Example 7.) In Exercises 77–82, sketch a graph of the function. 77. 78. 79. 80. 81. y 2 arccos x gt arccost 2 f x) arctan 2x f x arctan x 2 hv tanarccos v 82. f x arccos x 4 In Exercises 83– 88, use a graphing utility to graph the function. 59. 61. 63. 65. 66. 67. cotarctan x cosarcsin 2x sinarccos x tanarccos x 3 cotarctan 1 x cscarctan cosarcsin 68. x 2 x h r 60. 62. 64. sinarctan x secarctan 3x secarcsinx 1 83. 84. 85. 86. 87. 88. f x 2 arccos2x f x arcsin4x f x arctan2x 3 f x 3 arctanx f x sin12 3 cos1 1 f x 2 In Exercises 69 and 70, use a graphing utility to graph and g in the same viewing window to verify that the two functions are equal. Explain why they are equal. Identify any asymptotes of the graphs. f 69. f x sinarctan 2x, gx f x tanarccos 70. , x 2 gx 2x 1 4x2 4 x 2 x In Exercises 71–74, fill in the blank. 71. arctan arcsin, x 0 9 x 36 x 2 6 72. arcsin arccos, 0 ≤ x ≤ 6 73. arccos 74. arccos 3 x 2 2x 10 x 2 2 arcsin arctan, x 2 ≤ 2 In Exercises 75 and 76, sketch a graph of the function and f x arcsin x. compare the graph of with the graph of g 75. gx arcsinx 1 76. gx arcsin x 2 In Exercises 89 and 90, write the function in terms of the sine function by using the identity A cos t B sin t A2 B2 sint arctan . A B Use a graphing utility to graph both forms of the function. What does the graph imply? f t 3 cos 2t 3 sin 2t f t 4 cos t 3 sin t 90. 89. 91. Docking a Boat A boat is pulled in by means of a winch located on a dock 5 feet above the deck of the boat (see figure). Let be the angle of elevation from the boat to the winch and let be the length of the rope from the winch to the boat. s 5 ft s θ (a) Write as a function of s. (b) Find when s 40 feet and s 20 feet. 333202_0407.qxd 12/7/05 11:10 AM Page 351 92. Photography A television camera at ground level is filming the lift-off of a space shuttle at a point
750 meters from the launch pad (see figure). Let be the angle of elevation to the shuttle and let be the height of the shuttle. s Section 4.7 Inverse Trigonometric Functions 351 94. Granular Angle of Repose Different types of granular substances naturally settle at different angles when stored is called the angle of in cone-shaped piles. This angle repose (see figure). When rock salt is stored in a coneshaped pile 11 feet high, the diameter of the pile’s base is about 34 feet. (Source: Bulk-Store Structures, Inc.) 11 ft θ 17 ft s θ 750 m Not drawn to scale (a) Write as a function of s. (a) Find the angle of repose for rock salt. (b) How tall is a pile of rock salt that has a base diameter of 40 feet? 95. Granular Angle of Repose When whole corn is stored in a cone-shaped pile 20 feet high, the diameter of the pile’s base is about 82 feet. (a) Find the angle of repose for whole corn. (b) Find when s 300 meters and s 1200 meters. (b) How tall is a pile of corn that has a base diameter of 100 feet? Model It 93. Photography A photographer is taking a picture of a three-foot-tall painting hung in an art gallery. The camera lens is 1 foot below the lower edge of the painting (see figure). The angle subtended by the camera lens x feet from the painting is arctan 3x x 2 4 , x > 0. 3 ft 1 ft β θ α x Not drawn to scale 96. Angle of Elevation An airplane flies at an altitude of 6 miles toward a point directly over an observer. Consider and as shown in the figure. x θ x 6 mi Not drawn to scale (a) Write as a function of x. (b) Find when x 7 miles and x 1 mile. 97. Security Patrol A security car with its spotlight on is as parked 20 meters from a warehouse. Consider shown in the figure. and x (a) Use a graphing utility to graph as a function of x. θ 20 m (b) Move the cursor along the graph to approximate is maximum. the distance from the picture when (c) Identify the asymptote of the graph and discuss its meaning in the context of the problem. Not drawn to scale x (a) Write as a function of x. (b) Find when x 5 meters and x 12 meters. 333202_0407.qxd 12/7/05 11:10 AM Page 352 352 Chapter 4 Trigonometry Synthesis True or False? the statement is true or false. Justify your answer. In Exercises 98–100, determine whether 98. sin 99. tan 5 6 5 4 1 2 1 arcsin 5 1 6 2 arctan 1 5 4 100. arctan x arcsin x arccos x 101. Define the inverse cotangent function by restricting the 0, , domain of the cotangent function to the interval and sketch its graph. 102. Define the inverse secant function by restricting the 0, 2 domain of the secant function to the intervals and sketch its graph. and 2, , 103. Define the inverse cosecant function by restricting the domain of the cosecant function to the intervals 2, 0 and sketch its graph. 0, 2, and 107. Think About It Consider the functions given by f x sin x and f 1x arcsin x. functions (a) Use a graphing utility to graph the composite f 1 f. (b) Explain why the graphs in part (a) are not the graph and Why do the graphs of y x. f f 1 f f 1 and of the line f 1 f differ? (a) 108. Proof Prove each identity. arcsinx arcsin x arctanx arctan x , arctan x arctan (b) (c) (d) arcsin x arccos x (e) arcsin x arctan Skills Review 104. Use the results of Exercises 101–103 to evaluate each expression without using a calculator. (a) (c) arcsec 2 arccot3 (b) arcsec 1 (d) arccsc 2 In Exercises 109–112, evaluate the expression. Round your result to three decimal places. 109. 111. 8.23.4 1.150 110. 112. 10142 162 105. Area In calculus, the y 1x 2 1, region bounded by and x a, it is shown that the area of y 0, the graphs of is given by x b Area arctan b arctan a In Exercises 113–116, sketch a right triangle corresponding to the trigonometric function of the acute angle Use the Pythagorean Theorem to determine the third side. Then find the other five trigonometric functions of . . (see figure). Find the area for the following values of and a 113. 115. sin 3 4 cos 5 6 114. 116. tan 2 sec 3 b. a 0, b 1 a 0, b 3 (a) (c) (b) (d) a 1, b 1 a 12 106. Think About It Use a graphing utility to graph the functions f x x and gx 6 arctan x. x > 0, it appears that For that there exists a positive real number a. for Approximate the number g > f. x > a. Explain why you know g < f such that a 117. Partnership Costs A group of people agree to share equally in the cost of a $250,000 endowment to a college. If they could find two more people to join the group, each person’s share of the cost would decrease by $6250. How many people are presently in the group? 118. Speed A boat travels at a speed of 18 miles per hour in still water. It travels 35 miles upstream and then returns to the starting point in a total of 4 hours. Find the speed of the current. 119. Compound Interest A total of $15,000 is invested in an account that pays an annual interest rate of 3.5%. Find the balance in the account after 10 years, if interest is compounded (a) quarterly, (b) monthly, (c) daily, and (d) continuously. 120. Profit Because of a slump in the economy, a department store finds that its annual profits have dropped from $742,000 in 2002 to $632,000 in 2004. The profit follows an exponential pattern of decline. What is the expected profit for 2008? (Let represent 2002.) t 2 333202_0408.qxd 12/7/05 11:11 AM Page 353 4.8 Applications and Models Section 4.8 Applications and Models 353 What you should learn • Solve real-life problems involving right triangles. • Solve real-life problems involving directional bearings. • Solve real-life problems involving harmonic motion. Why you should learn it Right triangles often occur in real-life situations. For instance, in Exercise 62 on page 362, right triangles are used to determine the shortest grain elevator for a grain storage bin on a farm. Applications Involving Right Triangles In this section, the three angles of a right triangle are denoted by the letters and (where angles by the letters B,A, is the right angle), and the lengths of the sides opposite these is the hypotenuse). (where and b, a, C C c c Example 1 Solving a Right Triangle Solve the right triangle shown in Figure 4.78 for all unknown sides and angles. B a C c A 34.2° b = 19.4 FIGURE 4.78 it follows that A B 90 and B 90 34.2 55.8. Solution Because To solve for C 90, a, use the fact that tan A opp adj a b a b tan A. So, So, a 19.4 tan 34.2 13.18. cos A adj hyp c 19.4 23.46. b c cos 34.2 Similarly, to solve for use the fact that c b cos A c, . Now try Exercise 1. B Example 2 Finding a Side of a Right Triangle c = 110 ft A 72° C b A safety regulation states that the maximum angle of elevation for a rescue ladder A fire department’s longest ladder is 110 feet. What is the maximum safe is rescue height? 72. a Solution A sketch is shown in Figure 4.79. From the equation sin A ac, it follows that a c sin A 110 sin 72 104.6. So, the maximum safe rescue height is about 104.6 feet above the height of the fire truck. FIGURE 4.79 Now try Exercise 15. 333202_0408.qxd 12/7/05 11:11 AM Page 354 354 Chapter 4 Trigonometry Example 3 Finding a Side of a Right Triangle s a At a point 200 feet from the base of a building, the angle of elevation to the 53, bottom of a smokestack is as shown in Figure 4.80. Find the height of the smokestack alone. whereas the angle of elevation to the top is 35, s Solution Note from Figure 4.80 that this problem involves two right triangles. For the smaller right triangle, use the fact that tan 35 a 200 35° 53° 200 ft to conclude that the height of the building is a 200 tan 35. For the larger right triangle, use the equation FIGURE 4.80 tan 53 a s 200 a s 200 tan 53º. to conclude that s 200 tan 53 a 200 tan 53 200 tan 35 125.4 feet. Now try Exercise 19. So, the height of the smokestack is 20 m Angle of depression A FIGURE 4.81 Example 4 Finding an Acute Angle of a Right Triangle 1.3 m 2.7 m A swimming pool is 20 meters long and 12 meters wide. The bottom of the pool is slanted so that the water depth is 1.3 meters at the shallow end and 4 meters at the deep end, as shown in Figure 4.81. Find the angle of depression of the bottom of the pool. Solution Using the tangent function, you can see that tan A opp adj 2.7 20 0.135. So, the angle of depression is A arctan 0.135 0.13419 radian 7.69. Now try Exercise 25. 333202_0408.qxd 12/7/05 11:11 AM Page 355 Section 4.8 Applications and Models 355 Trigonometry and Bearings In surveying and navigation, directions are generally given in terms of bearings. A bearing measures the acute angle that a path or line of sight makes with a fixed E in north-south line, as shown in Figure 4.82. For instance, the bearing S Figure 4.82 means 35 degrees east of south. 35 N N 80° N 45° W E W E W E 35° S S 35° E S N 80° W S N 45° E FIGURE 4.82 Example 5 Finding Directions in Terms of Bearings A ship leaves port at noon and heads due west at 20 knots, or 20 nautical miles (nm) per hour. At 2 P.M. the ship changes course to N W, as shown in Figure 4.83. Find the ship’s bearing and distance from the port of departure at 3 P.M. 54 D b C FIGURE 4.83 20 nm d 54° B c Not drawn to scale E W N S 40 nm = 2(20 nm) A Solution For triangle can be determined to be b 20 sin 36 BCD, and d 20 cos 36. you have B 90 54 36. The two sides of this triangle For triangle ACD, tan A b you can find angle 20 sin 36 d 40 20 cos 36 40 A as follows. 0.2092494 A arctan 0.2092494 0.2062732 radian 11.82 90 11.82 78.18. you have Finally, from triangle The angle with the north-south line is the ship is yields N 78.18 W. ACD, So, the bearing of which sin A bc, c b sin A 20 sin 36 sin 11.82 57.4 nautical miles. Distance from port Now try Exercise 31. In air navigation, bearings are measured in degrees clockwise from north. Examples of air navigation bearings are shown below. 0° N 60° 270° W E 90° S 180° 0° N S 180° E 90° 225° 270° W 333202_0408.qxd 12/7/05 11:11 AM Page 356 356 Chapter 4 Trigonometry Harmonic Motion The periodic nature of the trigonometric functions is useful for describing the motion of a point on an object that vibrates, oscilla
tes, rotates, or is moved by wave motion. For example, consider a ball that is bobbing up and down on the end of a spring, as shown in Figure 4.84. Suppose that 10 centimeters is the maximum distance the ball moves vertically upward or downward from its equilibrium (at rest) position. Suppose further that the time it takes for the ball to move from its maximum displacement above zero to its maximum displacement below zero and back again is seconds. Assuming the ideal conditions of perfect elasticity and no friction or air resistance, the ball would continue to move up and down in a uniform and regular manner. t 4 10 cm 10 cm 10 cm 0 cm 0 cm 0 cm −10 cm −10 cm −10 cm Equilibrium FIGURE 4.84 Maximum negative displacement Maximum positive displacement From this spring you can conclude that the period (time for one complete cycle) of the motion is Period 4 seconds its amplitude (maximum displacement from equilibrium) is Amplitude 10 centimeters and its frequency (number of cycles per second) is Frequency 1 4 cycle per second. Motion of this nature can be described by a sine or cosine function, and is called simple harmonic motion. 333202_0408.qxd 12/7/05 11:11 AM Page 357 Section 4.8 Applications and Models 357 Definition of Simple Harmonic Motion A point that moves on a coordinate line is said to be in simple harmonic motion if its distance from the origin at time d a sin t d a cos t is given by either or d t a where and a, period 2, are real numbers such that and frequency 2. > 0. The motion has amplitude Example 6 Simple Harmonic Motion Write the equation for the simple harmonic motion of the ball described in Figure 4.84, where the period is 4 seconds. What is the frequency of this harmonic motion? Solution Because the spring is at equilibrium d a sin t. d 0 when t 0, you use the equation Moreover, because the maximum displacement from zero is 10 and the period is 4, you have Amplitude a 10 Period 2 4 . 2 Consequently, the equation of motion is d 10 sin t. 2 Note that the choice of initially moves up or down. The frequency is or a 10 a 10 depends on whether the ball Frequency 2 2 2 1 4 cycle per second. Now try Exercise 51. x One illustration of the relationship between sine waves and harmonic motion can be seen in the wave motion resulting when a stone is dropped into a calm pool of water. The waves move outward in roughly the shape of sine (or cosine) waves, as shown in Figure 4.85. As an example, suppose you are fishing and your fishing bob is attached so that it does not move horizontally. As the waves move outward from the dropped stone, your fishing bob will move up and down in simple harmonic motion, as shown in Figure 4.86. FIGURE 4.85 y FIGURE 4.86 333202_0408.qxd 12/7/05 11:11 AM Page 358 358 Chapter 4 Trigonometry Example 7 Simple Harmonic Motion Given the equation for simple harmonic motion d 6 cos 3 t 4 find (a) the maximum displacement, (b) the frequency, (c) the value of when t 4, t and (d) the least positive value of for which d 0. d Algebraic Solution The given equation has the form with 34. a 6 and d a cos t, Graphical Solution Use a graphing utility set in radian mode to graph a. The maximum displacement (from the point of equilibrium) is given by the amplitude. So, the maximum displacement is 6. y 6 cos 3 x. 4 a. Use the maximum feature of the graphing utility to estimate that y 0 the maximum displacement from the point of equilibrium is 6, as shown in Figure 4.87. b. c. 2 34 2 4 Frequency d 6 cos3 4 6 cos 3 61 6 3 8 cycle per unit of time y = 6 cos x3π ( ) 4 3 2 8 0 −8 FIGURE 4.87 d. To find the least positive value of for which t d 0, solve the equation 3 4 t 0. d 6 cos First divide each side by 6 to obtain cos 3 4 t 0. This equation is satisfied when Multiply these values by 43 to obtain t 2 3 , 2, 10 3 , . . . . 8 0 −8 So, the least positive value of t is t 2 3. FIGURE 4.88 Now try Exercise 55. b. The period is the time for the graph to complete one cycle, which You can estimate the frequency as follows. is x 2.667. Frequency 1 2.667 0.375 cycle per unit of time c. Use the trace feature to estimate that the value of when y is y 6, as shown in Figure 4.88. x 4 d. Use the zero or root feature to estimate that the least positive as shown in Figure is x 0.6667, for which y 0 x value of 4.89. y = 6 cos x3π ( ) 4 3 2 8 3 2 0 −8 FIGURE 4.89 333202_0408.qxd 12/7/05 11:11 AM Page 359 Section 4.8 Applications and Models 359 4.8 Exercises VOCABULARY CHECK: Fill in the blanks. 1. An angle that measures from the horizontal upward to an object is called the angle of ________, whereas an angle that measures from the horizontal downward to an object is called the angle of ________. 2. A ________ measures the acute angle a path or line of sight makes with a fixed north-south line. 3. A point that moves on a coordinate line is said to be in simple ________ ________ if its distance d a sin t from the origin at time is given by either d a cos t. or t d PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. In Exercises 1–10, solve the right triangle shown in the figure. Round your answers to two decimal places. 16. Length The sun is 20 above the horizon. Find the length of a shadow cast by a building that is 600 feet tall. b 10 b 24 b 10 c 52 1. 3. 5. 7. 9. 10. A 20, B 71, a 6, b 16, A 12 15, B 65 12, c 430.5 a 14.2 2. 4. 6. 8. B 54, A 8.4, a 25, b 1.32, c 15 a 40.5 c 35 c 9.45 17. Height A ladder 20 feet long leans against the side of a house. Find the height from the top of the ladder to the ground if the angle of elevation of the ladder is 80 . 18. Height The length of a shadow of a tree is 125 feet when Approximate the 33. the angle of elevation of the sun is height of the tree FIGURE FOR 1–10 FIGURE FOR 11–14 In Exercises 11–14, find the altitude of the isosceles triangle shown in the figure. Round your answers to two decimal places. 11. 12. 13. 14. 52, 18, 41, 27, b 4 b 10 b 46 b 11 inches meters inches feet 15. Length The sun is 25 above the horizon. Find the length of a shadow cast by a silo that is 50 feet tall (see figure). 19. Height From a point 50 feet in front of a church, the angles of elevation to the base of the steeple and the top of 40, the steeple are (a) Draw right triangles that give a visual representation of the problem. Label the known and unknown quantities. respectively. and 47 35 (b) Use a trigonometric function to write an equation involving the unknown quantity. (c) Find the height of the steeple. 20. Height You are standing 100 feet from the base of a platform from which people are bungee jumping. The angle of elevation from your position to the top of the platform from which they jump is From what height are the people jumping? 51. 21. Depth The sonar of a navy cruiser detects a submarine that is 4000 feet from the cruiser. The angle between the water line and the submarine is (see figure). How deep is the submarine? 34 34° 4000 ft 25° 50 ft Not drawn to scale 22. Angle of Elevation An engineer erects a 75-foot cellular telephone tower. Find the angle of elevation to the top of the tower at a point on level ground 50 feet from its base. 333202_0408.qxd 12/7/05 11:11 AM Page 360 360 Chapter 4 Trigonometry 121 2 23. Angle of Elevation The height of an outdoor basketball feet, and the backboard casts a shadow backboard is 171 feet long. 3 (a) Draw a right triangle that gives a visual representation of the problem. Label the known and unknown quantities. 30. Navigation A jet leaves Reno, Nevada and is headed The distance toward Miami, Florida at a bearing of between the two cities is approximately 2472 miles. 100. (a) How far north and how far west is Reno relative to Miami? (b) If the jet is to return directly to Reno from Miami, at (b) Use a trigonometric function to write an equation what bearing should it travel? involving the unknown quantity. (c) Find the angle of elevation of the sun. 24. Angle of Depression A Global Positioning System satellite orbits 12,500 miles above Earth’s surface (see figure). Find the angle of depression from the satellite to the horizon. Assume the radius of Earth is 4000 miles. 12,500 mi GPS satellite Angle of depression 4,000 mi Not drawn to scale 25. Angle of Depression A cellular telephone tower that is 150 feet tall is placed on top of a mountain that is 1200 feet above sea level. What is the angle of depression from the top of the tower to a cell phone user who is 5 horizontal miles away and 400 feet above sea level? 26. Airplane Ascent During takeoff, an airplane’s angle of ascent is 18 and its speed is 275 feet per second. (a) Find the plane’s altitude after 1 minute. (b) How long will it take the plane to climb to an altitude of 10,000 feet? 27. Mountain Descent A sign on a roadway at the top of a mountain indicates that for the next 4 miles the grade is 10.5 (see figure). Find the change in elevation over that distance for a car descending the mountain. Not drawn to scale 4 mi 10.5° 28. Mountain Descent A roadway sign at the top of a mountain indicates that for the next 4 miles the grade is 12%. Find the angle of the grade and the change in elevation over the 4 miles for a car descending the mountain. 29. Navigation An airplane flying at 600 miles per hour has a bearing of After flying for 1.5 hours, how far north and how far east will the plane have traveled from its point of departure? 52. 31. Navigation A ship leaves port at noon and has a bearing of S 29 W. The ship sails at 20 knots. (a) How many nautical miles south and how many nautical miles west will the ship have traveled by 6:00 P.M.? (b) At 6:00 P.M., the ship changes course to due west. Find the ship’s bearing and distance from the port of departure at 7:00 P.M. 32. Navigation A privately owned yacht leaves a dock in Myrtle Beach, South Carolina and heads toward Freeport The yacht averages in the Bahamas at a bearing of a speed of 20 knots over the 428 nautical-mile trip. S 1.4 E. (a) How long will it take the yacht to m
ake the trip? (b) How far east and south is the yacht after 12 hours? (c) If a plane leaves Myrtle Beach to fly to Freeport, what bearing should be taken? 33. Surveying A surveyor wants to find the distance across a 32 is N W. C to swamp (see figure). The bearing from A, The surveyor walks 50 meters from the bearing to C is N W. Find (a) the bearing from B. A and at the point A B and (b) the distance from 68 to to B A B N S E W C 50 m A 34. Location of a Fire Two fire towers are 30 kilometers is due west of tower A fire is apart, where tower spotted from the towers, and the bearings from and are E N, respectively (see figure). Find the distance of the fire from the line segment N and W B. A 34 14 AB. B A d W N S E A 14° d 34° B 30 km Not drawn to scale 333202_0408.qxd 12/7/05 11:11 AM Page 361 35. Navigation A ship is 45 miles east and 30 miles south of port. The captain wants to sail directly to port. What bearing should be taken? 36. Navigation An airplane is 160 miles north and 85 miles east of an airport. The pilot wants to fly directly to the airport. What bearing should be taken? 37. Distance An observer in a lighthouse 350 feet above sea level observes two ships directly offshore. The angles of (see figure). How far depression to the ships are apart are the ships? 6.5 and 4 Section 4.8 Applications and Models 361 41. 42. L1: L2: L1: L2: 3x 2y 5 x y 1 2x y x 5y 8 4 43. Geometry Determine the angle between the diagonal of a cube and the diagonal of its base, as shown in the figure. 6.5° 4° 350 ft θ a a θ a a a FIGURE FOR 43 FIGURE FOR 44 Not drawn to scale cube and its edge, as shown in the figure. 44. Geometry Determine the angle between the diagonal of a 38. Distance A passenger in an airplane at an altitude of 10 kilometers sees two towns directly to the east of the plane. (see The angles of depression to the towns are figure). How far apart are the towns? and 28 55 45. Geometry Find the length of the sides of a regular penta- gon inscribed in a circle of radius 25 inches. 46. Geometry Find the length of the sides of a regular hexa- gon inscribed in a circle of radius 25 inches. 47. Hardware Write the distance across the flat sides of a hexagonal nut as a function of as shown in the figure. y r, 55° 28° 10 km Not drawn to scale 39. Altitude A plane is observed approaching your home and you assume that its speed is 550 miles per hour. The angle 16 of elevation of the plane is one at one time and minute later. Approximate the altitude of the plane. 57 40. Height While traveling across flat land, you notice a mountain directly in front of you. The angle of elevation to After you drive 17 miles closer to the the peak is 9. mountain, the angle of elevation is Approximate the height of the mountain. 2.5. r 60° x y 48. Bolt Holes The figure shows a circular piece of sheet metal that has a diameter of 40 centimeters and contains 12 equally spaced bolt holes. Determine the straight-line distance between the centers of consecutive bolt holes. 30° Geometry between two nonvertical lines satisfies the equation In Exercises 41 and 42, find the angle L1 The angle and L2. tan m2 1 m2 m1 m1 40 cm 35 cm m1 where and (Assume that m2 m1m2 are the slopes of 1. ) L1 and L2, respectively. 333202_0408.qxd 12/7/05 11:11 AM Page 362 362 Chapter 4 Trigonometry Trusses unknown members of the truss. In Exercises 49 and 50, find the lengths of all the 49. 50. 35° 10 10 10 a b c 36 ft a 35° 10 6 ft 6 ft b 9 ft Harmonic Motion In Exercises 51–54, find a model for simple harmonic motion satisfying the specified conditions. Displacement t 0 51. 0 52. 0 53. 3 inches 54. 2 feet Amplitude Period 4 centimeters 3 meters 3 inches 2 feet 2 seconds 6 seconds 1.5 seconds 10 seconds Harmonic Motion In Exercises 55–58, for the simple harmonic motion described by the trigonometric function, find (a) the maximum displacement, (b) the frequency, t 5, and (d) the least positive (c) the value of when value of Use a graphing utility to verify your results. for which d 0. d t 55. 56. 57. 58. d 4 cos 8t d 1 2 cos 20t d 1 16 sin 120t d 1 64 sin 792t 59. Tuning Fork A point on the end of a tuning fork moves d a sin t. given that the tuning fork for middle C has a fre- in simple harmonic motion described by Find quency of 264 vibrations per second. 60. Wave Motion A buoy oscillates in simple harmonic motion as waves go past. It is noted that the buoy moves a total of 3.5 feet from its low point to its high point (see figure), and that it returns to its high point every 10 seconds. Write an equation that describes the motion of the buoy if its high point is at t 0. Equilibrium High point 3.5 ft Low point FIGURE FOR 60 61. Oscillation of a Spring A ball that is bobbing up and down on the end of a spring has a maximum displacement of 3 inches. Its motion (in ideal conditions) is modeled by y 1 y is the time in seconds. 4 cos 16t t > 0, is measured in feet and where t (a) Graph the function. (b) What is the period of the oscillations? (c) Determine the first time the weight passes the point of equilibrium y 0. Model It 62. Numerical and Graphical Analysis A two-meterhigh fence is 3 meters from the side of a grain storage bin. A grain elevator must reach from ground level outside the fence to the storage bin (see figure). The objective is to determine the shortest elevator that meets the constraints. L2 θ L1 2 m θ 3 m (a) Complete four rows of the table. 0.1 0.2 L1 2 sin 0.1 2 sin 0.2 L2 L1 L2 3 cos 0.1 3 cos 0.2 23.0 13.1 333202_0408.qxd 12/7/05 11:11 AM Page 363 Model It (co n t i n u e d ) (b) Use a graphing utility to generate additional rows of the table. Use the table to estimate the minimum length of the elevator. L1 (c) Write the length L2 (d) Use a graphing utility to graph the function. Use the graph to estimate the minimum length. How does your estimate compare with that of part (b)? as a function of . 63. Numerical and Graphical Analysis The cross section of an irrigation canal is an isosceles trapezoid of which three of the sides are 8 feet long (see figure). The objective is to find the angle that maximizes the area of the cross . section. Hint: The area of a trapezoid is h2b1 b2 8 ft θ 8 ft θ Section 4.8 Applications and Models 363 (a) Create a scatter plot of the data. (b) Find a trigonometric model that fits the data. Graph the model with your scatter plot. How well does the model fit the data? (c) What is the period of the model? Do you think it is reasonable given the context? Explain your reasoning. (d) Interpret the meaning of the model’s amplitude in the context of the problem. Synthesis True or False? the statement is true or false. Justify your answer. In Exercises 65 and 66, determine whether 65. The Leaning Tower of Pisa is not vertical, but if you know the exact angle of elevation to the 191-foot tower when you stand near it, then you can determine the exact distance to the tower by using the formula d tan 191 d . 66. For the harmonic motion of a ball bobbing up and down on the end of a spring, one period can be described as the length of one coil of the spring. 8 ft 67. Writing Is it true that N 24 E means 24 degrees north of east? Explain. 68. Writing Explain the difference between bearings used in nautical navigation and bearings used in air navigation. Skills Review In Exercises 69 –72, write the slope-intercept form of the equation of the line with the specified characteristics.Then sketch the line. 69. m 4, passes through m 1 2, 71. Passes through passes through 70. 1, 2 1 3, 0 3, 2 and and 1 2, 1 3 2, 6 1 4, 2 3 72. Passes through (a) Complete seven additional rows of the table. Base 1 Base 2 Altitude Area 8 8 8 16 cos 10 8 16 cos 20 8 sin 10 8 sin 20 22.1 42.5 (b) Use a graphing utility to generate additional rows of the table. Use the table to estimate the maximum crosssectional area. (c) Write the area A as a function of . (d) Use a graphing utility to graph the function. Use the graph to estimate the maximum cross-sectional area. How does your estimate compare with that of part (b)? 64. Data Analysis The table shows the average sales (in millions of dollars) of an outerwear manufacturer for each t, month where represents January. t 1 S Time, t 1 2 3 4 5 6 Sales, s 13.46 11.15 8.00 4.85 2.54 1.70 Time, t 7 8 9 10 11 12 Sales, s 2.54 4.85 8.00 11.15 13.46 14.3 333202_040R.qxd 12/7/05 11:13 AM Page 364 364 Chapter 4 Trigonometry 4 Chapter Summary What did you learn? Section 4.1 Describe angles (p. 282). Use radian measure (p. 283). Use degree measure (p. 285). Use angles to model and solve real-life problems (p. 287). Section 4.2 Identify a unit circle and describe its relationship to real numbers (p. 294). Evaluate trigonometric functions using the unit circle (p. 295). Use domain and period to evaluate sine and cosine functions (p. 297). Use a calculator to evaluate trigonometric functions (p. 298). Section 4.3 Evaluate trigonometric functions of acute angles (p. 301). Use the fundamental trigonometric identities (p. 304). Use a calculator to evaluate trigonometric functions (p. 305). Use trigonometric functions to model and solve real-life problems (p. 306). Section 4.4 Evaluate trigonometric functions of any angle (p. 312). Use reference angles to evaluate trigonometric functions (p. 314). Evaluate trigonometric functions of real numbers (p. 315). Section 4.5 Use amplitude and period to help sketch the graphs of sine and cosine functions (p. 323). Sketch translations of the graphs of sine and cosine functions (p. 325). Use sine and cosine functions to model real-life data (p. 327). Section 4.6 Sketch the graphs of tangent (p. 332) and cotangent (p. 334) functions. Sketch the graphs of secant and cosecant functions (p. 335). Sketch the graphs of damped trigonometric functions (p. 337). Review Exercises 1, 2 3–6, 11–18 7–18 19–24 25–28 29–32 33–36 37–40 41–44 45–48 49–54 55, 56 57–70 71–82 83–88 89–92 93–96 97, 98 99–102 103–106 107, 108 Section 4.7 Evaluate and graph the inverse sine function (p. 343). Evaluate and graph th
e other inverse trigonometric functions (p. 345). Evaluate compositions of trigonometric functions (p. 347). 109–114, 123, 126 115–122, 124, 125 127–132 Section 4.8 Solve real-life problems involving right triangles (p. 353). Solve real-life problems involving directional bearings (p. 355). Solve real-life problems involving harmonic motion (p. 356). 133, 134 135 136 333202_040R.qxd 12/7/05 11:13 AM Page 365 4 Review Exercises In Exercises 1 and 2, estimate the angle to the nearest 4.1 one-half radian. 1. 2. In Exercises 3 –10, (a) sketch the angle in standard position, (b) determine the quadrant in which the angle lies, and (c) determine one positive and one negative coterminal angle. 11 4 4 3 70 110 3. 5. 7. 9. 2 9 23 3 280 405 4. 6. 8. 10. Review Exercises 365 In Exercises 25–28, find the point 4.2 circle that corresponds to the real number x, y t. on the unit 25. 27. t 2 3 t 5 6 26. 28. t 3 4 t 4 3 In Exercises 29–32, evaluate (if possible) the six trigonometric functions of the real number. 29. 31. t 7 6 t 2 3 30. t 4 32. t 2 In Exercises 33–36, evaluate the trigonometric function using its period as an aid. In Exercises 11–14, convert the angle measure from degrees to radians. Round your answer to three decimal places. 11. 13. 480 33º 45 12. 14. 127.5 196 77 33. 35. sin 11 4 sin17 6 34. cos 4 36. cos13 3 In Exercises 15–18, convert the angle measure from radians to degrees. Round your answer to three decimal places. In Exercises 37–40, use a calculator to evaluate the trigonometric function. Round your answer to four decimal places. 5 7 3.5 15. 17. 16. 11 6 18. 5.7 37. tan 33 39. sec 12 5 38. 40. csc 10.5 sin 9 19. Arc Length Find the length of the arc on a circle with a 138. radius of 20 inches intercepted by a central angle of 20. Arc Length Find the length of the arc on a circle with a 60. radius of 11 meters intercepted by a central angle of 21. Phonograph Compact discs have all but replaced phonograph records. Phonograph records are vinyl discs that rotate on a turntable. A typical record album is 12 inches in 1 diameter and plays at 33 revolutions per minute. 3 (a) What is the angular speed of a record album? (b) What is the linear speed of the outer edge of a record album? 22. Bicycle At what speed is a bicyclist traveling when his 27-inch-diameter tires are rotating at an angular speed of 5 radians per second? 23. Circular Sector Find the area of the sector of a circle with a radius of 18 inches and central angle 120. 24. Circular Sector Find the area of the sector of a circle with a radius of 6.5 millimeters and central angle 56. In Exercises 41–44, find the exact values of the six 4.3 trigonometric functions of the angle shown in the figure. 41. 42. 4 θ 5 θ 6 43. 44. 6 8 θ 4 9 5 θ 333202_040R.qxd 12/7/05 11:13 AM Page 366 366 Chapter 4 Trigonometry In Exercises 45– 48, use the given function value and trigonometric identities (including the cofunction identities) to find the indicated trigonometric functions. 45. sin 1 3 46. tan 4 47. csc 4 48. csc 5 (a) csc (c) sec (a) cot (c) cos (a) sin (c) sec (a) sin (c) tan (b) cos (d) tan (b) sec (d) csc (b) cos (d) tan (b) cot (d) sec 90 In Exercises 65–70, find the values of the six trigonometric functions of . Function Value sec 6 5 csc 3 2 sin 3 8 tan 5 4 cos 2 5 sin 2 4 65. 66. 67. 68. 69. 70. Constraint tan < 0 cos < 0 cos < 0 cos < 0 sin > 0 cos > 0 In Exercises 71–74, find the reference angle in standard position. and , and sketch In Exercises 49– 54, use a calculator to evaluate the trigonometric function. Round your answer to four decimal places. 49. 50. 51. 52. 53. 54. tan 33 csc 11 sin 34.2 sec 79.3 cot 15 14 cos 78 11 58 55. Railroad Grade A train travels 3.5 kilometers on a (see figure). What is straight track with a grade of the vertical rise of the train in that distance? 1 10 3.5 km 1°10′ Not drawn to scale 56. Guy Wire A guy wire runs from the ground to the top of a 25-foot telephone pole. The angle formed between the wire and the ground is How far from the base of the pole is the wire attached to the ground? 52. 71. 73. 264 6 5 72. 74. 635 17 3 In Exercises 75– 82, evaluate the sine, cosine, and tangent of the angle without using a calculator. 3 7 3 495 240 75. 77. 79. 81. 4 5 4 150 315 76. 78. 80. 82. In Exercises 83– 88, use a calculator to evaluate the trigonometric function. Round your answer to four decimal places. 83. 85. 85. sin 4 sin3.2 12 5 sin 84. 86. 88. tan 3 cot4.8 tan25 7 In Exercises 89–96, sketch the graph of the function. 4.5 Include two full periods. In Exercises 57– 64, the point is on the terminal side in standard position. Determine the exact 4.4 of an angle values of the six trigonometric functions of the angle . 57. 58. 59. 60. 61. 62. 63. 64. 2 12, 16 3, 4 2 3, 5 10 3 , 2 0.5, 4.5 0.3, 0.4 x, 4x, x > 0 2x, 3x, 3 x > 0 89. y sin x 91. 93. 95. f x 5 sin 2x 5 y 2 sin x gt 5 2 sint 90. 92. 94. 96. y cos x f x 8 cos x 4 y 4 cos x gt 3 cost 97. Sound Waves Sound waves can be modeled by sine y a sin bx, is measured in where x functions of the form seconds. (a) Write an equation of a sound wave whose amplitude is 2 and whose period is second. 1 264 (b) What is the frequency of the sound wave described in part (a)? 333202_040R.qxd 12/7/05 11:13 AM Page 367 98. Data Analysis: Meteorology The times S of sunset (Greenwich Mean Time) at 40 north latitude on the 15th of each month are: 1(16:59), 2(17:35), 3(18:06), 4(18:38), 5(19:08), 6(19:30), 7(19:28), 8(18:57), 9(18:09), 10(17:21), 11(16:44), 12(16:36). The month is represented by with corresponding to January. A model (in which minutes have been converted to the decimal parts of an hour) for the data is St 18.09 1.41 sint 4.60. 6 t 1 t, (a) Use a graphing utility to graph the data points and the model in the same viewing window. (b) What is the period of the model? Is it what you expected? Explain. (c) What is the amplitude of the model? What does it represent in the model? Explain. In Exercises 99–106, sketch a graph of the function. 4.6 Include two full periods. 99. f x tan x 100. f t tant 4 Review Exercises 367 In Exercises 119–122, use a calculator to evaluate the expression. Round your answer to two decimal places. 119. 121. arccos 0.324 tan11.5 120. 122. arccos0.888 tan1 8.2 In Exercises 123–126, use a graphing utility to graph the function. 123. f x 2 arcsin x 124. f x 3 arccos x 125. f x arctan x 2 126. f x arcsin 2x In Exercises 127–130, find the exact value of the expression. 127. 128. 129. 130. 4 cosarctan 3 tanarccos 3 5 secarctan 12 5 cotarcsin12 13 In Exercises 131 and 132, write an algebraic expression that is equivalent to the expression. 131. tanarccos secarcsinx 1 x 2 132. 101. 102. 103. 104. 105. 106. f x cot x gt 2 cot 2t f x sec x ht sect f x csc x f t 3 csc2t 4 4 4.8 133. Angle of Elevation The height of a radio transmission tower is 70 meters, and it casts a shadow of length 30 meters (see figure). Find the angle of elevation of the sun. In Exercises 107 and 108, use a graphing utility to graph the function and the damping factor of the function in the same viewing window. Describe the behavior of increases without bound. the function as x 107. f x x cos x 108. gx x4 cos x In Exercises 109–114, evaluate the expression. If 4.7 necessary, round your answer to two decimal places. 109. 111. 113. arcsin1 2 arcsin 0.4 sin10.44 110. 112. 114. arcsin1 arcsin 0.213 sin1 0.89 In Exercises 115–118, evaluate the expression without the aid of a calculator. 115. arccos 3 2 117. cos11 116. arccos 118. cos1 2 2 3 2 70 m θ 30 m 134. Height Your football has landed at the edge of the roof of your school building. When you are 25 feet from the base of the building, the angle of elevation to your football is How high off the ground is your football? 21. 135. Distance From city at a bearing of 810 miles at a bearing of A to city 48. C A B, a plane flies 650 miles to city B From city the plane flies 115 . Find the distance from city to city C, and the bearing from city A to city C. 333202_040R.qxd 12/7/05 11:13 AM Page 368 368 Chapter 4 Trigonometry 136. Wave Motion Your fishing bobber oscillates in simple harmonic motion from the waves in the lake where you fish. Your bobber moves a total of 1.5 inches from its high point to its low point and returns to its high point every 3 seconds. Write an equation modeling the motion of your bobber if it is at its high point at time t 0. Synthesis True or False? the statement is true or false. Justify your answer. In Exercises 137–140, determine whether 137. The tangent function is often useful for modeling simple harmonic motion. 138. The inverse sine function y arcsin cannot be defined as a function over any interval that is greater than the 2 ≤ y ≤ 2. interval defined as y sin is not a function because sin 30 sin 150. 139. x 140. Because tan 34 1, arctan1 34. In Exercises 141–144, match the function with its graph. Base your selection solely on your interpretation a Explain your reasoning. [The of the constants graphs are labeled (a), (b), (c), and (d).] and y a sin bx b. (a) y (b) y x x π 2 −2 (c) y 3 2 1 −3 π 141. y 3 sin x 143. y 2 sin x 3 2 1 −d) 142. y 3 sin x 144. y 2 sin x 2 145. Writing Describe the behavior of g cos . zeros of Explain your reasoning. f sec at the (b) Make a conjecture about the relationship between tan 2 and cot . 147. Writing When graphing the sine and cosine functions, determining the amplitude is part of the analysis. Explain why this is not true for the other four trigonometric functions. 148. Oscillation of a Spring A weight is suspended from a ceiling by a steel spring. The weight is lifted (positive direction) from the equilibrium position and released. The resulting motion of the weight is modeled by y Aekt cos bt 1 5et10 cos 6t t y is the distance in feet from equilibrium and is where the time in seconds. The graph of the function is shown in the figure. For each of the following, describe the change in the system without graphing the resulting function. 1 3. 1 3. to is changed from 6 to 9. is
changed from is changed from 1 5 1 10 (b) (a) (c) to A b k y 0.2 0.1 −0.1 −0.2 t 5π 149. Graphical Reasoning The formulas for the area of a A 1 s r, and is the angle measured circular sector and arc length are r respectively. ( in radians.) is the radius and 2 r2 (a) For 0.8, r. write the area and arc length as functions of What is the domain of each function? Use a graphing utility to graph the functions. Use the graphs to determine which function changes more r rapidly as r 10 centimeters, write the area and arc length as functions of What is the domain of each function? Use a graphing utility to graph and identify the functions. increases. Explain. . (b) For 146. Conjecture (a) Use a graphing utility to complete the table. 0.1 0.4 0.7 1.0 1.3 tan 2 cot 150. Writing Describe a real-life application that can be represented by a simple harmonic motion model and is different from any that you’ve seen in this chapter. Explain which function you would use to model your application and why. Explain how you would determine the amplitude, period, and frequency of the model for your application. 333202_040R.qxd 12/7/05 11:13 AM Page 369 Chapter Test 369 4 Chapter Test Take this test as you would take a test in class. When you are finished, check your work against the answers given in the back of the book. 1. Consider an angle that measures 5 4 radians. (a) Sketch the angle in standard position. (b) Determine two coterminal angles (one positive and one negative). (c) Convert the angle to degree measure. x 2. A truck is moving at a rate of 90 kilometers per hour, and the diameter of its wheels is 1 meter. Find the angular speed of the wheels in radians per minute. 3. A water sprinkler sprays water on a lawn over a distance of 25 feet and rotates through an angle of 130. Find the area of the lawn watered by the sprinkler. 4. Find the exact values of the six trigonometric functions of the angle shown in the figure. (−2, 6) y θ FIGURE FOR 4 5. Given that tan 3 2, 6. Determine the reference angle standard position. 7. Determine the quadrant in which 8. Find two exact values of a calculator.) find the other five trigonometric functions of . of the angle 290 and sketch and in sec < 0 lies if 0 ≤ < 360 and tan > 0. cos 32. if in degrees (Do not use 9. Use a calculator to approximate two values of Round the results to two decimal places. csc 1.030. in radians 0 ≤ < 2 if In Exercises 10 and 11, find the remaining five trigonometric functions of satisfying the conditions. cos 3 5, sec 17 8 , tan < 0 sin > 0 11. 10. y 1 f − π −1 −2 In Exercises 12 and 13, sketch the graph of the function. (Include two full periods.) gx 2 sinx 12. 4 13. f 1 2 tan 2 x π π 2 In Exercises 14 and 15, use a graphing utility to graph the function. If the function is periodic, find its period. 14. y sin 2x 2 cos x 15. y 6e0.12t cos0.25t, 0 ≤ t ≤ 32 FIGURE FOR 16 and for the function c f x a sinbx c such that the graph of matches f 16. Find b,a, the figure. 17. Find the exact value of 18. Graph the function tanarccos 2 f x 2 arcsin 1 3 without the aid of a calculator. 2x. 19. A plane is 80 miles south and 95 miles east of Cleveland Hopkins International Airport. What bearing should be taken to fly directly to the airport? 20. Write the equation for the simple harmonic motion of a ball on a spring that starts at its lowest point of 6 inches below equilibrium, bounces to its maximum height of 6 inches above equilibrium, and returns to its lowest point in a total of 2 seconds. 333202_040R.qxd 12/7/05 11:13 AM Page 370 Proofs in Mathematics The Pythagorean Theorem The Pythagorean Theorem is one of the most famous theorems in mathematics. More than 100 different proofs now exist. James A. Garfield, the twentieth president of the United States, developed a proof of the Pythagorean Theorem in 1876. His proof, shown below, involved the fact that a trapezoid can be formed from two congruent right triangles and an isosceles right triangle. The Pythagorean Theorem In a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse, where and are the legs and c a b is the hypotenuse. a2 b2 c2 a c b O c b Q a P Area of NOQ Proof N a M c b Area of MNOP trapezoid a ba b 1 2 Area of MNQ ab 1 2 1 2 Area of PQO ab 1 2 c2 1 2 a ba b ab 1 2 c2 a ba b 2ab c2 a2 2ab b2 2ab c2 a2 b2 c2 370 333202_040R.qxd 12/7/05 11:13 AM Page 371 P.S. Problem Solving This collection of thought-provoking and challenging exercises further explores and expands upon concepts learned in this chapter. 1. The restaurant at the top of the Space Needle in Seattle, Washington is circular and has a radius of 47.25 feet. The dining part of the restaurant revolves, making about one complete revolution every 48 minutes. A dinner party was seated at the edge of the revolving restaurant at 6:45 P.M. and was finished at 8:57 P.M. (a) Find the angle through which the dinner party rotated. (b) Find the distance the party traveled during dinner. 2. A bicycle’s gear ratio is the number of times the freewheel turns for every one turn of the chainwheel (see figure). The table shows the numbers of teeth in the freewheel and chainwheel for the first five gears of an 18-speed touring bicycle. The chainwheel completes one rotation for each gear. Find the angle through which the freewheel turns for each gear. Give your answers in both degrees and radians. Gear number Number of teeth Number of teeth in freewheel in chainwheel 1 2 3 4 5 Freewheel 32 26 22 32 19 24 24 24 40 24 (a) What is the shortest distance d the helicopter would have to travel to land on the island? (b) What is the horizontal distance that the helicopter would have to travel before it would be directly over the nearer end of the island? x (c) Find the width of the island. Explain how you obtained w your answer. 4. Use the figure below. F D E G B C A (a) Explain why triangles. ABC, ADE, and AFG are similar (b) What does similarity imply about the ratios BC AB , DE AD , and FG AF ? A (c) Does the value of sin depend on which triangle from A part (a) is used to calculate it? Would the value of sin change if it were found using a different right triangle that was similar to the three given triangles? (d) Do your conclusions from part (c) apply to the other five trigonometric functions? Explain. 5. Use a graphing utility to graph h, and use the graph to decide Chainwheel (a) (b) 6. If f whether is even, odd, or neither. h hx cos2 x hx sin2 x is an even function and g is an odd function, use the 3. A surveyor in a helicopter is trying to determine the width of an island, as shown in the figure. 3000 ft 27° 39° d x w Not drawn to scale results of Exercise 5 to make a conjecture about where h, (a) hx f x2 hx gx2. 7. The model for the height (b) h (in feet) of a Ferris wheel car is h 50 50 sin 8t t where is the time (in minutes). (The Ferris wheel has a radius of 50 feet.) This model yields a height of 50 feet when t 0. Alter the model so that the height of the car is 1 foot when t 0. 371 333202_040R.qxd 12/7/05 11:13 AM Page 372 8. The pressure P (in millimeters of mercury) against the walls of the blood vessels of a patient is modeled by t P 100 20 cos8 3 where t is time (in seconds). (a) Use a graphing utility to graph the model. (b) What is the period of the model? What does the period tell you about this situation? (c) What is the amplitude of the model? What does it tell you about this situation? (d) If one cycle of this model is equivalent to one heart- beat, what is the pulse of this patient? (e) If a physician wants this patient’s pulse rate to be 64 beats per minute or less, what should the period be? What should the coefficient of be? t 9. A popular theory that attempts to explain the ups and downs of everyday life states that each of us has three cycles, called biorhythms, which begin at birth. These three cycles can be modeled by sine waves. Physical (23 days): P sin Emotional (28 days): E sin 2t 23 , 2t 28 , Intellectual (33 days): I sin 2t 33 , where person who was born on July 20, 1986. is the number of days since birth. Consider a (a) Use a graphing utility to graph the three models in the same viewing window for 7300 ≤ t ≤ 7380. (b) Describe the person’s biorhythms during the month of September 2006. (c) Calculate the person’s three energy levels on September 22, 2006. 10. (a) Use a graphing utility to graph the functions given by f x 2 cos 2x 3 sin 3x and gx 2 cos 2x 3 sin 4x. (b) Use the graphs from part (a) to find the period of each function. (c) If and are positive integers, is the function given by hx A cos x B sin x periodic? Explain your reasoning. 372 11. Two trigonometric functions and have periods of 2, and f x 5.35. (a) Give one smaller and one larger positive value of at their graphs intersect at g x which the functions have the same value. (b) Determine one negative value of at which the graphs x intersect. (c) Is it true that reasoning. f 13.35 g4.65? Explain your 12. The function f t c f t. (a) f is periodic, with period Are the following equal? Explain. 2t 2c f 1 f t 1 (b) c. f t 2c f t f 1 2t t c f 1 2 (c) So, 13. If you stand in shallow water and look at an object below the surface of the water, the object will look farther away from you than it really is. This is because when light rays pass between air and water, the water refracts, or bends, the light rays. The index of refraction for water is 1.333. This (see figure). is the ratio of the sine of and the sine of 2 1 θ 2 θ 1 2 ft x d y (a) You are standing in water that is 2 feet deep and are (measured from a 2. looking at a rock at angle line perpendicular to the surface of the water). Find 60 1 (b) Find the distances and x y. (c) Find the distance d between where the rock is and where it appears to be. (d) What happens to d as you move closer to the rock? Explain your reasoning. 14. In calculus, it can be shown that the arctangent function can be approximated by the polynomial arctan x x x3 3 x7 7 x5 5 where x is
in radians. (a) Use a graphing utility to graph the arctangent function and its polynomial approximation in the same viewing window. How do the graphs compare? (b) Study the pattern in the polynomial approximation of the arctangent function and guess the next term. Then repeat part (a). How does the accuracy of the approximation change when additional terms are added? 333202_0500.qxd 12/5/05 8:57 AM Page 373 Analytic Trigonometry 5.1 5.2 5.3 5.4 Using Fundamental Identities Verifying Trigonometric Identities Solving Trigonometric Equations Sum and Difference Formulas 5.5 Multiple-Angle and Product-to-Sum Formula 55 Concepts of trigonometry can be used to model the height above ground of a seat on a Ferris wheel AT I O N S Trigonometric equations and identities have many real-life applications. The applications listed below represent a small sample of the applications in this chapter. • Friction, • Data Analysis: Unemployment Rate, • Projectile Motion, Exercise 99, page 381 Exercise 76, page 398 Exercise 101, page 421 • Shadow Length, Exercise 56, page 388 • Ferris Wheel, Exercise 75, page 398 • Harmonic Motion, Exercise 75, page 405 • Mach Number, Exercise 121, page 417 • Ocean Depth, Exercise 10, page 428 373 333202_0501.qxd 12/5/05 9:15 AM Page 374 374 Chapter 5 Analytic Trigonometry 5.1 Using Fundamental Identities What you should learn • Recognize and write the fundamental trigonometric identities. • Use the fundamental trigonometric identities to evaluate trigonometric functions, simplify trigonometric expressions, and rewrite trigonometric expressions. Why you should learn it Fundamental trigonometric identities can be used to simplify trigonometric expressions. For instance, in Exercise 99 on page 381, you can use trigonometric identities to simplify an expression for the coefficient of friction. Introduction In Chapter 4, you studied the basic definitions, properties, graphs, and applications of the individual trigonometric functions. In this chapter, you will learn how to use the fundamental identities to do the following. 1. Evaluate trigonometric functions. 2. Simplify trigonometric expressions. 3. Develop additional trigonometric identities. 4. Solve trigonometric equations. Fundamental Trigonometric Identities Reciprocal Identities sin u 1 csc u csc u 1 sin u cos u 1 sec u sec u 1 cos u tan u 1 cot u cot u 1 tan u Quotient Identities tan u sin u cos u Pythagorean Identities cot u cos u sin u sin2 u cos2 u 1 1 tan2 u sec2 u 1 cot 2 u csc2 u Cofunction Identities sin 2 tan 2 sec 2 u cos u u cot u u csc u Even/Odd Identities cos 2 cot 2 csc 2 u sin u u tan u u sec u sinu sin u cscu csc u cosu cos u secu sec u tanu tan u cotu cot u Pythagorean identities are sometimes used in radical form such as sin u ± 1 cos2 u The HM mathSpace® CD-ROM and Eduspace® for this text contain additional resources related to the concepts discussed in this chapter. or tan u ± sec 2 u 1 where the sign depends on the choice of u. 333202_0501.qxd 12/5/05 9:15 AM Page 375 Section 5.1 Using Fundamental Identities 375 Using the Fundamental Identities One common use of trigonometric identities is to use given values of trigonometric functions to evaluate other trigonometric functions. Example 1 Using Identities to Evaluate a Function Use the values trigonometric functions. sec u 3 2 and tan u > 0 to find the values of all six You should learn the fundamental trigonometric identities well, because they are used frequently in trigonometry and they will also appear later in calculus. Note that u can be an angle, a real number, or a variable. Solution Using a reciprocal identity, you have 2 3 cos u 1 sec u 1 32 . Using a Pythagorean identity, you have sin2 u 1 cos . Pythagorean identity Substitute 2 3 for cos u. Simplify. sec u < 0 Because Moreover, because sin u 53. the negative root and obtain sine and cosine, you can find the values of all six trigonometric functions. lies in Quadrant III. is in Quadrant III, you can choose Now, knowing the values of the tan u > 0, is negative when it follows that and sin u u u sin u 5 3 cos u 2 3 tan u sin u cos u csc u 1 sin u sec u 1 cos u 3 5 35 5 3 2 53 23 5 2 cot u 1 tan u 2 5 25 5 Now try Exercise 11. Te c h n o l o g y You can use a graphing utility to check the result of Example 2. To do this, graph sin x cos 2 x sin x y1 and sin3 x y2 in the same viewing window, as shown below. Because Example 2 shows the equivalence algebraically and the two graphs appear to coincide, you can conclude that the expressions are equivalent. −π 2 −2 Example 2 Simplifying a Trigonometric Expression π Simplify sin x cos 2 x sin x. Solution First factor out a common monomial factor and then use a fundamental identity. sin x cos 2 x sin x sin xcos2 x 1 Factor out common monomial factor. sin x1 cos 2 x sin xsin2 x sin3 x Factor out 1. Pythagorean identity Multiply. Now try Exercise 45. 333202_0501.qxd 12/5/05 9:15 AM Page 376 376 Chapter 5 Analytic Trigonometry When factoring trigonometric expressions, it is helpful to find a special polynomial factoring form that fits the expression, as shown in Example 3. Example 3 Factoring Trigonometric Expressions Factor each expression. a. sec2 1 b. 4 tan2 tan 3 Solution a. Here you have the difference of two squares, which factors as sec2 1 sec 1sec 1). b. This expression has the polynomial form ax 2 bx c, and it factors as 4 tan2 tan 3 4 tan 3tan 1. Now try Exercise 47. On occasion, factoring or simplifying can best be done by first rewriting the expression in terms of just one trigonometric function or in terms of sine and cosine only. These strategies are illustrated in Examples 4 and 5, respectively. Example 4 Factoring a Trigonometric Expression Factor csc2 x cot x 3. Solution Use the identity cotangent. csc2 x 1 cot 2 x to rewrite the expression in terms of the csc2 x cot x 3 1 cot 2 x cot x 3 cot 2 x cot x 2 cot x 2cot x 1 Now try Exercise 51. Pythagorean identity Combine like terms. Factor. Example 5 Simplifying a Trigonometric Expression Simplify sin t cot t cos t. Remember that when adding rational expressions, you must first find the least common denominator (LCD). In Example 5, the LCD is sin t. Solution Begin by rewriting cot in terms of sine and cosine. t sin t cot t cos t sin t cos t sin t sin2 t cos 2 t sin t cos t 1 sin t csc t Now try Exercise 57. Quotient identity Add fractions. Pythagorean identity Reciprocal identity 333202_0501.qxd 12/5/05 9:15 AM Page 377 Section 5.1 Using Fundamental Identities 377 Example 6 Adding Trigonometric Expressions Perform the addition and simplify. sin 1 cos cos sin Solution sin 1 cos cos sin sin sin (cos 1 cos 1 cos sin sin2 cos2 cos 1 cos sin 1 cos 1 cos sin Multiply. Pythagorean identity: sin2 cos2 1 1 sin csc Now try Exercise 61. Divide out common factor. Reciprocal identity The last two examples in this section involve techniques for rewriting expres- sions in forms that are used in calculus. Example 7 Rewriting a Trigonometric Expression Rewrite 1 1 sin x so that it is not in fractional form. Solution cos 2 x 1 sin2 x 1 sin x1 sin x, From the Pythagorean identity you can see that multiplying both the numerator and the denominator by 1 sin x will produce a monomial denominator. 1 1 sin x 1 sin x 1 sin x 1 1 sin x 1 sin x 1 sin2 x 1 sin x cos 2 x 1 cos2 x sin x cos2 x 1 cos 2 x sin x cos x 1 cos x sec2 x tan x sec x Now try Exercise 65. Multiply numerator and denominator by 1 sin x. Multiply. Pythagorean identity Write as separate fractions. Product of fractions Reciprocal and quotient identities 333202_0501.qxd 12/5/05 9:15 AM Page 378 378 Chapter 5 Analytic Trigonometry Example 8 Trigonometric Substitution Use the substitution 4 x 2 x 2 tan , 0 < < 2, to write as a trigonometric function of . Solution Begin by letting x 2 tan . Then, you can obtain 4 x 2 4 2 tan 2 4 4 tan2 41 tan2 4 sec2 2 sec . Now try Exercise 77. Substitute 2 tan for x. Rule of exponents Factor. Pythagorean identity sec > 0 for 0 < < 2 4 + x2 x θ = arctan x 2 2 Angle whose tangent is x 2 . FIGURE 5.1 Figure 5.1 shows the right triangle illustration of the trigonometric substitux 2 tan tion in Example 8. You can use this triangle to check the solution of 0 < < 2, Example 8. For adj 2, opp x, hyp 4 x 2. you have and With these expressions, you can write the following. sec hyp adj 4 x 2 2 sec 2 sec 4 x 2 So, the solution checks. Example 9 Rewriting a Logarithmic Expression Rewrite lncsc lntan as a single logarithm and simplify the result. Solution lncsc lntan ln csc tan sin cos ln 1 ln 1 cos sin lnsec Now try Exercise 91. Product Property of Logarithms Reciprocal and quotient identities Simplify. Reciprocal identity 333202_0501.qxd 12/5/05 9:15 AM Page 379 5.1 Exercises The HM mathSpace® CD-ROM and Eduspace® for this text contain step-by-step solutions to all odd-numbered exercises. They also provide Tutorial Exercises for additional help. Section 5.1 Using Fundamental Identities 379 VOCABULARY CHECK: Fill in the blank to complete the trigonometric identity. sin u cos u ________ ________ 1 tan u 1 ________ csc2 u u ________ sin 2 cosu ________ 1. 3. 5. 7. 9. 1 sec u ________ ________ 1 sin u 1 tan2 u ________ u ________ sec 2 tanu ________ 2. 4. 6. 8. 10. PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. In Exercises 1–14, use the given values to evaluate (if possible) all six trigonometric functions. 19. sinx cosx 20. sin2 x cos2 x 1. sin x 2. tan x 3 2 3 3 , , 3. sec 2, cos x cos x 1 2 3 2 2 2 sin 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. csc 5 3, tan x 5 12, cot 3, sin tan 3 4 sec x 13 12 10 10 csc 35 5 cos x 4 5 2 4 tan x , , , sec 3 2 x 3 5 cos 2 sinx 1 3 sec x 4, tan 2, csc 5, sin 1, tan sin x > 0 sin < 0 cos < 0 cot 0 is undefined, sin > 0 In Exercises 15–20, match the trigonometric expression with one of the following. (a) sec x (d) 1 (b) (e) 1 tan x (c) cot x (f) sin x 15. 17. sec x cos x cot2 x csc 2 x 16. 18. tan x csc x 1 cos2 xcsc x In Exercises 21–26, match the trigonometric expre
ssion with one of the following. (b) tan x sec2 x (a) csc x (d) sin x x tan (e) 21. 23. 25. sin x sec x sec4 x tan4 x sec2 x 1 sin2 x (c) (f) sin2 x sec2 x tan2 x 22. 24. 26. cos2 xsec2 x 1 cot x sec x cos22 x cos x In Exercises 27–44, use the fundamental identities to simplify the expression. There is more than one correct form of each answer. 27. 29. 31. 33. 35. cot sec sin csc sin cot x csc x 1 sin2 x csc2 x 1 sec sin tan xsec x 39. 37. cos 2 cos2 y 1 sin y sin tan cos cot u sin u tan u cos u 43. 44. sin sec cos csc 41. 28. 30. 32. 34. 36. 38. cos tan sec2 x1 sin2 x csc sec 1 tan2 x 1 tan2 sec2 cot 2 xcos x 40. cos t1 tan2 t 42. csc tan sec 333202_0501.qxd 12/5/05 9:15 AM Page 380 380 Chapter 5 Analytic Trigonometry In Exercises 45–56, factor the expression and use the fundamental identities to simplify. There is more than one correct form of each answer. 71. 72. y1 y1 cos x 1 sin x , y2 sec4 x sec2 x, 1 sin x cos x tan2 x tan4 x y2 45. 47. 49. 51. 53. 55. 56. tan2 x tan2 x sin2 x sin2 x sec2 x sin2 x sec2 x 1 sec x 1 tan4 x 2 tan2 x 1 sin4 x cos4 x csc3 x csc2 x csc x 1 sec3 x sec2 x sec x 1 46. 48. 50. 52. 54. sin2 x csc2 x sin2 x cos2 x cos2 x tan2 x cos2 x 4 cos x 2 1 2 cos2 x cos4 x sec4 x tan4 x In Exercises 57– 60, perform the multiplication and use the fundamental identities to simplify. There is more than one correct form of each answer. 57. 58. 59. 60. sin x cos x2 cot x csc xcot x csc x 2 csc x 22 csc x 2 3 3 sin x3 3 sin x In Exercises 61–64, perform the addition or subtraction and use the fundamental identities to simplify. There is more than one correct form of each answer. 61. 63. 1 1 cos x cos x 1 sin x 1 1 cos x 1 sin x cos x 62. 64. 1 sec x 1 1 sec x 1 tan x sec2 x tan x In Exercises 65– 68, rewrite the expression so that it is not in fractional form. There is more than one correct form of each answer. 65. 67. sin2 y 1 cos y 3 sec x tan x 66. 68. 5 tan x sec x tan2 x csc x 1 Numerical and Graphical Analysis In Exercises 69 –72, use a graphing utility to complete the table and graph the functions. Make a conjecture about and y1 y2. In Exercises 73–76, use a graphing utility to determine which of the six trigonometric functions is equal to the expression. Verify your answer algebraically. 73. 74. 75. 76. cos x cot x sin x sec x csc x tan x cos x 1 1 cos x sin x 1 sin cos 1 sin cos 1 2 In Exercises 77– 82, use the trigonometric substitution to write the algebraic expression as a trigonometric function of where , 0 < < /2. x 3 cos 77. 78. 79. 80. 81. 82. 9 x 2, 64 16x 2, x 2 9, x 2 4, x 2 25, x 2 100, x 2 cos x 3 sec x 2 sec x 5 tan x 10 tan In Exercises 83– 86, use the trigonometric substitution to write the algebraic equation as a trigonometric function of where Then find sin and cos . , /2 < < /2. x 3 sin x 6 sin 3 9 x2, 3 36 x2, 22 16 4x2, 53 100 x2, 83. 84. 85. 86. x 2 cos x 10 cos In Exercises 87–90, use a graphing utility to solve the 0 ≤ < 2. equation for where , 87. 88. 89. 90. sin 1 cos2 cos 1 sin2 sec 1 tan2 csc 1 cot2 0.2 0.4 0.6 0.8 1.0 1.2 1.4 In Exercises 91–94, rewrite the expression as a single logarithm and simplify the result. x y1 y2 cos x, 2 sec x cos x, 69. 70. y1 y1 sin x y2 y2 sin x tan x 91. 92. lncos x lnsin x lnsec x lnsin x lncot t ln1 tan2 t 93. 94. lncos2 t ln1 tan2 t 333202_0501.qxd 12/5/05 9:15 AM Page 381 In Exercises 95–98, use a calculator to demonstrate the identity for each value of . 95. csc2 cot2 1 (a) 132, (b) 2 7 96. 97. 98. tan2 1 sec2 346, (a) (b) cos sin 2 80, (a) (b) sin sin 250, (a) (b) 3.1 0.8 1 2 99. Friction The forces acting on an object weighing W units on an inclined plane positioned at an angle of with the horizontal (see figure) are modeled by W cos W sin where for and simplify the result. is the coefficient of friction. Solve the equation W θ 100. Rate of Change The rate of change of the function f x csc x sin x is given by the expression csc x cot x cos x. Show that this expression can also be written as cos x cot2 x. Synthesis True or False? In Exercises 101 and 102, determine whether the statement is true or false. Justify your answer. 101. The even and odd trigonometric identities are helpful for determining whether the value of a trigonometric function is positive or negative. 102. A cofunction identity can be used to transform a tangent function so that it can be represented by a cosecant function. Section 5.1 Using Fundamental Identities 381 In Exercises 103–106, fill in the blanks. (Note: The notation x → c from the right and x → c indicates that indicates that approaches x approaches x c 103. As 104. As 105. As 106. As 2 , sin x → x → x → 0, cos x → , tan x → x → x → , sin x → 2 from the left.) c csc x →. sec x →. cot x →. csc x →. and and and and In Exercises 107–112, determine whether or not the equation is an identity, and give a reason for your answer. 108. cot csc2 1 is a constant. 107. 109. 110. 111. k tan , cos 1 sin2 sin k cos k 1 5 cos sin csc 1 5 sec 112. csc2 1 113. Use the definitions of sine and cosine to derive the Pythagorean identity sin2 cos2 1. 114. Writing Use the Pythagorean identity sin2 cos2 1 identities, to derive the 1 tan2 sec2 Discuss how to remember these identities and other fundamental identities. Pythagorean 1 cot2 csc2 . other and Skills Review In Exercises 115 and 116, perform the operation and simplify. 115. x 5x 5 116. 2z 32 In Exercises 117–120, perform the addition or subtraction and simplify. 117. 119. 1 x 5 2x x2 4 x x 8 7 x 4 118. 120. 6x x 4 x x2 25 3 4 x x2 x 5 In Exercises 121–124, sketch the graph of the function. (Include two full periods.) 121. 123. f x 1 2 f x 1 2 sin x 122. f x 2 tan x 2 secx 4 124. f x 3 2 cosx 3 333202_0502.qxd 12/5/05 9:01 AM Page 382 382 Chapter 5 Analytic Trigonometry 5.2 Verifying Trigonometric Identities What you should learn • Verify trigonometric identities. Why you should learn it You can use trigonometric identities to rewrite trigonometric equations that model real-life situations. For instance, in Exercise 56 on page 388, you can use trigonometric identities to simplify the equation that models the length of a shadow cast by a gnomon (a device used to tell time). Introduction In this section, you will study techniques for verifying trigonometric identities. In the next section, you will study techniques for solving trigonometric equations. The key to verifying identities and solving equations is the ability to use the fundamental identities and the rules of algebra to rewrite trigonometric expressions. Remember that a conditional equation is an equation that is true for only some of the values in its domain. For example, the conditional equation sin x 0 Conditional equation x n, is true only for are solving the equation. where n is an integer. When you find these values, you On the other hand, an equation that is true for all real values in the domain of the variable is an identity. For example, the familiar equation sin2 x 1 cos2 x Identity is true for all real numbers So, it is an identity. x. Verifying Trigonometric Identities Although there are similarities, verifying that a trigonometric equation is an identity is quite different from solving an equation. There is no well-defined set of rules to follow in verifying trigonometric identities, and the process is best learned by practice. Guidelines for Verifying Trigonometric Identities 1. Work with one side of the equation at a time. It is often better to work with the more complicated side first. Robert Ginn/PhotoEdit 2. Look for opportunities to factor an expression, add fractions, square a binomial, or create a monomial denominator. 3. Look for opportunities to use the fundamental identities. Note which functions are in the final expression you want. Sines and cosines pair up well, as do secants and tangents, and cosecants and cotangents. 4. If the preceding guidelines do not help, try converting all terms to sines and cosines. 5. Always try something. Even paths that lead to dead ends provide insights. Verifying trigonometric identities is a useful process if you need to convert a trigonometric expression into a form that is more useful algebraically. When you verify an identity, you cannot assume that the two sides of the equation are equal because you are trying to verify that they are equal. As a result, when verifying identities, you cannot use operations such as adding the same quantity to each side of the equation or cross multiplication. 333202_0502.qxd 12/5/05 9:01 AM Page 383 Section 5.2 Verifying Trigonometric Identities 383 Example 1 Verifying a Trigonometric Identity Verify the identity sec2 1 sec2 sin2 . Remember that an identity is only true for all real values in the domain of the variable. For instance, in Example 1 the identity is not true when 2 is because not defined when 2. sec2 Solution Because the left side is more complicated, start with it. tan2 1 1 sec2 sec2 1 sec2 Pythagorean identity tan2 sec2 tan2 cos 2 sin2 cos2 cos2 sin2 Simplify. Reciprocal identity Quotient identity Simplify. Notice how the identity is verified. You start with the left side of the equation (the more complicated side) and use the fundamental trigonometric identities to simplify it until you obtain the right side. Now try Exercise 5. There is more than one way to verify an identity. Here is another way to verify the identity in Example 1. sec2 1 sec2 sec2 1 sec2 sec2 1 cos2 sin2 Rewrite as the difference of fractions. Reciprocal identity Pythagorean identity Example 2 Combining Fractions Before Using Identities Verify the identity 1 1 sin 1 1 sin 2 sec2 . Solution 1 1 sin 1 1 sin 1 sin 1 sin 1 sin 1 sin 2 1 sin2 2 cos2 2 sec2 Now try Exercise 19. Add fractions. Simplify. Pythagorean identity Reciprocal identity 333202_0502.qxd 12/5/05 9:01 AM Page 384 384 Chapter 5 Analytic Trigonometry Example 3 Verifying Trigonometric Identity Verify the identity tan2 x 1cos2 x 1 tan2 x. Algebraic Solution By applying identities before multiplying, you obtain the following. tan2 x 1cos2 x 1 sec2 xsin2 x Pythagorean
identities sin2 x cos 2 x sin x cos x tan2 x 2 Reciprocal identity Rule of exponents Quotient identity Numerical Solution Use the table feature of a graphing utility set in radian mode to create a table that shows the tan2 x 1cos2 x 1 y1 and values of tan2 x y2 for different values of as shown in Figure 5.2. From the table you can see that the y2 values of appear to be identical, so tan2 x 1cos2 x 1 tan2 x appears to be an identity. and y1 x, Now try Exercise 39. FIGURE 5.2 Example 4 Converting to Sines and Cosines Verify the identity tan x cot x sec x csc x. Solution Try converting the left side into sines and cosines. Although a graphing utility can be useful in helping to verify an identity, you must use algebraic techniques to produce a valid proof. tan x cot x sin x cos x cos x sin x sin2 x cos 2 x cos x sin x Quotient identities Add fractions. Pythagorean identity 1 cos x sin x 1 cos x 1 sin x sec x csc x Reciprocal identities As shown at the right, csc2 x1 cos x is considered a 11 cos x simplified form of because the expression does not contain any fractions. Now try Exercise 29. Recall from algebra that rationalizing the denominator using conjugates is, on occasion, a powerful simplification technique. A related form of this technique, shown below, works for simplifying trigonometric expressions as well. 1 cos x 1 cos x 1 cos x sin2 x 1 cos x 1 cos2 x 1 1 cos x 1 1 cos x csc2 x1 cos x This technique is demonstrated in the next example. 333202_0502.qxd 12/5/05 9:01 AM Page 385 Section 5.2 Verifying Trigonometric Identities 385 Example 5 Verifying Trigonometric Identities Verify the identity sec y tan y cos y 1 sin y . Solution Begin with the right side, because you can create a monomial denominator by multiplying the numerator and denominator by 1 sin y. cos y 1 sin y 1 sin y cos y 1 sin y 1 sin y cos y cos y sin y 1 sin2 y cos y cos y sin y cos 2 y cos y sin y cos2 y cos y cos2 y 1 cos y sin y cos y sec y tan y Now try Exercise 33. Multiply numerator and denominator by 1 sin y. Multiply. Pythagorean identity Write as separate fractions. Simplify. Identities In Examples 1 through 5, you have been verifying trigonometric identities by working with one side of the equation and converting to the form given on the other side. On occasion, it is practical to work with each side separately, to obtain one common form equivalent to both sides. This is illustrated in Example 6. Example 6 Working with Each Side Separately Verify the identity cot 2 1 csc 1 sin sin . Solution Working with the left side, you have cot 2 1 csc csc2 1 1 csc csc 1csc 1 1 csc csc 1. Now, simplifying the right side, you have 1 sin sin 1 sin sin sin csc 1. Pythagorean identity Factor. Simplify. Write as separate fractions. Reciprocal identity The identity is verified because both sides are equal to csc 1. Now try Exercise 47. 333202_0502.qxd 12/5/05 9:01 AM Page 386 386 Chapter 5 Analytic Trigonometry In Example 7, powers of trigonometric functions are rewritten as more complicated sums of products of trigonometric functions. This is a common procedure used in calculus. Example 7 Three Examples from Calculus Verify each identity. a. b. c. tan4 x tan2 x sec2 x tan2 x sin3 x cos4 x cos4 x cos 6 x sin x csc4 x cot x csc2 xcot x cot3 x Solution a. tan4 x tan2 xtan2 x tan2 xsec2 x 1 tan2 x sec2 x tan2 x sin3 x cos4 x sin2 x cos4 x sin x b. 1 cos2 xcos4 x sin x cos4 x cos6 x sin x c. csc4 x cot x csc2 x csc2 x cot x csc2 x1 cot2 x cot x csc2 xcot x cot3 x Now try Exercise 49. Write as separate factors. Pythagorean identity Multiply. Write as separate factors. Pythagorean identity Multiply. Write as separate factors. Pythagorean identity Multiply. W RITING ABOUT MATHEMATICS Error Analysis You are tutoring a student in trigonometry. One of the homework problems your student encounters asks whether the following statement is an identity. tan2 x sin2 x ? 5 6 tan2 x Your student does not attempt to verify the equivalence algebraically, but mistakenly uses only a graphical approach. Using range settings of Xmin 3 Xmax 3 Xscl 2 Ymin 20 Ymax 20 Yscl 1 your student graphs both sides of the expression on a graphing utility and concludes that the statement is an identity. What is wrong with your student’s reasoning? Explain. Discuss the limitations of verifying identities graphically. 333202_0502.qxd 12/5/05 9:01 AM Page 387 Section 5.2 Verifying Trigonometric Identities 387 5.2 Exercises VOCABULARY CHECK: In Exercises 1 and 2, fill in the blanks. 1. An equation that is true for all real values in its domain is called an ________. 2. An equation that is true for only some values in its domain is called a ________ ________. In Exercises 3–8, fill in the blank to complete the trigonometric identity. 3. 1 cot u ________ 5. sin2 u ________ 1 7. cscu ________ 4. 6. 8. ________ cos u sin u cos 2 secu ________ u ________ PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. 1 csc x 1 1 1 sin x 1 cos x cos x sin x cos x sin x cos x 1 tan x tan 1 26. cot x tan 2 cscx secx 1 sin y1 siny cos2 y tan x cot x cos x sec x cos2 x sin2 x tan x In Exercises 1–38, verify the identity. 1. 3. 4. 5. 6. 7. 8. 9. 11. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 2. sec y cos y 1 sin t csc t 1 1 sin 1 sin cos 2 cot 2 ysec 2 y 1 1 cos2 sin2 1 2 sin2 cos2 sin2 2 cos 2 1 sin2 sin4 cos2 cos4 cos x sin x tan x sec x csc2 cot cot2 t csc t sin12 x cos x sin52 x cos x cos3 xsin x sec6 xsec x tan x sec4 xsec x tan x sec5 x tan3 x csc t sin t csc sec 1 tan cot3 t csc t 12. 10. cos t csc2 t 1 tan sec2 tan csc x sin x 1 sec x tan x sec 1 1 cos csc x sin x cos x cot x sec x cos x sin x tan x sec 1 tan x 1 cot x 1 csc x 1 sin x cos cot 1 sin 1 sin cos tan x cot x csc x sin x 1 csc cos 1 sin 2 sec 23. 24. 25. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. sin x sin y cos x cos y 0 cot x cot y cot x cot y 1 tan y cot x tan x tan y 1 tan x tan y tan x cot y tan x cot y cos x cos y sin x sin y 1 sin 1 sin 1 cos 1 cos cos2 cos2 2 sec2 y cot 2 2 sin t csc 2 x 1 cot2 x 1 sin cos 1 cos sin 1 t tan t y 1 38. sec2 2 333202_0502.qxd 12/5/05 9:01 AM Page 388 388 Chapter 5 Analytic Trigonometry In Exercises 39– 46, (a) use a graphing utility to graph each side of the equation to determine whether the equation is an identity, (b) use the table feature of a graphing utility to determine whether the equation is an identity, and (c) confirm the results of parts (a) and (b) algebraically. 39. 40. 41. 42. 43. 44. 45. 2 sec2 x 2 sec2 x sin2 x sin2 x cos 2 x 1 csc xcsc x sin x sin x cos x cot x csc2 x sin x 2 cos 2 x 3 cos4 x sin2 x3 2 cos2 x tan 4 x tan2 x 3 sec2 x4 tan2 x 3 csc4 x 2 csc2 x 1 cot4 x sin4 2 sin2 1 cos cos5 cot csc 1 1 sin x cos x cos x 1 sin x 46. csc 1 cot 47. 48. In Exercises 47–50, verify the identity. tan5 x tan3 x sec2 x tan3 x sec4 x tan2 x tan2 x tan4 x sec2 x cos3 x sin2 x sin2 x sin4 x cos x sin4 x cos4 x 1 2 cos2 x 2 cos4 x 49. 50. In Exercises 51–54, use the cofunction identities to evaluate the expression without the aid of a calculator. 51. 53. 54. sin2 25 sin2 65 cos2 20 cos2 52 cos2 38 cos2 70 sin2 12 sin2 40 sin2 50 sin2 78 52. cos2 55 cos2 35 55. Rate of Change The rate of change of the function f x sin x csc x with respect to change in the variable x is given by the expression Show that the expression for the rate of change can also be cos x cot2 x. cos x csc x cot x. Model It s 56. Shadow Length The length of a shadow cast by a h (see vertical gnomon (a device used to tell time) of height when the angle of the sun above the horizon is figure) can be modeled by the equation s h sin90 . sin h ft θ s Model It (co n t i n u e d ) (a) Verify that the equation for s is equal to h cot . (b) Use a graphing utility to complete the table. Let h 5 feet. 10 20 30 40 50 60 70 80 90 s s (c) Use your table from part (b) to determine the angles of the sun for which the length of the shadow is the greatest and the least. (d) Based on your results from part (c), what time of day do you think it is when the angle of the sun above the horizon is 90? Synthesis True or False? the statement is true or false. Justify your answer. In Exercises 57 and 58, determine whether sin2 cos2 1 tan2 is an identity, 57. The equation because sin20 cos20 1 and 1 tan2 1 cot2 1 tan20 1. 58. The equation is not 1 tan26 11 3, an and identity, because it is true that 1 cot26 4. Think About It In Exercises 59 and 60, explain why the equation is not an identity and find one value of the variable for which the equation is not true. 59. 60. sin 1 cos2 tan sec2 1 Skills Review In Exercises 61–64, perform the operation and simplify. 61. 63. 2 3i 26 16 1 4 62. 64. 2 5i 2 3 2i 3 In Exercises 65–68, use the Quadratic Formula to solve the quadratic equation. 65. 67. x2 6x 12 0 3x2 6x 12 0 x2 5x 7 0 66. 68. 8x2 4x 3 0 333202_0503.qxd 12/5/05 9:03 AM Page 389 5.3 Solving Trigonometric Equations Section 5.3 Solving Trigonometric Equations 389 What you should learn • Use standard algebraic techniques to solve trigonometric equations. • Solve trigonometric equations of quadratic type. • Solve trigonometric equations involving multiple angles. • Use inverse trigonometric functions to solve trigonometric equations. Why you should learn it You can use trigonometric equations to solve a variety of real-life problems. For instance, in Exercise 72 on page 398, you can solve a trigonometric equation to help answer questions about monthly sales of skiing equipment. Tom Stillo/Index Stock Imagery Introduction To solve a trigonometric equation, use standard algebraic techniques such as collecting like terms and factoring. Your preliminary goal in solving a trigonometric equation is to isolate the trigonometric function involved in the equation. 2 sin x 1, For example, to solve the equation divide each side by 2 to obtain sin x 1 2 . x, x 56 To solve for note in Figure 5.3 that the equation 0, 2. and there are infinitely many other solutions, which can be written as sin x 1 2 sin x Moreover, because in
the interval has solutions x 6 2, has a period of x 6 2n and x 5 6 2n General solution where n is an integer, as shown in Figure 5.3 = sin x FIGURE 5.3 Another way to show that the equation sin x 1 2 solutions is indicated in Figure 5.4. Any angles that are coterminal with 56 will also be solutions of the equation. has infinitely many or 6 π 5 sin + 2 6 ( ) π =n 1 2 π 5 6 π sin + 2 6 ( ) π =n 1 2 π 6 FIGURE 5.4 When solving trigonometric equations, you should write your answer(s) using exact values rather than decimal approximations. 333202_0503.qxd 12/5/05 9:03 AM Page 390 390 Chapter 5 Analytic Trigonometry Example 1 Collecting Like Terms Solve sin x 2 sin x. Solution Begin by rewriting the equation so that equation. sin x 2 sin x sin x sin x 2 0 sin x sin x 2 2 sin x 2 sin x is isolated on one side of the Write original equation. Add sin x to each side. Subtract 2 from each side. Combine like terms. Divide each side by 2. sin x 2 2 2, and sin x Because x 74. These solutions are each of these solutions to get the general form has a period of x 54 first find all solutions in the interval Finally, add multiples of 0, 2. 2 to x 5 4 2n and x 7 4 2n General solution where n is an integer. Now try Exercise 7. Example 2 Extracting Square Roots Solve 3 tan2 x 1 0. Solution Begin by rewriting the equation so that equation. tan x is isolated on one side of the 3 tan2 x 1 0 3 tan2 x 1 tan2 x 1 3 tan x ± 1 3 ± 3 3 , and tan x has a period of x 6 Because These solutions are of these solutions to get the general form x 5 6 n x and 6 where n is an integer. Now try Exercise 11. Write original equation. Add 1 to each side. Divide each side by 3. Extract square roots. first find all solutions in the interval x 56. Finally, add multiples of 0, . to each n General solution 333202_0503.qxd 12/5/05 9:03 AM Page 391 Exploration Using the equation from Example 3, explain what would happen if you divided each side of the equation by Is this a correct method to use when solving equations? cot x. y 1 −1 −2 −3 − π π x y = cot x cos 2 x − 2 cot x FIGURE 5.5 Section 5.3 Solving Trigonometric Equations 391 The equations in Examples 1 and 2 involved only one trigonometric function. When two or more functions occur in the same equation, collect all terms on one side and try to separate the functions by factoring or by using appropriate identities. This may produce factors that yield no solutions, as illustrated in Example 3. Example 3 Factoring Solve cot x cos2 x 2 cot x. Solution Begin by rewriting the equation so that all terms are collected on one side of the equation. cot x cos2 x 2 cot x Write original equation. cot x cos2 x 2 cot x 0 cot xcos2 x 2 0 Subtract 2 cot x from each side. Factor. By setting each of these factors equal to zero, you obtain cot x 0 x 2 and cos2 x 2 0 cos2 x 2 cos x ± 2. cot x 0 The equation solution is obtained for cosine function. Because is obtained by adding multiples of x 2 ± 2 because , has a period of x 2, to has the solution cos x ± 2 cot x to get [in the interval No are outside the range of the the general form of the solution 0, ]. x 2 n General solution is an integer. You can confirm this graphically by sketching the graph of as shown in Figure 5.5. From the graph you can see and so on. These intercepts occur at 32, 2, 2, x- n where y cot x cos 2 x 2 cot x, that the x- 32, cot x cos2 x 2 cot x 0. intercepts correspond to the solutions of Now try Exercise 15. Equations of Quadratic Type Many trigonometric equations are of quadratic type a couple of examples. ax2 bx c 0. Here are Quadratic in sin x 2 sin2 x sin x 1 0 2sin x2 sin x 1 0 Quadratic in sec x sec2 x 3 sec x 2 0 sec x2 3sec x 2 0 To solve equations of this type, factor the quadratic or, if this is not possible, use the Quadratic Formula. 333202_0503.qxd 12/5/05 9:03 AM Page 392 392 Chapter 5 Analytic Trigonometry Example 4 Factoring an Equation of Quadratic Type Find all solutions of 2 sin2 x sin x 1 0 in the interval 0, 2. Algebraic Solution Begin by treating the equation as a quadratic in factoring. sin x and 2 sin2 x sin x 1 0 2 sin x 1sin x 1 0 Write original equation. Factor. Setting each factor equal to zero, you obtain the following solutions in the interval 0, 2. 2 sin x 1 0 and sin x 1 0 sin x 1 2 x 7 , 6 11 6 sin x 1 x 2 Graphical Solution Use a graphing utility set in radian mode to graph y 2 sin2 x sin x 1 as shown in Figure 5.6. Use the zero or root feature or the zoom and trace xfeatures to approximate the 0 ≤ x < 2, intercepts to be for x 1.571 , 2 x 3.665 7 , 6 and x 5.760 11 . 6 These values are 2 sin2 x sin x 1 0 the approximate in the interval 0, 2. solutions of y = 2 sin2x − sin x − 1 2 3 0 −2 Now try Exercise 29. FIGURE 5.6 Example 5 Rewriting with a Single Trigonometric Function Solve 2 sin2 x 3 cos x 3 0. Solution This equation contains both sine and cosine functions. You can rewrite the equa1 cos 2 x. tion so that it has only cosine functions by using the identity sin2 x 2 sin2 x 3 cos x 3 0 21 cos 2 x 3 cos x 3 0 2 cos 2 x 3 cos x 1 0 2 cos x 1cos x 1 0 Write original equation. Pythagorean identity Multiply each side by 1. Factor. Set each factor equal to zero to find the solutions in the interval 5 3 cos x 1 2 2 cos x 1 0 , x 3 0, 2. cos x 1 0 x 0 cos x 1 2, the general form of the solution is obtained by cos x Because adding multiples of has a period of 2 to get x 2n, x 2n, 3 where n is an integer. Now try Exercise 31. x 5 3 2n General solution 333202_0503.qxd 12/5/05 9:03 AM Page 393 Section 5.3 Solving Trigonometric Equations 393 Sometimes you must square each side of an equation to obtain a quadratic, as demonstrated in the next example. Because this procedure can introduce extraneous solutions, you should check any solutions in the original equation to see whether they are valid or extraneous. Example 6 Squaring and Converting to Quadratic Type Find all solutions of cos x 1 sin x in the interval 0, 2. Solution It is not clear how to rewrite this equation in terms of a single trigonometric function. Notice what happens when you square each side of the equation. cos x 1 sin x cos 2 x 2 cos x 1 sin2 x cos 2 x 2 cos x 1 1 cos 2 x cos 2 x cos2 x 2 cos x 1 1 0 2 cos2 x 2 cos x 0 2 cos xcos x 1 0 Setting each factor equal to zero produces Write original equation. Square each side. Pythagorean identity Rewrite equation. Combine like terms. Factor. 2 cos x 0 cos x 0 x , 2 and cos x 1 0 cos x 1 x . 3 2 Because you squared the original equation, check for extraneous solutions. Check x /2 1 ? cos sin 2 2 0 1 1 x 3/2 3 2 0 1 1 1 ? sin Check cos Check x cos 1 ? 1 1 0 sin Substitute 2 for x. Solution checks. ✓ 3 2 Substitute 32 for x. Solution does not check. x. for Substitute Solution checks. ✓ x 32 x 2 Of the three possible solutions, 0, 2, the only two solutions are is extraneous. So, in the interval and x . Now try Exercise 33. You square each side of the equation in Example 6 because the squares of the sine and cosine functions are related by a Pythagorean identity. The same is true for the squares of the secant and tangent functions and the cosecant and cotangent functions. Exploration Use a graphing utility to confirm the solutions found in Example 6 in two different ways. Do both methods produce the same -values? Which method do you prefer? Why? x 1. Graph both sides of the equation and find the x -coordinates of the points at which the graphs intersect. y cos x 1 y sin x Right side: Left side: 2. Graph the equation y cos x 1 sin x x and find the -intercepts of the graph. 333202_0503.qxd 12/5/05 9:03 AM Page 394 394 Chapter 5 Analytic Trigonometry Functions Involving Multiple Angles The next two examples involve trigonometric functions of multiple angles of the ku forms sin To solve equations of these forms, first solve the equation for then divide your result by and cos ku, ku. k. Example 7 Functions of Multiple Angles Solve 2 cos 3t 1 0. Solution 2 cos 3t 1 0 2 cos 3t 1 cos 3t 1 2 0, 2, In the interval solutions, so, in general, you have you know that Write original equation. Add 1 to each side. Divide each side by 2. 3t 3 and 3t 53 are the only 3t 3 2n and 3t 5 3 2n. Dividing these results by 3, you obtain the general solution t 9 2n 3 and t 5 9 2n 3 General solution where n is an integer. Now try Exercise 35. Example 8 Functions of Multiple Angles Solve 3 tan x 2 3 0. Solution 3 tan x 2 3 0 3 tan tan 3 x 2 x 2 0, , 1 In the interval general, you have 3 4 x 2 n. Write original equation. Subtract 3 from each side. Divide each side by 3. you know that x2 34 is the only solution, so, in Multiplying this result by 2, you obtain the general solution x 3 2 2n where n is an integer. Now try Exercise 39. General solution 333202_0503.qxd 12/5/05 9:03 AM Page 395 Section 5.3 Solving Trigonometric Equations 395 Using Inverse Functions In the next example, you will see how inverse trigonometric functions can be used to solve an equation. Example 9 Using Inverse Functions Solve sec2 x 2 tan x 4. Solution sec2 x 2 tan x 4 1 tan2 x 2 tan x 4 0 tan2 x 2 tan x 3 0 tan x 3tan x 1 0 Write original equation. Pythagorean identity Combine like terms. Factor. Setting each factor equal to zero, you obtain two solutions in the interval 2, 2. [Recall that the range of the inverse tangent function is 2, 2.] tan x 3 0 tan x 3 x arctan 3 and tan x 1 0 tan x 1 x 4 Finally, because adding multiples of tan x has a period of , you obtain the general solution by x arctan 3 n and x 4 n General solution n where arctan 3. is an integer. You can use a calculator to approximate the value of Now try Exercise 59. W RITING ABOUT MATHEMATICS Equations with No Solutions One of the following equations has solutions and the other two do not. Which two equations do not have solutions? a. b. c. sin2 x 5 sin x 6 0 sin2 x 4 sin x 6 0 sin2 x 5 sin x 6 0 Find conditions involving the constants b and c that will guarantee that the equation sin2 x b sin x c 0 has at least one solution on some interval of length 2. 333202_0503.qxd 12/5/05 9:03 AM Page 396 396 Chapter 5
Analytic Trigonometry 5.3 Exercises VOCABULARY CHECK: Fill in the blanks. 1. The equation 2 sin 1 0 has the solutions 7 6 2n and 11 6 2n, which are called ________ solutions. 2. The equation 2 tan2 x 3 tan x 1 0 is a trigonometric equation that is of ________ type. 3. A solution to an equation that does not satisfy the original equation is called an ________ solution. PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. In Exercises 1– 6, verify that the the equation. x -values are solutions of 1. 2 cos x 1 0 (a) x 2. sec x 2 0 3 3. 4. (a) x 3 3 tan2 2x 1 0 (a) x 12 2 cos2 4x 1 0 (a) x 16 5. 2 sin2 x sin x 1 0 (a) x 6. 2 csc4 x 4 csc2 x 0 (a) x 6 (b) x 5 3 (b) x 5 3 (b) x 5 12 (b) x 3 16 (b) x 7 6 (b) x 5 6 In Exercises 7–20, solve the equation. 7. 9. 11. 13. 14. 15. 17. 19. 8. 10. 12. 2 cos x 1 0 3 csc x 2 0 3 sec2 x 4 0 sin xsin x 1 0 3 tan2 x 1tan2 x 3 0 4 cos2 x 1 0 16. 2 sin2 2x 1 tan 3xtan x 1 0 20. 18. 2 sin x 1 0 tan x 3 0 3 cot2 x 1 0 sin2 x 3 cos2 x tan2 3x 3 cos 2x2 cos x 1 0 26. 28. sec x csc x 2 csc x sec x tan x 1 25. 27. 29. 30. 31. 32. 33. 34. sec2 x sec x 2 2 sin x csc x 0 2 cos2 x cos x 1 0 2 sin2 x 3 sin x 1 0 2 sec2 x tan2 x 3 0 cos x sin x tan x 2 csc x cot x 1 sin x 2 cos x 2 In Exercises 35– 40, solve the multiple-angle equation. 35. 37. 39. cos 2x 1 2 tan 3x 1 x 2 cos 2 2 36. sin 2x 3 2 38. 40. sec 4x 2 x 2 sin 3 2 In Exercises 41– 44, find the -intercepts of the graph. x 41. y sin x 2 1 42. y sin x cos x y 3 2 1 −2 −1 1 2 3 4 −2 43. 3 y tan2x 44. 4 y sec4x 8 y 2 1 −3 −1 1 3 −2 x x x x In Exercises 21–34, find all solutions of the equation in the interval [0, 2. −3 −1 1 3 −2 21. 23. cos3 x cos x 3 tan3 x tan x 22. 24. sec2 x 1 0 2 sin2 x 2 cos x 333202_0503.qxd 12/5/05 9:03 AM Page 397 In Exercises 45– 54, use a graphing utility to approximate the solutions (to three decimal places) of the equation in the interval [0, 2. 50. x cos x 1 0 − π 45. 46. 47. 48. 49. 51. 52. 53. 54. 2 sin x cos x 0 4 sin3 x 2 sin2 x 2 sin x 1 0 1 sin x cos x cos x 1 sin x 4 3 cos x cot x 1 sin x x tan x 1 0 sec2 x 0.5 tan x 1 0 csc2 x 0.5 cot x 5 0 2 tan2 x 7 tan x 15 0 6 sin2 x 7 sin x 2 0 In Exercises 55–58, use the Quadratic Formula to solve the equation in the interval Then use a graphing utility to approximate the angle [0, 2. x. 55. 56. 57. 58. 12 sin2 x 13 sin x 3 0 3 tan2 x 4 tan x 4 0 tan2 x 3 tan x 1 0 4 cos2 x 4 cos x 1 0 In Exercises 59–62, use inverse functions where needed to find all solutions of the equation in the interval [0, 2. 59. 60. 61. 62. tan2 x 6 tan x 5 0 sec2 x tan x 3 0 2 cos2 x 5 cos x 2 0 2 sin2 x 7 sin x 3 0 In Exercises 63 and 64, (a) use a graphing utility to graph the function and approximate the maximum and minimum points on the graph in the interval and (b) solve the trigonometric equation and demonstrate that its x -coordinates of the maximum and solutions are the f. minimum points of (Calculus is required to find the trigonometric equation.) [0, 2, Function f x sin x cos x f x 2 sin x cos 2x 63. 64. Trigonometric Equation cos x sin x 0 2 cos x 4 sin x cos x 0 Fixed Point positive fixed point of the function function [ f. f c c.] is a real number such that In Exercises 65 and 66, find the smallest A fixed point of a c f 65. f x tan x 4 66. f x cos x Section 5.3 Solving Trigonometric Equations 397 67. Graphical Reasoning Consider the function given by f x cos 1 x and its graph shown in the figure. y 2 1 −2 π x (a) What is the domain of the function? (b) Identify any symmetry and any asymptotes of the graph. (c) Describe the behavior of the function as x → 0. (d) How many solutions does the equation cos 1 x 0 have in the interval 1, 1? Find the solutions. (e) Does the equation have a greatest solution? If so, approximate the solution. If not, explain why. cos1x 0 68. Graphical Reasoning Consider the function given by f x sin x x and its graph shown in the figure. y 3 2 −1 −2 −3 − π π x (a) What is the domain of the function? (b) Identify any symmetry and any asymptotes of the graph. (c) Describe the behavior of the function as x → 0. (d) How many solutions does the equation sin x x 0 have in the interval 8, 8? Find the solutions. 333202_0503.qxd 12/5/05 9:03 AM Page 398 398 Chapter 5 Analytic Trigonometry 69. Harmonic Motion A weight is oscillating on the end of a spring (see figure). The position of the weight relative to the cos 8t 3 sin 8t, point of equilibrium is given by where is the displacement (in meters) and is the time (in seconds). Find the times when the weight is at the point of 0 ≤ t ≤ 1. equilibrium y 0 y 1 12 for y t 74. Projectile Motion A sharpshooter intends to hit a target at a distance of 1000 yards with a gun that has a muzzle velocity of 1200 feet per second (see figure). Neglecting air resistance, determine the gun’s minimum angle of elevation if the range is given by r Equilibrium y r 1 32 2 sin 2. v0 θ r = 1000 yd Not drawn to scale 70. Damped Harmonic Motion The displacement from equilibrium of a weight oscillating on the end of a spring is given by is the where displacement (in feet) and is the time (in seconds). Use a graphing utility to graph the displacement function for 0 ≤ t ≤ 10. Find the time beyond which the displacement does not exceed 1 foot from equilibrium. y 1.56e0.22t cos 4.9t, t y 71. Sales The monthly sales S (in thousands of units) of a seasonal product are approximated by S 74.50 43.75 sin t 6 t is the time (in months), with where corresponding to January. Determine the months when sales exceed 100,000 units. t 1 75. Ferris Wheel A Ferris wheel is built such that the height (in feet) above ground of a seat on the wheel at time (in t h minutes) can be modeled by ht 53 50 sin 16 t . 2 The wheel makes one revolution every 32 seconds. The ride begins when t 0. (a) During the first 32 seconds of the ride, when will a person on the Ferris wheel be 53 feet above ground? (b) When will a person be at the top of the Ferris wheel for the first time during the ride? If the ride lasts 160 seconds, how many times will a person be at the top of the ride, and at what times? 72. Sales The monthly sales S (in hundreds of units) of skiing equipment at a sports store are approximated by Model It S 58.3 32.5 cos t 6 t is the time (in months), with where corresponding to January. Determine the months when sales exceed 7500 units. t 1 73. Projectile Motion A batted baseball leaves the bat at an angle of with the horizontal and an initial velocity of v0 feet per second. The ball is caught by an outfielder 300 feet from home plate (see figure). Find if 2 sin 2. the range of a projectile is given by 100 r 1 r 32 v0 θ r = 300 ft Not drawn to scale 76. Data Analysis: Unemployment Rate The table in the United States shows the unemployment rates t for selected years from 1990 through 2004. The time corresponding to is measured in years, with 1990. (Source: U.S. Bureau of Labor Statistics) t 0 r Time, t Rate, r Time, t Rate, r 0 2 4 6 5.6 7.5 6.1 5.4 8 10 12 14 4.5 4.0 5.8 5.5 (a) Create a scatter plot of the data. 333202_0503.qxd 12/5/05 9:03 AM Page 399 Model It (co n t i n u e d ) (b) Which of the following models best represents the data? Explain your reasoning. (1) (2) (3) (4) r 1.24 sin0.47t 0.40 5.45 r 1.24 sin0.47t 0.01 5.45 r sin0.10t 5.61 4.80 r 896 sin0.57t 2.05 6.48 (c) What term in the model gives the average unemployment rate? What is the rate? (d) Economists study the lengths of business cycles such as unemployment rates. Based on this short span of time, use the model to find the length of this cycle. (e) Use the model to estimate the next time the unemployment rate will be 5% or less. Section 5.3 Solving Trigonometric Equations 399 80. If you correctly solve a trigonometric equation to the statethen you can finish solving the equation sin x 3.4, ment by using an inverse function. In Exercises 81 and 82, use the graph to approximate the number of points of intersection of the graphs of and y2. y1 81. 2 sin x 3x 1 y1 y2 y 4 3 2 1 y1 y2 π 2 x 82. y1 y2 2 sin x 1 2 x 1 y y2 y1 π 2 4 3 2 1 −3 −4 . x 77. Geometry The area of a rectangle (see figure) inscribed Skills Review is given by In Exercises 83 and 84, solve triangle missing angle measures and side lengths. ABC by finding all in one arc of the graph of A 2x cos x, 0 < x < y cos 1 83. B 66° 22.3 x 84. B A 71° A 14.6 C C (a) Use a graphing utility to graph the area function, and approximate the area of the largest inscribed rectangle. (b) Determine the values of x for which A ≥ 1. 78. Quadratic Approximation Consider the function given by f x 3 sin0.6x 2. (a) Approximate the zero of the function in the interval 0, 6. (b) A quadratic approximation agreeing with at gx 0.45x 2 5.52x 13.70. f utility to graph Describe the result. and g is Use a graphing in the same viewing window. x 5 f (c) Use the Quadratic Formula to find the zeros of g. with the result 0, 6 Compare the zero in the interval of part (a). Synthesis In Exercises 85–88, use reference angles to find the exact values of the sine, cosine, and tangent of the angle with the given measure. 85. 87. 390 1845 86. 88. 600 1410 89. Angle of Depression Find the angle of depression from the top of a lighthouse 250 feet above water level to the water line of a ship 2 miles offshore. 90. Height From a point 100 feet in front of a public library, the angles of elevation to the base of the flagpole and the 39 respectively. The flagtop of the pole are pole is mounted on the front of the library’s roof. Find the height of the flagpole. 45, and 28 True or False? the statement is true or false. Justify your answer. In Exercises 79 and 80, determine whether 79. The equation 2 sin 4t 1 0 of solutions in the interval 2 sin t 1 0. has four times the number 0, 2 as the equation 91. Make a Decision To work an extended application analyzing the normal daily high temperatures in Phoenix and in Seattle, visit this text’s website at college.hmco.com. (Data Source: NOAA) 333202_0504.qxd 12/5/05 9:04 AM Page 400
400 Chapter 5 Analytic Trigonometry 5.4 Sum and Difference Formulas What you should learn • Use sum and difference formulas to evaluate trigonometric functions, verify identities, and solve trigonometric equations. Why you should learn it You can use identities to rewrite trigonometric expressions. For instance, in Exercise 75 on page 405, you can use an identity to rewrite a trigonometric expression in a form that helps you analyze a harmonic motion equation. Using Sum and Difference Formulas In this and the following section, you will study the uses of several trigonometric identities and formulas. Sum and Difference Formulas sinu v sin u cos v cos u sin v sinu v sin u cos v cos u sin v cosu v cos u cos v sin u sin v cosu v cos u cos v sin u sin v tanu v tan u tan v 1 tan u tan v tanu v tan u tan v 1 tan u tan v For a proof of the sum and difference formulas, see Proofs in Mathematics on page 424. Exploration cos x cos 2 Use a graphing utility to graph in the same viewing window. What can you conclude about the graphs? Is it true that cos 2? sin x sin 4 y1 in the same viewing window. What can you conclude about the graphs? Is it true that cosx 2 cos x Use a graphing utility to graph cosx 2 sin x sin 4? sinx 4 sinx 4 and and y1 y2 y2 Richard Megna/Fundamental Photographs Examples 1 and 2 show how sum and difference formulas can be used to find exact values of trigonometric functions involving sums or differences of special angles. Example 1 Evaluating a Trigonometric Function Find the exact value of cos 75. Solution To find the exact value of cos Consequently, the formula for cos 75 cos30 45 75, cosu v use the fact that yields 75 30 45. cos 30 cos 45 sin 30 sin 45 . Try checking this result on your calculator. You will find that cos 75 0.259. Now try Exercise 1. 333202_0504.qxd 12/5/05 9:04 AM Page 401 Historical Note Hipparchus, considered the most eminent of Greek astronomers, was born about 160 B.C. in Nicaea. He was credited with the invention of trigonometry. He also derived the sum and difference sinA ± B formulas for cosA ± B. and FIGURE 5.7 Section 5.4 Sum and Difference Formulas 401 Example 2 Evaluating a Trigonometric Expression Find the exact value of sin 12 . Solution Using the fact that 12 3 4 sinu v, you obtain together with the formula for sin 3 4 sin 12 sin cos cos sin . Now try Exercise 3. Example 3 Evaluating a Trigonometric Expression Find the exact value of sin 42 cos 12 cos 42 sin 12. Solution Recognizing that this expression fits the formula for sinu v, you can write sin 42 cos 12 cos 42 sin 12 sin42 12 sin 30 1 2. Now try Exercise 31. Example 4 An Application of a Sum Formula Write cosarctan 1 arccos x as an algebraic expression. Solution This expression fits the formula for v arccos x are shown in Figure 5.7. So cosu v. Angles u arctan 1 and cosu v cosarctan 1 cosarccos x sinarctan 1 sinarccos . Now try Exercise 51. 333202_0504.qxd 12/5/05 9:04 AM Page 402 402 Chapter 5 Analytic Trigonometry Example 5 shows how to use a difference formula to prove the cofunction identity cos 2 x sin x. Example 5 Proving a Cofunction Identity Prove the cofunction identity cos 2 x sin x. Solution Using the formula for x cos cos 2 cosu v, you have 2 cos x sin 2 sin x 0cos x 1sin x sin x. Now try Exercise 55. Sum and difference formulas can be used to rewrite expressions such as sin n cos n , 2 2 is an integer where and n as expressions involving only reduction formulas. sin or cos . The resulting formulas are called Example 6 Deriving Reduction Formulas Simplify each expression. a. cos 3 2 b. tan 3 Solution a. Using the formula for cos 3 2 you have cos cos cosu v, 3 2 cos 0 sin 1 sin . sin sin 3 2 b. Using the formula for tanu v, you have tan 3 tan tan 3 1 tan tan 3 tan 0 1 tan 0 tan . Now try Exercise 65. 333202_0504.qxd 12/5/05 9:04 AM Page 403 Section 5.4 Sum and Difference Formulas 403 Example 7 Solving a Trigonometric Equation Find all solutions of sinx sinx 4 1 4 in the interval 0, 2. Solution Using sum and difference formulas, rewrite the equation as sin x cos cos x sin sin x cos cos x sin 1 4 4 4 4 1 2 sin x cos 4 1 2sin x2 2 sin x 1 2 2 2 sin x . π π 2 x 2 π So, the only solutions in the interval 0, 2 are x 5 4 and x 7 . 4 y 3 2 1 −1 −2 −3 ( y = sin x + + sin FIGURE 5.8 You can confirm this graphically by sketching the graph of y sinx sinx 4 1 4 for 0 ≤ x < 2, as shown in Figure 5.8. From the graph you can see that the and 74. x- intercepts are 54 Now try Exercise 69. The next example was taken from calculus. It is used to derive the derivative of the sine function. Example 8 An Application from Calculus Verify that sinx h sin x h h 0. where cos x sin h h sin x1 cos h h Solution Using the formula for sinx h sin x h you have sinu v, sin x cos h cos x sin h sin x h cos x sin h sin x1 cos h h sin x1 cos h cos xsin h h Now try Exercise 91. h . 333202_0504.qxd 12/5/05 9:04 AM Page 404 404 Chapter 5 Analytic Trigonometry 5.4 Exercises VOCABULARY CHECK: Fill in the blank to complete the trigonometric identity. 2. 1. sinu v ________ tanu v ________ cosu v ________ 3. 5. cosu v ________ sinu v ________ tanu v ________ 4. 6. PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. 27. 28. 29. 30. sin 3 cos 1.2 cos 3 sin 1.2 cos cos 7 sin sin 7 5 5 tan 2x tan x 1 tan 2x tan x cos 3x cos 2y sin 3x sin 2y In Exercises 31–36, find the exact value of the expression. 31. 32. sin 330 cos 30 cos 330 sin 30 cos 15 cos 60 sin 15 sin 60 33. sin cos cos 12 16 4 3 16 12 sin 4 3 16 16 cos cos sin sin 34. 35. 36. tan 25 tan 110 1 tan 25 tan 110 tan54 tan12 1 tan54 tan12 In Exercises 1– 6, find the exact value of each expression. 1. (a) 2. (a) 3. (a) 4. (a) 5. (a) 6. (a) cos120 45 sin135 30 cos 3 4 5 sin3 6 4 sin7 6 sin315 60 3 (b) (b) (b) cos 120 cos 45 sin 135 cos 30 cos cos (b) sin (b) sin sin sin (b) sin 315 sin 60 3 5 6 3 4 3 4 7 6 In Exercises 7–22, find the exact values of the sine, cosine, and tangent of the angle by using a sum or difference formula. 165 135 30 255 300 45 7 12 4 3 7. 9. 11. 13. 15. 17. 19. 21. 6 5 6 105 60 45 195 225 30 11 3 4 12 9 17 4 12 285 165 13 12 13 12 8. 10. 12. 14. 16. 18. 20. 22. 12 105 15 7 12 5 12 6 4 In Exercises 37–44, find the exact value of the trigonometric u (Both and function given that v are in Quadrant II.) cos v 3 5. sin u 5 13 and 37. 39. 41. 43. sinu v cosu v tanu v secv u 38. 40. 42. 44. cosu v sinv u cscu v cotu v In Exercises 23–30, write the expression as the sine, cosine, or tangent of an angle. 23. 24. 25. 26. cos 25 cos 15 sin 25 sin 15 sin 140 cos 50 cos 140 sin 50 tan 325 tan 86 1 tan 325 tan 86 tan 140 tan 60 1 tan 140 tan 60 In Exercises 45–50, find the exact value of the trigonometric u function given that and are in Quadrant III.) cos v 4 5. sin u 7 25 (Both and v cosu v tanu v secu v 45. 47. 49. 46. sinu v cotv u 48. 50. cosu v 333202_0504.qxd 12/5/05 9:04 AM Page 405 Section 5.4 Sum and Difference Formulas 405 In Exercises 51–54, write the trigonometric expression as an algebraic expression. Model It 51. 53. 54. sinarcsin x arccos x cosarccos x arcsin x cosarccos x arctan x 52. sinarctan 2x arccos x In Exercises 55– 64, verify the identity. 55. sin3 x sin x 56. sin 2 x cos x 75. Harmonic Motion A weight is attached to a spring suspended vertically from a ceiling. When a driving the weight moves force is applied to the system, vertically from its equilibrium position, and this motion is modeled by y 1 3 sin 2t 1 4 cos 2t cos x sin x cos x 3 sin x sin x 1 2 6 cos5 x 2 2 4 cos sin 0 2 1 tan tan 1 tan 4 cosx y cosx y cos2 x sin2 y sinx y sinx y) sin2 x sin2 y sinx y sinx y 2 sin x cos y cosx y cosx y 2 cos x cos y 57. 58. 59. 60. 61. 62. 63. 64. In Exercises 65 –68, simplify the expression algebraically and use a graphing utility to confirm your answer graphically. cos3 2 sin3 2 cos x tan x 67. 66. 65. 68. In Exercises 69 –72, find all solutions of the equation in the interval [0, 2. sinx sinx 1 sinx 3 3 1 sinx 2 6 6 1 cosx cosx tanx 2 sinx 0 4 4 69. 70. 71. 72. In Exercises 73 and 74, use a graphing utility to approximate the solutions in the interval 1 cosx [0, 2. 73. cosx 4 0 4 tanx cosx 2 74. is the distance from equilibrium (in feet) and y where is the time (in seconds). t (a) Use the identity a sin B b cos B a2 b2 sinB C C arctanba, where in the form y a2 b2 sinBt C. a > 0, to write the model (b) Find the amplitude of the oscillations of the weight. (c) Find the frequency of the oscillations of the weight. 76. Standing Waves The equation of a standing wave is obtained by adding the displacements of two waves traveling in opposite directions (see figure). Assume that each of the waves has amplitude . If the models for these waves are and wavelength period A, T, y1 A cos 2 t T x and y2 A cos 2 t T x show that y1 y2 2A cos 2t T cos 2x . = T1 8 t = T2 8 y + y y1 2 2 1 y 2 y + y y1 2 2 1 y + y y1 2 2 1 y 2 y 2 333202_0504.qxd 12/5/05 9:04 AM Page 406 406 Chapter 5 Analytic Trigonometry Synthesis True or False? statement is true or false. Justify your answer. In Exercises 77–80, determine whether the 77. 78. 79. sinu ± v sin u ± sin v cosu ± v cos u ± cos v sin x cosx 2 sinx 80. cos x 2 n 81. 82. is an integer In Exercises 81–84, verify the identity. cosn 1n cos , sinn 1n sin , a sin B b cos Ba2 b2 sinB C, where a sin B b cos Ba2 b2 cosB C, where C arctanab C arctanba is an integer a > 0 b > 0 and and 83. 84. n In Exercises 85–88, use the formulas given in Exercises 83 and 84 to write the trigonometric expression in the following forms. (a) 85. 87. a2 b2 sinB C (b) a2 b2 cosB C sin cos 12 sin 3 5 cos 3 86. 88. 3 sin 2 4 cos 2 sin 2 cos 2 In Exercises 89 and 90, use the formulas given in Exercises 83 and 84 to write the trigonometric expression in the form a sin B b cos B. 2 sin 89. 90. 5 cos 3 4 2 (c) Use a graphing utility to graph the functions and f g. (d) Use the table and the graphs to make a conjecture about the values of the functions and as g f h → 0. In Exercises 93 and 94, use the figure, which shows
two lines whose equations are y1 m1x b1 and y2 m2 x b2. Assume that both lines have positive slopes. Derive a formula for the angle between the two lines.Then use your formula to find the angle between the given pair of lines. y 6 4 θ y1 = m1x + b1 −2 2 4 x y2 = m2x + b2 93. y x and 94. y x and y 3 x y 1 3 x 95. Conjecture Consider the function given by f sin2 sin2 4 . 4 Use a graphing utility to graph the function and use the graph to create an identity. Prove your conjecture. 91. Verify the following identity used in calculus. 96. Proof cosx h cos x h cos xcos h 1 h sin x sin h h 92. Exploration Let in the identity in Exercise 91 f x 6 g and define the functions and as follows. f h cos6 h cos6 h sin cos h 1 sin h h h gh cos 6 6 (a) What are the domains of the functions and f g? (b) Use a graphing utility to complete the table. 0.01 0.02 0.05 0.1 0.2 0.5 h f h gh (a) Write a proof of the formula for (b) Write a proof of the formula for sinu v. sinu v. Skills Review In Exercises 97–100, find the inverse function of Verify f 1f x x. that f f 1x x and f. 97. f x 5x 3 98. f x 7 x 8 99. f x x2 8 100. f x x 16 In Exercises 101–104, apply the inverse properties of and to simplify the expression. e x ln x 101. 103. log3 34x3 eln6x3 log8 83x 2 102. 104. 12x eln xx2 333202_0505.qxd 12/5/05 9:06 AM Page 407 Section 5.5 Multiple-Angle and Product-to-Sum Formulas 407 5.5 Multiple Angle and Product-to-Sum Formulas What you should learn • Use multiple-angle formulas to rewrite and evaluate trigonometric functions. • Use power-reducing formulas to rewrite and evaluate trigonometric functions. • Use half-angle formulas to rewrite and evaluate trigonometric functions. • Use product-to-sum and sum-to-product formulas to rewrite and evaluate trigonometric functions. • Use trigonometric formulas to rewrite real-life models. Why you should learn it You can use a variety of trigonometric formulas to rewrite trigonometric functions in more convenient forms. For instance, in Exercise 119 on page 417, you can use a double-angle formula to determine at what angle an athlete must throw a javelin. Multiple-Angle Formulas In this section, you will study four other categories of trigonometric identities. 1. The first category involves functions of multiple angles such as sin ku and cos ku. 2. The second category involves squares of trigonometric functions such as sin2 u. 3. The third category involves functions of half-angles such as 4. The fourth category involves products of trigonometric functions such as sinu2. sin u cos v. You should learn the double-angle formulas because they are used often in trigonometry and calculus. For proofs of the formulas, see Proofs in Mathematics on page 425. Double-Angle Formulas sin 2u 2 sin u cos u tan 2u 2 tan u 1 tan2 u cos 2u cos2 u sin2 u 2 cos 2 u 1 1 2 sin2 u Example 1 Solving a Multiple-Angle Equation Solve 2 cos x sin 2x 0. Solution Begin by rewriting the equation so that it involves functions of Then factor and solve as usual. x rather than 2x. Mark Dadswell/Getty Images 2 cos x sin 2x 0 2 cos x 2 sin x cos x 0 2 cos x1 sin x 0 2 cos x 0 and 1 sin Write original equation. Double-angle formula Factor. Set factors equal to zero. Solutions in 0, 2 So, the general solution is x 2 2n and x 3 2 2n where n is an integer. Try verifying these solutions graphically. Now try Exercise 9. 333202_0505.qxd 12/5/05 9:06 AM Page 408 408 Chapter 5 Analytic Trigonometry Example 2 Using Double-Angle Formulas to Analyze Graphs Use a double-angle formula to rewrite the equation y 4 cos2 x 2. Then sketch the graph of the equation over the interval 0, 2. Solution Using the double-angle formula for as cos 2u, you can rewrite the original equation y 4 cos2 x 2 22 cos2 x 1 2 cos 2x. Write original equation. Factor. Use double-angle formula. Using the techniques discussed in Section 4.5, you can recognize that the graph of this function has an amplitude of 2 and a period of The key points in the interval are as follows. 0, . Maximum 0, 2 Intercept , 0 4 Minimum , 2 2 Intercept , 0 3 4 Maximum , 2 y = 4 cos2x − 2 π x π2 y 2 1 −1 −2 FIGURE 5.9 Two cycles of the graph are shown in Figure 5.9. Now try Exercise 21. Example 3 Evaluating Functions Involving Double Angles y θ −4 −2 2 4 6 x −2 −4 −6 −8 −10 −12 13 (5, −12) Use the following to find 3 2 cos 5 13 , < < 2 sin 2, cos 2, and tan 2. Solution From Figure 5.10, you can see that each of the double-angle formulas, you can write sin yr 1213. Consequently, using sin 2 2 sin cos 212 13 cos 2 2 cos2 1 2 25 169 tan 2 sin 2 cos 2 120 119 . 120 5 169 13 1 119 169 FIGURE 5.10 Now try Exercise 23. The double-angle formulas are not restricted to angles and . and Other are also valid. Here are two and 3, 6 2 4 2 or double combinations, such as examples. sin 4 2 sin 2 cos 2 and cos 6 cos2 3 sin2 3 By using double-angle formulas together with the sum formulas given in the preceding section, you can form other multiple-angle formulas. 333202_0505.qxd 12/5/05 9:06 AM Page 409 Section 5.5 Multiple-Angle and Product-to-Sum Formulas 409 Example 4 Deriving a Triple-Angle Formula sin 3x sin2x x sin 2x cos x cos 2x sin x 2 sin x cos x cos x 1 2 sin2 xsin x 2 sin x cos2 x sin x 2 sin3 x 2 sin x1 sin2 x sin x 2 sin3 x 2 sin x 2 sin3 x sin x 2 sin3 x 3 sin x 4 sin3 x Now try Exercise 97. Power-Reducing Formulas The double-angle formulas can be used to obtain the following power-reducing formulas. Example 5 shows a typical power reduction that is used in calculus. Power-Reducing Formulas sin2 u 1 cos 2u 2 cos2 u 1 cos 2u 2 tan2 u 1 cos 2u 1 cos 2u For a proof of the power-reducing formulas, see Proofs in Mathematics on page 425. Example 5 Reducing a Power Rewrite sin4 x as a sum of first powers of the cosines of multiple angles. Solution Note the repeated use of power-reducing formulas. sin4 x sin2 x2 1 cos 2x 2 2 Property of exponents Power-reducing formula cos 2x cos2 2x Expand. 1 2 cos 2x 1 cos 4x 2 1 2 cos 2x 1 8 1 8 cos 4x Power-reducing formula Distributive Property 3 4 cos 2x cos 4x Factor out common factor. Now try Exercise 29. 333202_0505.qxd 12/5/05 9:06 AM Page 410 410 Chapter 5 Analytic Trigonometry Half-Angle Formulas You can derive some useful alternative forms of the power-reducing formulas by replacing with The results are called half-angle formulas. u2. u Half-Angle Formulas ±1 cos u ±1 cos u cos sin u 2 2 u 2 2 tan u 2 1 cos u sin u sin u 1 cos u The signs of sin u 2 and cos u 2 depend on the quadrant in which u 2 lies. Example 6 Using a Half-Angle Formula Find the exact value of sin 105. DMS To find the exact value of a trigonometric function with an angle measure in form using a half-angle formula, first convert the angle measure to decimal degree form. Then multiply the resulting angle measure by 2. Solution Begin by noting that sinu2 105 and the fact that is half of 105 210. Then, using the half-angle formula for lies in Quadrant II, you have 2 sin 105 1 cos 210 1 cos 30 1 32 2 2 3 2 2 . The positive square root is chosen because sin is positive in Quadrant II. Now try Exercise 41. Use your calculator to verify the result obtained in Example 6. That is, evaluate 2 3 2. sin 105 and sin 105 0.9659258 2 3 2 0.9659258 You can see that both values are approximately 0.9659258. 333202_0505.qxd 12/5/05 9:06 AM Page 411 Section 5.5 Multiple-Angle and Product-to-Sum Formulas 411 Example 7 Solving a Trigonometric Equation Find all solutions of 2 sin2 x 2 cos 2 x 2 in the interval 0, 2. Algebraic Solution Write original equation. 2 Half-angle formula Simplify. Simplify. Pythagorean identity Simplify. Factor. equal to zero, you find 2 sin2 x 2 cos 2 2 sin2 x 2±1 cos x 2 sin2 x 21 cos x 2 x 2 2 2 sin2 x 1 cos x 2 1 cos2 x 1 cos x cos2 x cos x 0 cos xcos x 1 0 By setting the factors and that the solutions in the interval cos x cos x 1 0, 2 are x , 2 x 3 , 2 and x 0. Now try Exercise 59. Graphical Solution Use a graphing utility set in radian mode to graph y 2 sin2 x 2 cos2x2, as shown in Figure 5.11. Use the zero or root feature or the zoom and trace features to approximate the intercepts in the interval 0, 2 to be x- x 0, x 1.571 , 2 and x 4.712 3 . 2 These values are the approximate solutions of 2 sin2 x 2 cos2x2 0 interval 0, 2. the in 3 2( ) y = 2 − sin2x − 2 cos 2 x − 2 −1 FIGURE 5.11 2 Product-to-Sum Formulas Each of the following product-to-sum formulas is easily verified using the sum and difference formulas discussed in the preceding section. Product-to-Sum Formulas sin u sin v 1 2 cos u cos v 1 2 sin u cos v 1 2 cos u sin v 1 2 cosu v cosu v cosu v cosu v sinu v sinu v sinu v sinu v Product-to-sum formulas are used in calculus to evaluate integrals involving the products of sines and cosines of two different angles. 333202_0505.qxd 12/5/05 9:06 AM Page 412 412 Chapter 5 Analytic Trigonometry Example 8 Writing Products as Sums Rewrite the product cos 5x sin 4x as a sum or difference. Solution Using the appropriate product-to-sum formula, you obtain cos 5x sin 4x 1 2 1 sin5x 4x sin5x 4x 2 sin 9x 1 2 sin x. Now try Exercise 67. Occasionally, it is useful to reverse the procedure and write a sum of trigonometric functions as a product. This can be accomplished with the following sum-to-product formulas. Sum-to-Product Formulas sin u sin v 2 sinu v 2 sin u sin v 2 cosu v 2 cos u cos v 2 cosu v 2 cos u cos v 2 sinu v 2 cosu v 2 sinu v 2 cosu v 2 sinu v 2 For a proof of the sum-to-product formulas, see Proofs in Mathematics on page 426. Example 9 Using a Sum-to-Product Formula Find the exact value of cos 195 cos 105. Solution Using the appropriate sum-to-product formula, you obtain cos 195 cos 105 2 cos195 105 2 cos195 105 2 2 cos 150 cos 45 2 2 2 3 2 6 2 . Now try Exercise 83. 333202_0505.qxd 12/7/05 4:13 PM Page 413 y = sin 5x + sin 3x x π3 2 y 2 1 FIGURE 5.12 Section 5.5 Multiple-Angle and Product-to-Sum Formulas 413 Example 10 Solving a Trigonometric Equation Solve sin 5x sin 3x 0. Solution 2 sin5x 3x 2 sin 5x sin 3x 0 cos5x 3x 0 2 2 sin 4x cos x 0 Write original equation. Sum-to-prod
uct formula Simplify. 2 sin 4x equal to zero, you can find that the solutions in the By setting the factor 0, 2 interval , x 0, are cos x 0 The equation the solutions are of the form yields no additional solutions, and you can conclude that x n 4 is an integer. You can confirm this graphically by sketching the graph of as shown in Figure 5.12. From the graph you can see that n where y sin 5x sin 3x, the x- intercepts occur at multiples of 4. Now try Exercise 87. Example 11 Verifying a Trigonometric Identity Verify the identity sin t sin 3t cos t cos 3t tan 2t. Solution Using appropriate sum-to-product formulas, you have sin t sin 3t cos t cos 3t 2 2 sint 3t 2 cost 3t 2 cost 3t cost 3t 2 2 2 sin2t cost 2 cos2t cost sin 2t cos 2t tan 2t. Now try Exercise 105. 333202_0505.qxd 12/5/05 9:06 AM Page 414 414 Chapter 5 Analytic Trigonometry Application Example 12 Projectile Motion Ignoring air resistance, the range of a projectile fired at an angle with the horizontal and with an initial velocity of feet per second is given by v0 θ FIGURE 5.13 r 1 16 2 sin cos v0 r where is the horizontal distance (in feet) that the projectile will travel. A place kicker for a football team can kick a football from ground level with an initial velocity of 80 feet per second (see Figure 5.13). Not drawn to scale a. Write the projectile motion model in a simpler form. b. At what angle must the player kick the football so that the football travels 200 feet? c. For what angle is the horizontal distance the football travels a maximum? Solution a. You can use a double-angle formula to rewrite the projectile motion model as r 1 32 1 32 22 sin cos v0 Rewrite original projectile motion model. 2 sin 2. v0 Rewrite model using a double-angle formula. b. 2 sin 2 v0 r 1 32 200 1 32 200 200 sin 2 1 sin 2 802 sin 2 Write projectile motion model. Substitute 200 for and 80 for r v0. Simplify. Divide each side by 200. You know that 4 45, Because 45 at an angle of c. From the model 2 2, so dividing this result by 2 produces 4. you can conclude that the player must kick the football so that the football will travel 200 feet. r 200 sin 2 r 200 maximum range is 45. to an angle of produce a maximum horizontal distance of 200 feet. you can see that the amplitude is 200. So the feet. From part (b), you know that this corresponds will Therefore, kicking the football at an angle of 45 Now try Exercise 119. W RITING ABOUT MATHEMATICS Deriving an Area Formula Describe how you can use a double-angle formula or a half-angle formula to derive a formula for the area of an isosceles triangle. Use a labeled sketch to illustrate your derivation. Then write two examples that show how your formula can be used. 333202_0505.qxd 12/5/05 9:06 AM Page 415 Section 5.5 Multiple-Angle and Product-to-Sum Formulas 415 5.5 Exercises VOCABULARY CHECK: Fill in the blank to complete the trigonometric formula. 1. sin 2u ________ 3. cos 2u ________ 5. sin u 2 ________ 2. 4. 6. 1 cos 2u 2 1 cos 2u 1 cos 2u u 2 tan ________ ________ ________ 7. cos u cos v ________ sin u sin v ________ sin u cos v ________ cos u cos v ________ PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. 10. 8. 9. In Exercises 1– 8, use the figure to find the exact value of the trigonometric function. 25. tan u 3 4 , 0 < u < 2 26. cot u 4, 27. sec u 5 2 , 28. csc u 3 < In Exercises 29–34, use the power-reducing formulas to rewrite the expression in terms of the first power of the cosine. 29. 31. 33. cos4 x sin2 x cos2 x sin2 x cos4 x 30. 32. 34. sin8 x sin4 x cos4 x sin4 x cos2 x In Exercises 35–40, use the figure to find the exact value of the trigonometric function. 1 θ 4 2. 4. 6. 8. tan sin 2 sec 2 cot 2 1. 3. 5. 7. sin cos 2 tan 2 csc 2 In Exercises 9–18, find the exact solutions of the equation in the interval [0, 2. 9. 11. 13. 15. 17. sin 2x sin x 0 4 sin x cos x 1 cos 2x cos x 0 tan 2x cot x 0 sin 4x 2 sin 2x 10. 12. 14. 16. 18. sin 2x cos x 0 sin 2x sin x cos x cos 2x sin x 0 tan 2x 2 cos x 0 sin 2x cos 2x2 1 In Exercises 19–22, use a double-angle formula to rewrite the expression. 19. 21. 22. 6 sin x cos x 4 8 sin2 x cos x sin xcos x sin x 20. 6 cos2 x 3 In Exercises 23–28, find the exact values of using the double-angle formulas. and tan 2u sin 2u, cos 2u, 35. cos 23. 24. , sin u 4 5 cos 15 2 2 2 37. tan 39. csc 8 36. sin 38. sec 40. cot 2 2 2 333202_0505.qxd 12/5/05 9:06 AM Page 416 416 Chapter 5 Analytic Trigonometry In Exercises 41– 48, use the half-angle formulas to determine the exact values of the sine, cosine, and tangent of the angle. 71. 73. sinx y sinx y cos sin 72. 74. sinx y cosx y sin sin 41. 43. 45. 47. 75 112 30 8 3 8 42. 44. 46. 48. 165 67 30 12 7 12 In Exercises 49–54, find the exact values of cosu/2, using the half-angle formulas. and sinu/2, tanu/2 49. 50. 51. , sin u 5 13 cos u 3 5 tan 52. cot u 3, < u < 3 2 53. 54. csc u 5 , 3 sec < In Exercises 55–58, use the half-angle formulas to simplify the expression. 55. 57. 2 1 cos 6x 1 cos 8x 1 cos 8x 56. 58. 1 cos 4x 1 cosx 1 2 2 In Exercises 59–62, find all solutions of the equation in [0, 2. the interval Use a graphing utility to graph the equation and verify the solutions. 59. sin x 2 61. cos x 2 cos x 0 sin x 0 60. sin x 2 62. tan x 2 cos x 1 0 sin x 0 In Exercises 63–74, use the product-to-sum formulas to write the product as a sum or difference. 63. 6 sin cos 4 4 65. 67. 69. 10 cos 75 cos 15 cos 4 sin 6 5 cos5 cos 3 64. 66. 68. 70. 5 6 4 cos sin 3 6 sin 45 cos 15 3 sin 2 sin 3 cos 2 cos 4 In Exercises 75–82, use the sum-to-product formulas to write the sum or difference as a product. 76. 78. sin 3 sin sin x sin 5x 75. 77. 79. 80. 81. 82. sin 5 sin 3 cos 6x cos 2x sin sin cos 2 cos cos cos 2 2 sinx sinx 2 2 In Exercises 83–86, use the sum-to-product formulas to find the exact value of the expression. 83. 85. sin 60 sin 30 cos cos 3 4 4 84. cos 120 cos 30 86. sin 5 4 sin 3 4 In Exercises 87–90, find all solutions of the equation in the interval Use a graphing utility to graph the equation and verify the solutions. [0, 2. 87. sin 6x sin 2x 0 88. cos 2x cos 6x 0 89. cos 2x sin 3x sin x 1 0 90. sin2 3x sin2 x 0 In Exercises 91–94, use the figure and trigonometric identities to find the exact value of the trigonometric function in two ways. 3 β 4 α 12 5 91. 93. sin2 sin cos 92. 94. cos2 cos sin In Exercises 95–110, verify the identity. 95. 96. csc 2 csc 2 cos sec 2 sec2 2 sec2 cos2 2 sin2 2 cos 4 cos4 x sin4 x cos 2x 98. 99. sin x cos x2 1 sin 2x 97. 333202_0505.qxd 12/5/05 9:06 AM Page 417 100. sin cos 3 3 1 sin 2 2 3 101. 102. 103. 1 cos 10y 2 cos 2 5y cos 3 cos u 2 1 4 sin2 ± 2 tan u sec tan u sin u 104. tan u 2 csc u cot u Section 5.5 Multiple-Angle and Product-to-Sum Formulas 417 120. Geometry The length of each of the two equal sides of an isosceles triangle is 10 meters (see figure). The angle between the two sides is . θ 10 m 10 m tan cot x ± y 2 x y 2 cot 3x cot t 105. 106. 107. 108. 109. sin x ± sin y cos x cos y sin x sin y cos x cos y cos 4x cos 2x sin 4x sin 2x cos t cos 3t sin 3t sin t sin 6 x sin 6 x cos x 110. cos 3 x cos 3 x cos x (a) Write the area of the triangle as a function of 2. (b) Write the area of the triangle as a function of . such that the area is a Determine the value of maximum. Model It M 121. Mach Number The mach number of an airplane is the ratio of its speed to the speed of sound. When an airplane travels faster than the speed of sound, the sound waves form a cone behind the airplane (see figure). The mach number is related to the apex angle of the cone by In Exercises 111–114, use a graphing utility to verify the identity. Confirm that it is an identity algebraically. sin 2 1 M . 111. 112. 113. 114. cos 3 cos3 3 sin2 cos sin 4 4 sin cos 1 2 sin2 cos 4x cos 2x2 sin 3x sin x cos 3x cos xsin 3x sin x tan 2x θ In Exercises 115 and 116, graph the function by hand in the interval by using the power-reducing formulas. [0, 2] f x sin2 x 115. 116. f x) cos2 x (a) Find the angle that corresponds to a mach number In Exercises 117 and 118, write the trigonometric expression as an algebraic expression. 117. sin2 arcsin x 118. cos2 arccos x 119. Projectile Motion The range of a projectile fired at an angle with the horizontal and with an initial velocity of v0 r 1 32 feet per second is 2 sin 2 v0 r where is measured in feet. An athlete throws a javelin at 75 feet per second. At what angle must the athlete throw the javelin so that the javelin travels 130 feet? of 1. (b) Find the angle that corresponds to a mach number of 4.5. (c) The speed of sound is about 760 miles per hour. Determine the speed of an object with the mach numbers from parts (a) and (b). (d) Rewrite the equation in terms of . 333202_0505.qxd 12/5/05 9:06 AM Page 418 418 Chapter 5 Analytic Trigonometry 122. Railroad Track When two railroad tracks merge, the overlapping portions of the tracks are in the shapes of circular arcs (see figure). The radius of each arc (in feet) and the angle are related by r x 2 2r sin2 . 2 Write a formula for x in terms of cos . r θ r θ x Synthesis True or False? In Exercises 123 and 124, determine whether the statement is true or false. Justify your answer. 123. Because the sine function is an odd function, for a negative number u, sin 2u 2 sin u cos u. 124. sin 1 cos u 2 u 2 u when is in the second quadrant. In Exercises 125 and 126, (a) use a graphing utility to graph the function and approximate the maximum and minimum points on the graph in the interval and (b) solve the trigonometric equation and verify that its solutions are the -coordinates of the maximum and minimum points of x f. (Calculus is required to find the trigonometric equation.) [0, 2 Function Trigonometric Equation 125. x f x 4 sin 2 cos x x 2 cos 2 sin x 0 126. f xcos 2x 2 sin x 2 cos x2 sin x 1 0 127. Exploration Consider the function given by f x sin4 x cos4 x. (a) Use the power-reducing formulas to write the function in terms of cosine to the first power. (b) Determine another way of rewriting the
function. Use a graphing utility to rule out incorrectly rewritten functions. (c) Add a trigonometric term to the function so that it becomes a perfect square trinomial. Rewrite the function as a perfect square trinomial minus the term that you added. Use a graphing utility to rule out incorrectly rewritten functions. (d) Rewrite the result of part (c) in terms of the sine of a double angle. Use a graphing utility to rule out incorrectly rewritten functions. (e) When you rewrite a trigonometric expression, the result may not be the same as a friend’s. Does this mean that one of you is wrong? Explain. 128. Conjecture Consider the function given by f x 2 sin x2 cos2 1. x 2 (a) Use a graphing utility to graph the function. (b) Make a conjecture about the function that is an identity with f. (c) Verify your conjecture analytically. Skills Review In Exercises 129–132, (a) plot the points, (b) find the distance between the points, and (c) find the midpoint of the line segment connecting the points. 129. 130. 131. 132. 5, 2, 1, 4 4, 3, 6, 10 , 4 0, 1 3, 5 , 1, 3 1 3, 2 2 3 2 2 In Exercises 133–136, find (if possible) the complement and supplement of each angle. 133. (a) 134. (a) 135. (a) 55 109 18 (b) (b) (b) 162 78 9 20 136. (a) 0.95 (b) 2.76 137. Profit The total profit for a car manufacturer in October was 16% higher than it was in September. The total profit for the 2 months was $507,600. Find the profit for each month. 138. Mixture Problem A 55-gallon barrel contains a mixture with a concentration of 30%. How much of this mixture must be withdrawn and replaced by 100% concentrate to bring the mixture up to 50% concentration? 139. Distance A baseball diamond has the shape of a square in which the distance between each of the consecutive bases is 90 feet. Approximate the straight-line distance from home plate to second base. 333202_050R.qxd 12/5/05 9:07 AM Page 419 5 Chapter Summary What did you learn? Section 5.1 Recognize and write the fundamental trigonometric identities (p. 374). Use the fundamental trigonometric identities to evaluate trigonometric functions, simplify trigonometric expressions, and rewrite trigonometric expressions (p. 375). Section 5.2 Verify trigonometric identities (p. 382). Section 5.3 Use standard algebraic techniques to solve trigonometric equations (p. 389). Solve trigonometric equations of quadratic type (p. 391). Solve trigonometric equations involving multiple angles (p. 394). Use inverse trigonometric functions to solve trigonometric equations (p. 395). Section 5.4 Use sum and difference formulas to evaluate trigonometric functions, verify identities, and solve trigonometric equations (p. 400). Section 5.5 Use multiple-angle formulas to rewrite and evaluate trigonometric functions (p. 407). Use power-reducing formulas to rewrite and evaluate trigonometric functions (p. 409). Use half-angle formulas to rewrite and evaluate trigonometric functions (p. 410). Use product-to-sum and sum-to-product formulas to rewrite and evaluate trigonometric functions (p. 411). Use trigonometric formulas to rewrite real-life models (p. 414). Chapter Summary 419 Review Exercises 1–6 7–24 25–32 33–38 39–42 43–46 47–50 51–74 75–78 79–82 83–92 93–100 101–106 333202_050R.qxd 12/5/05 9:07 AM Page 420 420 Chapter 5 Analytic Trigonometry 5 Review Exercises In Exercises 1–6, name the trigonometric function 5.1 that is equivalent to the expression. 1. 3. 5. 1 cos x 1 sec x cos x sin x 2. 4. 1 sin x 1 tan x 6. 1 tan 2 x In Exercises 7–10, use the given values and trigonometric identities to evaluate (if possible) all six trigonometric functions. 7. 8. 9. 10. cos x 4 5 13 3 sec , sin x 3 5, tan 2 3 x 2 2 9, sin 2 csc 2 , sin x 2 2 sin 45 9 In Exercises 11–22, use the fundamental trigonometric identities to simplify the expression. 11. 13. 15. 17. 18. 19. 21. 1 cot2 x 1 tan2 xcsc2 x 1 sin 2 sin cos2 x cos2 x cot2 x tan2 csc2 tan2 tan x 12 cos x 12. 14. 16. tan 1 cos2 cot2 xsin2 x u cot 2 cos u 20. sec x tan x2 1 csc 1 1 csc 1 22. cos2 x 1 sin x 23. Rate of Change The rate of change of the function the expression Show that this expression can also be is given by f x csc x cot x csc2 x csc x cot x. written as 1 cos x sin2 x . 5.2 In Exercises 25–32, verify the identity. 25. 26. 27. 28. 29. 30. 31. 32. cos xtan2 x 1 sec x sec2 x cot x cot x tan x cosx cot 2 sin x x tan x 2 1 tan csc cos cot x 1 tan x csc x sin x sin5 x cos2 x cos2 x 2 cos4 x cos6 x sin x cos3 x sin2 x sin2 x sin4 x cos x 5.3 In Exercises 33–38, solve the equation. 34. 4 cos 1 2 cos 33. 35. 36. 37. 38. sin x 3 sin x 33 tan u 3 1 2 sec x 1 0 3 csc2 x 4 4 tan2 u 1 tan2 u In Exercises 39– 46, find all solutions of the equation in the interval [0, 2. 39. 40. 41. 43. 45. 2 cos2 x cos x 1 2 sin2 x 3 sin x 1 cos2 x sin x 1 2 sin 2x 2 0 cos 4xcos x 1 0 42. 44. 46. sin2 x 2 cos x 2 3 tan 3x 0 3 csc2 5x 4 In Exercises 47–50, use inverse functions where needed to find all solutions of the equation in the interval [0, 2. 47. 49. 50. sin2 x 2 sin x 0 tan2 tan 12 0 sec2 x 6 tan x 4 0 48. 2 cos2 x 3 cos x 0 5.4 In Exercises 51–54, find the exact values of the sine, cosine, and tangent of the angle by using a sum or difference formula. 24. Rate of Change The rate of change of the function f x 2sin x sin12 x cos x. is given by the expression Show that this expression can also be written as cot xsin x. 51. 53. 285 315 30 25 11 6 12 4 52. 54. 345 300 45 19 11 6 12 4 333202_050R.qxd 12/5/05 9:07 AM Page 421 In Exercises 55–58, write the expression as the sine, cosine, or tangent of an angle. 55. 56. 57. sin 60 cos 45 cos 60 sin 45 cos 45 cos 120 sin 45 sin 120 tan 25 tan 10 1 tan 25 tan 10 58. tan 68 tan 115 1 tan 68 tan 115 In Exercises 59–64, find the exact value of the trigonometsin u 3 u ric function given that 4 and are in Quadrant II.) cos v 5 13. (Both and v sinu v cosu v cosu v 59. 61. 63. 60. 62. 64. tanu v sinu v tanu v In Exercises 65–70, verify the identity. sin x 66. 65. 2 cosx cot x tan x 2 cos 3x 4 cos3 x 3 cos x sin cos cos tan tan 67. 69. 70. sinx 3 2 cos x 68. sin x sin x In Exercises 71–74, find all solutions of the equation in the interval [0, 2. 71. 72. 73. 74. sinx cosx sinx 2 cosx 3 4 1 sinx 4 4 1 cosx 6 6 3 sinx cosx 3 0 4 2 5.5 cos 2u, In Exercises 75 and 76, find the exact values of using the double-angle formulas. and tan 2u sin 2u, 75. 76. , sin u 4 5 cos < In Exercises 77 and 78, use double-angle formulas to verify the identity algebraically and use a graphing utility to confirm your result graphically. 77. 78. sin 4x 8 cos3 x sin x 4 cos x sin x tan2 x 1 cos 2x 1 cos 2x Review Exercises 421 In Exercises 79– 82, use the power-reducing formulas to rewrite the expression in terms of the first power of the cosine. 79. 81. tan2 2x sin2 x tan2 x 80. 82. cos2 3x cos2 x tan2 x In Exercises 83– 86, use the half-angle formulas to determine the exact values of the sine, cosine, and tangent of the angle. 83. 85. 75 19 12 84. 86. 15 17 12 sinu/2, In Exercises 87–90, find the exact values of cosu/2, 87. using the half-angle formulas. tanu/2 5, 0 < u < 2 8, < u < 32 and sin u 3 tan u 5 cos u 2 7, 2 < u < sec u 6, 2 < u < 88. 89. 90. In Exercises 91 and 92, use the half-angle formulas to simplify the expression. 91. 1 cos 10x 2 92. sin 6x 1 cos 6x In Exercises 93–96, use the product-to-sum formulas to write the product as a sum or difference. 93. cos sin 6 6 94. 6 sin 15 sin 45 95. cos 5 cos 3 96. 4 sin 3 cos 2 In Exercises 97–100, use the sum-to-product formulas to write the sum or difference as a product. 97. 98. 99. 100. sin 4 sin 2 cos 3 cos 2 cosx sinx cosx 6 6 sinx 4 4 101. Projectile Motion A baseball leaves the hand of the person at first base at an angle of with the horizontal and 80 feet per second. The ball at an initial velocity of is caught by the person at second base 100 feet away. Find if the range of a projectile is v0 r r 1 32 2 sin 2. v0 333202_050R.qxd 12/5/05 9:07 AM Page 422 422 Chapter 5 Analytic Trigonometry 102. Geometry A trough for feeding cattle is 4 meters long and its cross sections are isosceles triangles with the two equal sides being meter (see figure). The angle between the two sides is a) Write the trough’s volume as a function of . 2 (b) Write the volume of the trough as a function of and such that the volume is determine the value of maximum. Harmonic Motion In Exercises 103–106, use the following information. A weight is attached to a spring suspended vertically from a ceiling. When a driving force is applied to the system, the weight moves vertically from its equilibrium position, and this motion is described by the model y 1.5 sin 8t 0.5 cos 8t where y is the distance from equilibrium (in feet) and the time (in seconds). t is 103. Use a graphing utility to graph the model. 104. Write the model in the form y a 2 b2 sinBt C. 105. Find the amplitude of the oscillations of the weight. 106. Find the frequency of the oscillations of the weight. Synthesis True or False? the statement is true or false. Justify your answer. In Exercises 107–110, determine whether 107. If < < , then cos < 0. 2 2 sinx y sin x sin y 4 sinx cosx 2 sin 2x 4 sin 45 cos 15 1 3 108. 109. 110. 111. List the reciprocal identities, quotient identities, and Pythagorean identities from memory. 112. Think About It If a trigonometric equation has an infinite number of solutions, is it true that the equation is an identity? Explain. 113. Think About It Explain why you know from observahas no solution if a sin x b 0 tion that the equation a < b. 114. Surface Area The surface area of a honeycomb is given by the equation S 6hs 3 2 h 2.4 where shown in the figure. sin , 0 < ≤ 90 s23 cos s 0.75 inches, inch, and is the angle θ h = 2.4 in. s = 0.75 in. (a) For what value(s) of is the surface area 12 square inches? (b) What value of gives the minimum surface area? In Exercises 115 and 116, use the graphs of to determine how to change one function to form the identity y1 y2. and y2 y1 115. y1 y2 sec2 2 cot2 x y 4 x 116. y1 y2 cos 3x cos x 2 sin x2 y y1 y2 x π− π y2
π y1 x 4 −4 In Exercises 117 and 118, use the zero or root feature of a graphing utility to approximate the solutions of the equation. y x 3 4 cos x 117. 118. y 2 1 2 x2 3 sin x 2 333202_050R.qxd 12/5/05 9:07 AM Page 423 5 Chapter Test Chapter Test 423 Take this test as you would take a test in class. When you are finished, check your work against the answers given in the back of the book. 1. If tan 3 2 cos < 0, . trigonometric functions of and use the fundamental identities to evaluate the other five 2. Use the fundamental identities to simplify csc2 1 cos2 . 3. Factor and simplify sec4 x tan4 x sec2 x tan2 x . 4. Add and simplify cos sin sin cos . 5. Determine the values of , 6. Use a graphing utility to graph the functions Make a conjecture about y1 0 ≤ < 2, for which y1 and Verify the result algebraically. tan sec2 1 y2 and cos x sin x tan x y2. is true. sec x. In Exercises 7–12, verify the identity. 7. 9. 11. 12. sin sec tan csc sec sin cos sinn 1n sin , n sin x cos x2 1 sin 2x cot tan is an integer. 8. 10. sec2 x tan2 x sec2 x sec4 x sin x cosx 2 13. Rewrite sin4 x tan2 x 14. Use a half-angle formula to simplify the expression in terms of the first power of the cosine. sin 4 1 cos 4. 15. Write 16. Write 4 cos 2 sin 4 sin 3 sin 4 as a sum or difference. as a product. y (1, 2) u x In Exercises 17–20, find all solutions of the equation in the interval [0, 2. 17. 19. tan2 x tan x 0 4 cos2 x 3 0 18. 20. sin 2 cos 0 csc2 x csc x 2 0 21. Use a graphing utility to approximate the solutions of the equation 3 cos x x 0 accurate to three decimal places. cos 105 22. Find the exact value of using the fact that 105 135 30. FIGURE FOR 23 23. Use the figure to find the exact values of sin 2u, cos 2u, and tan 2u. 24. Cheyenne, Wyoming has a latitude of 41N. At this latitude, the position of the sun at sunrise can be modeled by t 1.4 D 31 sin 2 365 t is the time (in days) and D where represents the number of degrees north or south of due east that the sun rises. Use a graphing utility to determine the days on which the sun is more than north of due east at sunrise. represents January 1. In this model, 20 t 1 25. The heights modeled by h (in feet) of two people in different seats on a Ferris wheel can be h1 28 cos 10t 38 and h2 28 cos10t 38, 0 ≤ t ≤ 2 6 where t is the time (in minutes). When are the two people at the same height? 333202_050R.qxd 12/5/05 9:07 AM Page 424 Proofs in Mathematics Sum and Difference Formulas (p. 400) sinu v sin u cos v cos u sin v sinu v sin u cos v cos u sin v cosu v cos u cos v sin u sin v cosu v cos u cos v sin u sin v tanu v tan u tan v 1 tan u tan v tanu v tan u tan v 1 tan u tan 1, 0 ) Proof You can use the figures at the left for the proofs of the formulas for v In the top figure, let be the point (1, 0) and then use and , B x1, y1 and i 1, 2, and 3. for bottom figure, note that arcs AC cosu ± v. to locate the points 2 1 2 yi xi In the have the same length. So, line segments D x3, y3 For convenience, assume that BD and are also equal in length, which implies that on the unit circle. So, 0 < v < u < 2. , C x2, y2 BD AC A u and 12 y2 x2 2 2x2 x2 x2 2 y2 1 y2 2 1 2x2 1 1 2x2 x2 2 y3 02 x3 x1 x1 2 2x1x3 2 x3 2 x1 x3 2 y3 1 1 2x1x3 x3x1 Finally, by substituting the values y3 y1 and The for formula u v u v y3y1. x2 you obtain cosu v sin u, sin v, y1 2 2 y3 2 y1 2y1y3 2 2y1y3 2 2x1x3 2 y1 2y1y3 cosu v, cos v, cosu v cos u cos v sin u sin v. can be established by considering cos u, x1 x3 and using the formula just derived to obtain A = (1, 0) x cosu v cosu v cos u cos v sin u sinv cos u cos v sin u sin v. You can use the sum and difference formulas for sine and cosine to prove the formulas for tanu ± v. tanu ± v sinu ± v cosu ± v sin u cos v ± cos u sin v cos u cos v sin u sin v sin u cos v ± cos u sin v cos u cos v cos u cos v sin u sin v cos u cos v Quotient identity Sum and difference formulas Divide numerator and denominator by cos u cos v. D x y = ( , 3 3 ) 424 333202_050R.qxd 12/5/05 9:07 AM Page 425 sin u cos v cos u cos v cos u cos v cos u cos v ± cos u sin v cos u cos v sin u sin v cos u cos v Write as separate fractions. sin u cos u ± sin v cos v sin v cos v 1 sin u cos u tan u ± tan v 1 tan u tan v Product of fractions Quotient identity Double-Angle Formulas sin 2u 2 sin u cos u tan 2u 2 tan u 1 tan2 u (p. 407) cos 2u cos2 u sin2 u 2 cos2 u 1 1 2 sin2 u Proof To prove all three formulas, let sin 2u sinu u v u in the corresponding sum formulas. sin u cos u cos u sin u 2 sin u cos u cos 2u cosu u cos u cos u sin u sin u cos2 u sin2 u tan 2u tanu u tan u tan u 1 tan u tan u 2 tan u 1 tan2 u Trigonometry and Astronomy Trigonometry was used by early astronomers to calculate measurements in the universe. Trigonometry was used to calculate the circumference of Earth and the distance from Earth to the moon. Another major accomplishment in astronomy using trigonometry was computing distances to stars. Power-Reducing Formulas sin2 u 1 cos 2u 2 (p. 409) cos2 u 1 cos 2u 2 tan2 u 1 cos 2u 1 cos 2u Proof To prove the first formula, solve for cos 2u 1 2 sin2 u, as follows. sin2 u in the double-angle formula cos 2u 1 2 sin2 u 2 sin2 u 1 cos 2u sin2 u 1 cos 2u 2 Write double-angle formula. Subtract cos 2u from and add 2 sin2 u to each side. Divide each side by 2. 425 333202_050R.qxd 12/7/05 4:14 PM Page 426 In a similar way you can prove the second formula, by solving for double-angle formula cos2 u in the cos 2u 2 cos2 u 1. To prove the third formula, use a quotient identity, as follows. tan2 u sin2 u cos2 u 1 cos 2u 2 1 cos 2u 2 1 cos 2u 1 cos 2u Sum-to-Product Formulas sin u sin v 2 sinu v 2 sin u sin v 2 cosu v 2 cos u cos v 2 cosu v 2 cos u cos v 2 sinu v 2 (p. 412) cosu v 2 sinu v 2 cosu v 2 sinu v 2 Proof To prove the first formula, u x y2 and x u v in the product-to-sum formula. y u v. and Then substitute let v x y2 sin u cos v 1 2 1 2 sinu v sinu v sin x sin y sinx y 2 2 sinx y 2 cosx y 2 cosx y 2 sin x sin y The other sum-to-product formulas can be proved in a similar manner. 426 333202_050R.qxd 12/5/05 9:07 AM Page 427 P.S. Problem Solving This collection of thought-provoking and challenging exercises further explores and expands upon concepts learned in this chapter. 1. (a) Write each of the other trigonometric functions of terms of sin . (b) Write each of the other trigonometric functions of in in terms of cos . 2. Verify that for all integers n, cos2n 1 2 0. 3. Verify that for all integers n, sin12n 1 6 1 2 . 4. A particular sound wave is modeled by t 30p2 t p3 t p5 t 30p6 t pt 1 4 p1 t 1 n pn where sin524nt, and t is the time (in seconds). (a) Find the sine components and use a graphing utility to graph each component. Then verify the graph of that is shown. pn p t y 1.4 y = p(t) s u s FIGURE FOR 5 v s w s 6. The path traveled by an object (neglecting air resistance) that feet, an initial velocity is given by is projected at an initial height of of feet per second, and an initial angle h0 v0 y 16 v0 2 cos2 x2 tan x h0 x y where and are measured in feet. Find a formula for the maximum height of an object projected from ground level at velocity and angle To do this, find half of the horizontal distance v0 . 1 32 2 sin 2 v0 and then substitute it for where h0 of a projectile 7. Use the figure to derive the formulas for x 0. in the general model for the path sin , cos , and tan 2 2 2 t 0.006 where is an acute angle. −1.4 (b) Find the period of each sine component of p. Is p periodic? If so, what is its period? (c) Use the zero or root feature or the zoom and trace features of a graphing utility to find the -intercepts of the graph of over one cycle. p t (d) Use the maximum and minimum features of a graphing utility to approximate the absolute maximum and absolute minimum values of over one cycle. p s 5. Three squares of side are placed side by side (see figure). Make a conjecture about the relationship between the sum u v Prove your conjecture by using the identity for the tangent of the sum of two angles. and w. θ 2 1 1 cos θ θ sin θ (in pounds) on a person’s back when he or she is modeled by 8. The force F bends over at an angle F 0.6W sin 90 sin 12 where W is the person’s weight (in pounds). (a) Simplify the model. (b) Use a graphing utility to graph the model, where W 185 and 0 < < 90. (c) At what angle is the force a maximum? At what angle is the force a minimum? 427 333202_050R.qxd 12/5/05 9:07 AM Page 428 9. The number of hours of daylight that occur at any location on Earth depends on the time of year and the latitude of the location. The following equations model the numbers of and New hours of daylight in Seward, Alaska Orleans, Louisiana 60 latitude n 12. The index of refraction of a transparent material is the ratio of the speed of light in a vacuum to the speed of light in the material. Some common materials and their indices are air (1.00), water (1.33), and glass (1.50). Triangular prisms are often used to measure the index of refraction based on the formula 30 latitude. D 12.2 6.4 cost 0.2 182.6 D 12.2 1.9 cost 0.2 182.6 Seward New Orleans n 2 . sin 2 sin 2 For the prism shown in the figure, 60. α θ Air ht L ig Prism (a) Write the index of refraction as a function of cot2. (b) Find for a prism made of glass. 13. (a) Write a sum formula for (b) Write a sum formula for sinu v w. tanu v w. 14. (a) Derive a formula for (b) Derive a formula for cos 3. cos 4. h 15. The heights engine can be modeled by (in inches) of pistons 1 and 2 in an automobile h1 3.75 sin 733t 7.5 and h2 3.75 sin 733t 4 3 7.5 where t is measured in seconds. (a) Use a graphing utility to graph the heights of these two pistons in the same viewing window for 0 ≤ t ≤ 1. (b) How often are the pistons at the same height? D In these models, daylight and represents the day, with to January 1. represents the number of hours of t 0 corresponding t (a) Use a graphing utility to graph both models in the same viewing window. Use a viewing window of 0 ≤ t ≤ 365. (b) Find the days of the year on which both cities receive th
e same amount of daylight. What are these days called? (c) Which city has the greater variation in the number of daylight hours? Which constant in each model would you use to determine the difference between the greatest and least numbers of hours of daylight? (d) Determine the period of each model. 10. The tide, or depth of the ocean near the shore, changes (in feet) of a bay can d throughout the day. The water depth be modeled by d 35 28 cos 6.2 t is the time in hours, with t where 12:00 A.M. t 0 corresponding to (a) Algebraically find the times at which the high and low tides occur. (b) Algebraically find the time(s) at which the water depth is 3.5 feet. (c) Use a graphing utility to verify your results from parts (a) and (b). 11. Find the solution of each inequality in the interval 0, 2. (a) (c) sin x ≥ 0.5 tan x < sin x (b) (d) cos x ≤ 0.5 cos x ≥ sin x 428 333202_0600.qxd 12/5/05 10:38 AM Page 429 Additional Topics in Trigonometry 6.1 6.2 6.3 6.4 6.5 Law of Sines Law of Cosines Vectors in the Plane Vectors and Dot Products Trigonometric Form of a Complex Number 66 The work done by a force, such as pushing and pulling objects, can be calculated using vector operations AT I O N S Triangles and vectors have many real-life applications. The applications listed below represent a small sample of the applications in this chapter. • Bridge Design, Exercise 39, page 437 • Glide Path, Exercise 41, page 437 • Surveying, Exercise 31, page 444 • Paper Manufacturing, Exercise 45, page 445 • Revenue, Exercise 65, page 468 • Cable Tension, • Work, Exercises 79 and 80, page 458 Exercise 73, page 469 • Navigation, Exercise 84, page 459 429 333202_0601.qxd 12/5/05 10:40 AM Page 430 430 Chapter 6 Additional Topics in Trigonometry 6.1 Law of Sines What you should learn • Use the Law of Sines to solve oblique triangles (AAS or ASA). • Use the Law of Sines to solve oblique triangles (SSA). • Find the areas of oblique triangles. • Use the Law of Sines to model and solve real-life problems. Why you should learn it You can use the Law of Sines to solve real-life problems involving oblique triangles. For instance, in Exercise 44 on page 438, you can use the Law of Sines to determine the length of the shadow of the Leaning Tower of Pisa. Introduction In Chapter 4, you studied techniques for solving right triangles. In this section and the next, you will solve oblique triangles—triangles that have no right and angles. As standard notation, the angles of a triangle are labeled a, their opposite sides are labeled as shown in Figure 6.1. B,A, and and C, b, c, C b a A c FIGURE 6.1 B To solve an oblique triangle, you need to know the measure of at least one side and any two other parts of the triangle—either two sides, two angles, or one angle and one side. This breaks down into the following four cases. 1. Two angles and any side (AAS or ASA) 2. Two sides and an angle opposite one of them (SSA) 3. Three sides (SSS) 4. Two sides and their included angle (SAS) The first two cases can be solved using the Law of Sines, whereas the last two cases require the Law of Cosines (see Section 6.2). Law of Sines If ABC is a triangle with sides a, b, and c, then Hideo Kurihara/Getty Images a sin A b sin B c sin is acute. A is obtuse. The HM mathSpace® CD-ROM and Eduspace® for this text contain additional resources related to the concepts discussed in this chapter. The Law of Sines can also be written in the reciprocal form sin A a sin B b sin C c . For a proof of the Law of Sines, see Proofs in Mathematics on page 489. 333202_0601.qxd 12/5/05 10:40 AM Page 431 C b = 27.4 ft 102.3° a 28.7° c A FIGURE 6.2 When solving triangles, a careful sketch is useful as a quick test for the feasibility of an answer. Remember that the longest side lies opposite the largest angle, and the shortest side lies opposite the smallest angle. C b a 8° 43° Section 6.1 Law of Sines 431 Example 1 Given Two Angles and One Side—AAS For the triangle in Figure 6.2, the remaining angle and sides. C 102.3, B 28.7, and b 27.4 feet. Find B Solution The third angle of the triangle is A 180 B C 180 28.7 102.3 49.0. By the Law of Sines, you have a sin A b sin B c sin C . Using b 27.4 a b sin B and produces sin A 27.4 sin 28.7 sin 49.0 43.06 feet c b sin B sin C 27.4 sin 28.7 sin 102.3 55.75 feet. Now try Exercise 1. Example 2 Given Two Angles and One Side—ASA A pole tilts toward the sun at an angle from the vertical, and it casts a 22-foot shadow. The angle of elevation from the tip of the shadow to the top of the pole is How tall is the pole? 43. 8 Solution From Figure 6.3, note that is C 180 A B A 43 and B 90 8 98. So, the third angle 180 43 98 39. By the Law of Sines, you have a sin A c sin C . B c = 22 ft A Because c 22 FIGURE 6.3 a c sin C feet, the length of the pole is sin A 22 sin 43 23.84 feet. sin 39 Now try Exercise 35. For practice, try reworking Example 2 for a pole that tilts away from the sun under the same conditions. 333202_0601.qxd 12/5/05 10:40 AM Page 432 432 Chapter 6 Additional Topics in Trigonometry The Ambiguous Case (SSA) In Examples 1 and 2 you saw that two angles and one side determine a unique triangle. However, if two sides and one opposite angle are given, three possible situations can occur: (1) no such triangle exists, (2) one such triangle exists, or (3) two distinct triangles may satisfy the conditions. The Ambiguous Case (SSA) Consider a triangle in which you are given a, b, and A. h b sin A A is acute. A is acute. A is acute. A is acute. A is obtuse. A is obtuse. b h a b ah None One One Two a ≤ b None a b A a > b One Sketch Necessary condition Triangles possible C b = 12 in. 42° A c One solution: FIGURE 6.4 a > b Example 3 Single-Solution Case—SSA a = 22 in. For the triangle in Figure 6.4, the remaining side and angles. a 22 inches, b 12 inches, and A 42. Find Solution By the Law of Sines, you have B sin A a sin B b sin B bsin A a sin B 12sin 42 22 B 21.41. Reciprocal form Multiply each side by b. Substitute for A, a, and b. B is acute. Now, you can determine that C 180 42 21.41 116.59. Then, the remaining side is c sin C a sin A c a sin A sin C 22 sin 42 sin 116.59 29.40 inches. Now try Exercise 19. 333202_0601.qxd 12/5/05 10:40 AM Page 433 a = 15 b = 25 h 85° A No solution: FIGURE 6.5 a < h Section 6.1 Law of Sines 433 Example 4 No-Solution Case—SSA Show that there is no triangle for which a 15, b 25, and A 85. Solution Begin by making the sketch shown in Figure 6.5. From this figure it appears that no triangle is formed. You can verify this using the Law of Sines. sin A a sin B b sin B bsin A a sin B 25sin 85 15 1.660 > 1 This contradicts the fact that and sides a 15 b 25 sin B ≤ 1. and an angle of Reciprocal form Multiply each side by b. So, no triangle can be formed having A 85. Now try Exercise 21. Example 5 Two-Solution Case—SSA Find two triangles for which a 12 meters, b 31 meters, and A 20.5. Solution By the Law of Sines, you have sin A a sin B b sin B bsin A a 31sin 20.5 12 0.9047. Reciprocal form B2 64.8, 180 64.8 115.2 you obtain between 0 There are two angles and 64.8 and B1 B1 180 whose sine is 0.9047. For C 180 20.5 64.8 94.7 c a sin C 12 sin A 115.2, B2 you obtain C 180 20.5 115.2 44.3 c a sin C 12 sin A sin 20.5 sin 20.5 For sin 94.7 34.15 meters. sin 44.3 23.93 meters. The resulting triangles are shown in Figure 6.6. b = 31 m a = 12 m 20.5° 64.8° B1 A A FIGURE 6.6 Now try Exercise 23. b = 31 m 20.5° 115.2° B2 a = 12 m 333202_0601.qxd 12/5/05 10:40 AM Page 434 434 Chapter 6 Additional Topics in Trigonometry Area of an Oblique Triangle 180 A To see how to obtain the height of the obtuse triangle in Figure 6.7, notice the use of the reference angle and the difference formula for sine, as follows. h b sin180 A bsin 180 cos A cos 180 sin A b0 cos A 1 sin A b sin A b = 52 m 102° C a = 90 m FIGURE 6.8 The procedure used to prove the Law of Sines leads to a simple formula for the area of an oblique triangle. Referring to Figure 6.7, note that each triangle has a height of h b sin A. Consequently, the area of each triangle is cb sin A 1 2 baseheight 1 2 bc sin A. Area 1 2 By similar arguments, you can develop the formulas Area 1 2 ab sin C 1 2 ac sin B is acute FIGURE 6.7 c B A c B A is obtuse Area of an Oblique Triangle The area of any triangle is one-half the product of the lengths of two sides times the sine of their included angle. That is, Area 1 2 bc sin A 1 2 ab sin C 1 2 ac sin B. is Note that if angle Area 1 2 90, A bcsin 90 1 2 the formula gives the area for a right triangle: bc 1 2 baseheight. sin 90 1 Similar results are obtained for angles C and B equal to 90. Example 6 Finding the Area of a Triangular Lot Find the area of a triangular lot having two sides of lengths 90 meters and 52 meters and an included angle of 102. a 90 Solution Consider meters, Figure 6.8. Then, the area of the triangle is ab sin C 1 2 Area 1 2 b 52 Now try Exercise 29. meters, and angle C 102, as shown in 9052sin 102 2289 square meters. 333202_0601.qxd 12/5/05 10:40 AM Page 435 W N S B E 40° A 52° 8 km D C FIGURE 6.9 A c 52° B b = 8 km a 40° C FIGURE 6.10 Section 6.1 Law of Sines 435 Application Example 7 An Application of the Law of Sines The course for a boat race starts at point 52 direction S W to point back to Point total distance of the race course. in Figure 6.9 and proceeds in the and finally lies 8 kilometers directly south of point Approximate the E to point A. then in the direction S 40 C, A. B, C A Solution Because lines Consequently, triangle you have BD and ABC B 180 52 40 88. are parallel, it follows that has the measures shown in Figure 6.10. For angle Using the Law of Sines BCA DBC. B, AC a sin 52 b sin 88 c sin 40 b 8 and obtain you can let a 8 sin 88 sin 52 6.308 and c 8 sin 88 sin 40 5.145. The total length of the course is approximately Length 8 6.308 5.145 19.453 kilometers. Now try Exercise 39. W RITING ABOUT MATHEMATICS In this section, you have been using the Law of Sines to Using the Law of Sines solve oblique triangles. Can the
Law of Sines also be used to solve a right triangle? If so, write a short paragraph explaining how to use the Law of Sines to solve each triangle. Is there an easier way to solve these triangles? a. AAS B 50° c = 20 b. ASA B 50° a = 10 C A C A 333202_0601.qxd 12/5/05 10:40 AM Page 436 436 Chapter 6 Additional Topics in Trigonometry 6.1 Exercises The HM mathSpace® CD-ROM and Eduspace® for this text contain step-by-step solutions to all odd-numbered exercises. They also provide Tutorial Exercises for additional help. VOCABULARY CHECK: Fill in the blanks. 1. An ________ triangle is a triangle that has no right angle. 2. For triangle ABC, the Law of Sines is given by 3. The area of an oblique triangle is given by a sin A 2 bc sin A 1 1 ________ c sin C . 2ab sin C ________ . PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. In Exercises 1–18, use the Law of Sines to solve the triangle. Round your answers to two decimal places. C c C 105° b 30° b c = 20 C a = 20 45° a 40° B B 14. 15. 16. 17. 18. A 100, A 110 15, C 85 20, A 55, B 28, a 125, c 10 a 48, a 35, b 16 c 50 B 42, C 104, c 3 4 a 35 8 In Exercises 19–24, use the Law of Sines to solve (if possible) the triangle. If two solutions exist, find both. Round your answers to two decimal places. 19. 20. 21. 22. 23. 24. A 110, A 110, A 76, A 76, A 58, A 58, a 125, a 125, a 18, a 34, a 11.4, a 4.5, b 100 b 200 b 20 b 21 b 12.8 b 12.8 b 25° C 135° b a = 3.5 35° B B c a 10° c = 45 b 5 c 10 In Exercises 25–28, find values for b such that the triangle has (a) one solution, (b) two solutions, and (c) no solution. 25. 26. 27. 28. A 36, A 60, A 10, A 88, a 5 a 10 a 10.8 a 315.6 In Exercises 29–34, find the area of the triangle having the indicated angle and sides. a 8, a 9, A 36, A 60, A 102.4, A 24.3, A 83 20, A 5 40, B 15 30, B 2 45, C 145, C 16.7, C 54.6, a 21.6 c 2.68 C 54.6, c 18.1 b 4.8 B 8 15, a 4.5, b 6.2, b 4, c 14 b 6.8 c 5.8 29. 30. 31. 32. 33. 34. C 120, B 130, A 43 45, A 5 15, B 72 30, C 84 30, a 4, a 62, b 6 c 20 b 57, b 4.5, a 105, a 16, c 85 c 22 c 64 b 20 A A A A 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 333202_0601.qxd 12/5/05 10:40 AM Page 437 Section 6.1 Law of Sines 437 35. Height Because of prevailing winds, a tree grew so that it was leaning 4 from the vertical. At a point 35 meters from the tree, the angle of elevation to the top of the tree is 23 (see figure). Find the height of the tree. h 39. Bridge Design A bridge is to be built across a small lake from a gazebo to a dock (see figure). The bearing from the gazebo to the dock is S W. From a tree 100 meters from the gazebo, the bearings to the gazebo and the dock are S 74 E, respectively. Find the distance from the gazebo to the dock. E and S 41 28 h 94° Tree 74° 28° 100 m W N S E Gazebo 41° 23° 35 m 36. Height A flagpole at a right angle to the horizontal is located on a slope that makes an angle of with the horizontal. The flagpole’s shadow is 16 meters long and points directly up the slope. The angle of elevation from the tip of the shadow to the sun is 20. 12 (a) Draw a triangle that represents the problem. Show the known quantities on the triangle and use a variable to indicate the height of the flagpole. (b) Write an equation involving the unknown quantity. (c) Find the height of the flagpole. 37. Angle of Elevation A 10-meter telephone pole casts a 17-meter shadow directly down a slope when the angle of the angle of elevation of the sun is elevation of the ground. (see figure). Find 42 , A 10 m C 42° 42° − θ 1 7 m θ B 316 38. Flight Path A plane flies 500 kilometers with a bearing of from Naples to Elgin (see figure). The plane then flies 720 kilometers from Elgin to Canton. Find the bearing of the flight from Elgin to Canton. W N S E Elgin N 720 km 500 km Canton Not drawn to scale 44° Naples Dock 40. Railroad Track Design The circular arc of a railroad curve has a chord of length 3000 feet and a central angle of 40. (a) Draw a diagram that visually represents the problem. Show the known quantities on the diagram and use the s variables and to represent the radius of the arc and the length of the arc, respectively. r r (b) Find the radius of the circular arc. s (c) Find the length of the circular arc. 41. Glide Path A pilot has just started on the glide path for landing at an airport with a runway of length 9000 feet. The angles of depression from the plane to the ends of the runway are 18.8. 17.5 and (a) Draw a diagram that visually represents the problem. (b) Find the air distance the plane must travel until touching down on the near end of the runway. (c) Find the ground distance the plane must travel until touching down. (d) Find the altitude of the plane when the pilot begins the descent. 42. Locating a Fire The bearing from the Pine Knob fire tower to the Colt Station fire tower is N E, and the two towers are 30 kilometers apart. A fire spotted by rangers in N 80 E each tower has a bearing of from Pine Knob and S 70 E from Colt Station (see figure). Find the distance of the fire from each tower. 65 N S W 80° 65° E Colt Station 30 km 70° Fire Pine Knob Not drawn to scale 333202_0601.qxd 12/5/05 10:40 AM Page 438 438 Chapter 6 Additional Topics in Trigonometry 43. Distance A boat is sailing due east parallel to the shoreline at a speed of 10 miles per hour. At a given time, the bearing to the lighthouse is S E, and 15 minutes later the bearing is S E (see figure). The lighthouse is located at the shoreline. What is the distance from the boat to the shoreline? 63 70 70° d 63° W N S E Model It 44. Shadow Length The Leaning Tower of Pisa in Italy is characterized by its tilt. The tower leans because it was built on a layer of unstable soil—clay, sand, and water. The tower is approximately 58.36 meters tall from its foundation (see figure). The top of the tower leans about 5.45 meters off center. 5.45 m β α 58.36 m Synthesis True or False? the statement is true or false. Justify your answer. In Exercises 45 and 46, determine whether 45. If a triangle contains an obtuse angle, then it must be oblique. 46. Two angles and one side of a triangle do not necessarily determine a unique triangle. 47. Graphical and Numerical Analysis In the figure, and are positive angles. (a) Write . (b) Use a graphing utility to graph the function. Determine as a function of its domain and range. (c) Use the result of part (a) to write as a function of . (d) Use a graphing utility to graph the function in part (c). c Determine its domain and range. (e) Complete the table. What can you infer? 0.4 0.8 1.2 1.6 2.0 2.4 2.8 c 20 cm θ 2 8 cm θ 30 cm FIGURE FOR 48 γ 9 β 18 α c FIGURE FOR 47 θ d Not drawn to scale 48. Graphical Analysis (a) Find the angle of lean of the tower. (b) Write as a function of d and where , is the angle of elevation to the sun. (c) Use the Law of Sines to write an equation for the length of the shadow cast by the tower. d (d) Use a graphing utility to complete the table. (a) Write the area of the shaded region in the figure as a A function of . (b) Use a graphing utility to graph the area function. (c) Determine the domain of the area function. Explain how the area of the region and the domain of the function would change if the eight-centimeter line segment were decreased in length. 10 20 30 40 50 60 Skills Review d In Exercises 49–52, use the fundamental trigonometric identities to simplify the expression. 49. 51. sin x cot x 1 sin2 2 x 50. 52. 1 cot2 2 tan x cos x sec x x 333202_0602.qxd 12/5/05 10:41 AM Page 439 6.2 Law of Cosines Section 6.2 Law of Cosines 439 What you should learn • Use the Law of Cosines to solve oblique triangles (SSS or SAS). • Use the Law of Cosines to model and solve real-life problems. • Use Heron’s Area Formula to find the area of a triangle. Why you should learn it You can use the Law of Cosines to solve real-life problems involving oblique triangles. For instance, in Exercise 31 on page 444, you can use the Law of Cosines to approximate the length of a marsh. Introduction Two cases remain in the list of conditions needed to solve an oblique triangle— SSS and SAS. If you are given three sides (SSS), or two sides and their included angle (SAS), none of the ratios in the Law of Sines would be complete. In such cases, you can use the Law of Cosines. Law of Cosines Standard Form a2 b2 c 2 2bc cos A b2 a 2 c 2 2ac cos B c 2 a 2 b2 2ab cos C Alternative Form cos A b2 c 2 a 2 2bc cos B a 2 c 2 b2 2ac cos C a 2 b2 c 2 2ab For a proof of the Law of Cosines, see Proofs in Mathematics on page 490. Example 1 Three Sides of a Triangle—SSS Find the three angles of the triangle in Figure 6.11. B c = 14 ft b = 19 ft A a = 8 ft C FIGURE 6.11 © Roger Ressmeyer/Corbis Solution It is a good idea first to find the angle opposite the longest side—side case. Using the alternative form of the Law of Cosines, you find that b in this cos B a 2 c 2 b2 2ac 82 142 192 2814 0.45089. Because B is negative, you know that cos B 116.80 . At this point, it is simpler to use the Law of Sines to determine sin A asin B b is an obtuse angle given by A. 8sin 116.80 0.37583 19 B B Because obtuse angle. So, A A 22.08 is obtuse, must be acute, because a triangle can have, at most, one and C 180 22.08 116.80 41.12. Now try Exercise 1. 333202_0602.qxd 12/5/05 10:41 AM Page 440 440 Chapter 6 Additional Topics in Trigonometry Exploration What familiar formula do you obtain when you use the third form of the Law of Cosines c2 a2 b2 2ab cos C C 90? What is and you let the relationship between the Law of Cosines and this formula? Do you see why it was wise to find the largest angle first in Example 1? Knowing the cosine of an angle, you can determine whether the angle is acute or obtuse. That is, cos > 0 cos < 0 0 < < 90 90 < < 180. Obtuse Acute for for So, in Example 1, once you found that angle was obtuse, you knew that angles A C and were both acute. If the largest angle is acute, the remaining two angles are acute also. B Example 2 Two Sides and the Incl
uded Angle—SAS Find the remaining angles and side of the triangle in Figure 6.12. C b = 15 cm FIGURE 6.12 a 115° A c = 10 cm B Solution Use the Law of Cosines to find the unknown side a in the figure. a2 b2 c2 2bc cos A a2 152 102 21510 cos 115 a2 451.79 a 21.26 a 21.26 Because the reciprocal form of the Law of Sines to solve for centimeters, you now know the ratio B. sin Aa and you can use When solving an oblique triangle given three sides, you use the alternative form of the Law of Cosines to solve for an angle. When solving an oblique triangle given two sides and their included angle, you use the standard form of the Law of Cosines to solve for an unknown. sin A a sin B b sin B bsin A a 15sin 115 21.26 0.63945 So, B arcsin 0.63945 39.75 and C 180 115 39.75 25.25. Now try Exercise 3. 333202_0602.qxd 12/5/05 10:41 AM Page 441 Section 6.2 Law of Cosines 441 Applications Example 3 An Application of the Law of Cosines The pitcher’s mound on a women’s softball field is 43 feet from home plate and the distance between the bases is 60 feet, as shown in Figure 6.13. (The pitcher’s mound is not halfway between home plate and second base.) How far is the pitcher’s mound from first base? Solution In triangle p 60. H 45 HPF, bisects the right angle at Using the Law of Cosines for this SAS case, you have (line HP H ), f 43, and 60 ft 60 ft P h F f = 43 ft 60 ft 45° p = 60 ft H FIGURE 6.13 h2 f 2 p2 2fp cos H 432 602 24360 cos 45º 1800.3 So, the approximate distance from the pitcher’s mound to first base is h 1800.3 42.43 feet. Now try Exercise 31. Example 4 An Application of the Law of Cosines A ship travels 60 miles due east, then adjusts its course northward, as shown in Figure 6.14. After traveling 80 miles in that direction, the ship is 139 miles from its point of departure. Describe the bearing from point to point C. B N S E W A FIGURE 6.14 c = 60 mi Solution You have of Cosines, you have a 80, b 139, and c 60; so, using the alternative form of the Law cos B a 2 c 2 b2 2ac 802 602 1392 28060 0.97094. B arccos0.97094 166.15, So, north from point B to point C is and thus the bearing measured from due or 166.15 90 76.15, N 76.15 E. Now try Exercise 37. 333202_0602.qxd 12/5/05 10:41 AM Page 442 442 Chapter 6 Additional Topics in Trigonometry Historical Note Heron of Alexandria (c. 100 B.C.) was a Greek geometer and inventor. His works describe how to find the areas of triangles, quadrilaterals, regular polygons having 3 to 12 sides, and circles as well as the surface areas and volumes of three-dimensional objects. Heron’s Area Formula The Law of Cosines can be used to establish the following formula for the area of a triangle. This formula is called Heron’s Area Formula after the Greek mathematician Heron (c. 100 B.C.). Heron’s Area Formula Given any triangle with sides of lengths triangle is Area ss as bs c where s a b c2. a, b, and c, the area of the For a proof of Heron’s Area Formula, see Proofs in Mathematics on page 491. Example 5 Using Heron’s Area Formula Find the area of a triangle having sides of lengths and c 72 meters. a 43 meters, b 53 meters, Solution Because s a b c2 1682 84 , Heron’s Area Formula yields Area ss as bs c 84413112 1131.89 square meters. Now try Exercise 47. You have now studied three different formulas for the area of a triangle. Standard Formula Area 1 Area 1 2 ab sin C 1 Oblique Triangle Heron’s Area Formula Area ss as bs c 2 bh 2 bc sin A 1 2 ac sin B W RITING ABOUT MATHEMATICS The Area of a Triangle Use the most appropriate formula to find the area of each triangle below. Show your work and give your reasons for choosing each formula. 2 ft 50° a. c. 4 ft 2 ft 4 ft b. d. 2 ft 3 ft 4 ft 3 ft 4 ft 5 ft 333202_0602.qxd 12/5/05 10:41 AM Page 443 Section 6.2 Law of Cosines 443 6.2 Exercises VOCABULARY CHECK: Fill in the blanks. 1. If you are given three sides of a triangle, you would use the Law of ________ to find the three angles of the triangle. 2. The standard form of the Law of Cosines for cos B a2 c2 b2 2ac is ________ . 3. The Law of Cosines can be used to establish a formula for finding the area of a triangle called ________ ________ Formula. PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. In Exercises 1–16, use the Law of Cosines to solve the triangle. Round your answers to two decimal places. 1. 3. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 2. 4 105° b = 4.5 A a = 10 c B C b = 10 A c = 15 a = 7 B b = 15 30° A C a c = 30 B c 20 c 72 b 14, b 25, a 11, a 55, a 75.4, a 1.42, A 135, A 55, B 10 35, B 75 20, B 125 40, C 15 15, C 43, C 103, c 52 c 1.25 b 52, b 0.75, b 4, b 3, c 9 c 10 c 30 c 9.5 c 32 b 2.15 a 40, a 6.2, a 32, a 6.25, b 7 a 4 9, 9 a 3 b 3 8, 4 In Exercises 17–22, complete the table by solving the parallelogram shown in the figure. (The lengths of the d. ) diagonals are given by and c φ θ a c d b a 5 17. 18. 25 19. 10 20. 40 21. 15 22. b 8 35 14 60 25 c d 45 120 20 80 25 20 35 50 In Exercises 23–28, use Heron’s Area Formula to find the area of the triangle. 23. 24. 25. 26. 27. 28. c 10 c 9 a 5, a 12, a 2.5, a 75.4, a 12.32, a 3.05, b 7, b 15, b 10.2, b 52, b 8.46, b 0.75, c 9 c 52 c 15.05 c 2.45 B,A, 29. Navigation A boat race runs along a triangular course C. marked by buoys and The race starts with the boats headed west for 3700 meters. The other two sides of the course lie to the north of the first side, and their lengths are 1700 meters and 3000 meters. Draw a figure that gives a visual representation of the problem, and find the bearings for the last two legs of the race. 30. Navigation A plane flies 810 miles from Franklin to Then it flies 648 miles Centerville with a bearing of 32. from Centerville to Rosemount with a bearing of Draw a figure that visually represents the problem, and find the straight-line distance and bearing from Franklin to Rosemount. 75. 333202_0602.qxd 12/5/05 10:41 AM Page 444 444 Chapter 6 Additional Topics in Trigonometry 31. Surveying To approximate the length of a marsh, a B , then (see figure). surveyor walks 250 meters from point turns Approximate the length and walks 220 meters to point AC of the marsh. to point C 75 A B 75° 220 m 250 m C A 32. Surveying A triangular parcel of land has 115 meters of frontage, and the other boundaries have lengths of 76 meters and 92 meters. What angles does the frontage make with the two other boundaries? 33. Surveying A triangular parcel of ground has sides of lengths 725 feet, 650 feet, and 575 feet. Find the measure of the largest angle. 34. Streetlight Design Determine the angle in the design of the streetlight shown in the figure. 3 θ 1 4 2 2 35. Distance Two ships leave a port at 9 A.M. One 53 travels at a bearing of N W at 12 miles per hour, and the other travels at a bearing of S W at 16 miles per hour. Approximate how far apart they are at noon that day. 67 36. Length A 100-foot vertical tower is to be erected on the 6 side of a hill that makes a angle with the horizontal (see figure). Find the length of each of the two guy wires that will be anchored 75 feet uphill and downhill from the base of the tower. 37. Navigation On a map, Orlando is 178 millimeters due south of Niagara Falls, Denver is 273 millimeters from Orlando, and Denver is 235 millimeters from Niagara Falls (see figure). 235 mm 235 mm Niagara Falls Niagara Falls Denver Denver 273 mm 273 mm 178 mm 178 mm Orlando Orlando (a) Find the bearing of Denver from Orlando. (b) Find the bearing of Denver from Niagara Falls. 38. Navigation On a map, Minneapolis is 165 millimeters due west of Albany, Phoenix is 216 millimeters from Minneapolis, and Phoenix is 368 millimeters from Albany (see figure). Minneapolis Minneapolis 165 mm 165 mm Albany Albany 216 mm 216 mm Phoenix Phoenix 368 mm 368 mm (a) Find the bearing of Minneapolis from Phoenix. (b) Find the bearing of Albany from Phoenix. 39. Baseball On a baseball diamond with 90-foot sides, the pitcher’s mound is 60.5 feet from home plate. How far is it from the pitcher’s mound to third base? 40. Baseball The baseball player in center field is playing approximately 330 feet from the television camera that is behind home plate. A batter hits a fly ball that goes to the wall 420 feet from the camera (see figure). The camera turns to follow the play. Approximately how far does the center fielder have to run to make the catch? 8 100 ft 6° 75 f t 75 f t 330 ft 8° 420 ft 333202_0602.qxd 12/8/05 9:22 AM Page 445 Section 6.2 Law of Cosines 445 41. Aircraft Tracking To determine the distance between two aircraft, a tracking station continuously determines the distance to each aircraft and the angle between them (see figure). Determine the distance between the planes when A 42, a c 20 miles, and b 35 miles. A a C b B c A FIGURE FOR 41 42. Aircraft Tracking Use the figure for Exercise 41 to a between the planes when c 20 miles, and determine the distance A 11, b 20 43. Trusses Q is the midpoint of the line segment in the truss rafter shown in the figure. What are the lengths of the RS ? line segments miles. and PQ, QS, PR R Q 10 45. Paper Manufacturing In a process with continuous paper, the paper passes across three rollers of radii 3 inches, 4 inches, and 6 inches (see figure). The centers of the three-inch and six-inch rollers are inches apart, and the length of the arc in contact with the paper on the four-inch roller is inches. Complete the table. d s 3 in. s θ d 4 in. 6 in. d (inches) 9 10 12 13 14 15 16 (degrees) s (inches) 46. Awning Design A retractable awning above a patio door from the exterior wall at a height lowers at an angle of of 10 feet above the ground (see figure). No direct sunlight is to enter the door when the angle of elevation of the sun is greater than What is the length of the awning? 70. 50 x P 8 S 8 8 8 50° x Sun’s rays Model It 44. Engine Design An engine has a seven-inch connecting rod fastened to a crank (see figure). 10 ft 70° 1.5 in. 7 in. θ x (a) Use the Law of Cosines to write an equation giving the relationsh
ip between and x (b) Write x . as a function of x. ) yields positive values of . (Select the sign that 47. Geometry The lengths of the sides of a triangular parcel of land are approximately 200 feet, 500 feet, and 600 feet. Approximate the area of the parcel. 48. Geometry A parking lot has the shape of a parallelogram (see figure). The lengths of two adjacent sides are 70 meters and 100 meters. The angle between the two sides is 70 What is the area of the parking lot? . (c) Use a graphing utility to graph the function in part (b). (d) Use the graph in part (c) to determine the maximum distance the piston moves in one cycle. 70 m 70° 100 m 333202_0602.qxd 12/5/05 10:41 AM Page 446 446 Chapter 6 Additional Topics in Trigonometry 49. Geometry You want to buy a triangular lot measuring 510 yards by 840 yards by 1120 yards. The price of the land is $2000 per acre. How much does the land cost? (Hint: 1 acre 4840 square yards) 50. Geometry You want to buy a triangular lot measuring 1350 feet by 1860 feet by 2490 feet. The price of the land is $2200 per acre. How much does the land cost? (Hint: 1 acre 43,560 square feet) 57. Proof Use the Law of Cosines to prove that bc 1 cos . 58. Proof Use the Law of Cosines to prove that a b c 2 bc 1 cos A a b c 1 2 2 . Synthesis Skills Review True or False? statement is true or false. Justify your answer. In Exercises 51–53, determine whether the In Exercises 59– 64, evaluate the expression without using a calculator. 51. In Heron’s Area Formula, s is the average of the lengths of the three sides of the triangle. 52. In addition to SSS and SAS, the Law of Cosines can be used to solve triangles with SSA conditions. 53. A triangle with side lengths of 10 centimeters, 16 centimeters, and 5 centimeters can be solved using the Law of Cosines. 54. Circumscribed and Inscribed Circles Let r be the radii of the circumscribed and inscribed circles of a ABC, triangle s a b c 2 respectively (see figure), and let and a) Prove that (b) Prove that b c 2R a sin A r s as bs c sin C sin B . . s Circumscribed and Inscribed Circles 56, use the results of Exercise 54. In Exercises 55 and 55. Given a triangle with a 25, b 55, and c 72 find the areas of (a) the triangle, (b) the circumscribed circle, and (c) the inscribed circle. 56. Find the length of the largest circular running track that can be built on a triangular piece of property with sides of lengths 200 feet, 250 feet, and 325 feet. 59. 60. 61. 62. 63. 64. arcsin1 arccos 0 arctan 3 arctan3 arcsin 2 3 2 arccos 3 In Exercises 65– 68, write an algebraic expression that is equivalent to the expression. 65. 66. 67. 68. secarcsin 2x tanarccos 3x cotarctanx 2 x 1 cosarcsin 2 In Exercises 69–72, use trigonometric substitution to write , the algebraic equation as a trigonometric function of where Then find /2 < < /2. sec csc . and 69. 70. 71. 72. 5 25 x2, 2 4 x2, 3 x2 9, 12 36 x2, x 5 sin x 2 cos x 3 sec x 6 tan In Exercises 73 and 74, write the sum or difference as a product. 73. cos 5 6 74. sinx cos 3 sinx 2 2 333202_0603.qxd 12/5/05 10:42 AM Page 447 6.3 Vectors in the Plane Section 6.3 Vectors in the Plane 447 What you should learn • Represent vectors as directed line segments. • Write the component forms of vectors. • Perform basic vector operations and represent them graphically. • Write vectors as linear combinations of unit vectors. • Find the direction angles of vectors. • Use vectors to model and solve real-life problems. Why you should learn it You can use vectors to model and solve real-life problems involving magnitude and direction. For instance, in Exercise 84 on page 459, you can use vectors to determine the true direction of a commercial jet. Bill Bachman/Photo Researchers, Inc. Introduction Quantities such as force and velocity involve both magnitude and direction and cannot be completely characterized by a single real number. To represent such a quantity, you can use a directed line segment, as shown in Figure 6.15. The PQ Its magnitude directed line segment (or length) is denoted by and terminal point and can be found using the Distance Formula. has initial point \ PQ Q. P \ Terminal point Q PQ P Initial point FIGURE 6.15 FIGURE 6.16 Two directed line segments that have the same magnitude and direction are equivalent. For example, the directed line segments in Figure 6.16 are all equivalent. The set of all directed line segments that are equivalent to the directed line segment Vectors are denoted by lowercase, boldface letters such as v is a vector v in the plane, written w . PQ , and vu , PQ . \ \ Example 1 Vector Representation by Directed Line Segments u Let be represented by the directed line segment from v and let S 4, 4, P 0, 0 be represented by the directed line segment from as shown in Figure 6.17. Show that u v. Q to R 1, 2 3, 2, to y 5 4 3 2 1 P (0, 0) (4, 4) v S (3, 2) Q (1, 2) R u 1 2 3 4 x FIGURE 6.17 Solution From the Distance Formula, it follows that PQ \ 3 0 2 2 02 13 \ 4 12 4 22 13 PQ RS \ \ and RS have the same magnitude. Moreover, both line segments have the same direction because they are both directed toward the upper right on lines having a slope of So, have the same magnitude and direction, and it follows that 2 3. u v. and PQ RS \ \ Now try Exercise 1. 333202_0603.qxd 12/8/05 9:23 AM Page 448 448 Chapter 6 Additional Topics in Trigonometry Component Form of a Vector The directed line segment whose initial point is the origin is often the most convenient representative of a set of equivalent directed line segments. This representative of the vector v is in standard position. 0, 0 A vector whose initial point is the origin . v1, v2 can be uniquely represented by This is the component form of a the coordinates of its terminal point vector v, written as v v1, v2 v1 The coordinates the terminal point lie at the origin, 0 0, 0. and v2 . are the components of v v . If both the initial point and is the zero vector and is denoted by P p1, p2 and termi- Component Form of a Vector The component form of the vector with initial point nal point Q q1, q2 PQ \ q1 p1, q2 is given by v1, v2 p2 v The magnitude (or length) of is given by 2 q2 p2 is a unit vector. Moreover, v q1 v 1, p1 v v. 2 v1 2 v2 2. If vector 0. v 0 if and only if v is the zero u2 and v v1, v2 u u1, u2 Two vectors v2. u PQ For instance, in Example 1, the vector \ 3 0, 2 0 3, 2 R 1, 2 S 4, 4 and the vector v RS \ v from 4 1, 4 2 to 3, 2. is are equal if and only if u to P 0, 0 from v1 u1 Q 3, 2 and is Example 2 Finding the Component Form of a Vector Find the component form and magnitude of the vector 4, 7 and terminal point 1, 5. v that has initial point and let Q 1, 5 q1, q2 , as shown in Figure v v1, v2 are v1 Solution P 4, 7 p1, p2 Let 6.18. Then, the components of p1 q1 p2 q2 v2 v 5, 12 v 52 122 169 13. So, 1 4 5 5 7 12. and the magnitude of v is Te c h n o l o g y You can graph vectors with a graphing utility by graphing directed line segments. Consult the user’s guide for your graphing utility for specific instructions. Q = (−1, 5) y 6 2 −8 −6 −4 −2 2 4 6 x −2 −4 −6 −8 v P = (4, −7) FIGURE 6.18 Now try Exercise 9. 333202_0603.qxd 12/5/05 10:42 AM Page 449 v 1 2 v 2v −v − v3 2 Vector Operations Section 6.3 Vectors in the Plane 449 The two basic vector operations are scalar multiplication and vector addition. In operations with vectors, numbers are usually referred to as scalars. In this text, v scalars will always be real numbers. Geometrically, the product of a vector and k has the a scalar v same direction as , as shown in Figure 6.19. times as long as kv has the direction opposite that of k is negative, is the vector that is is positive, v , and if v . If kv k k To add two vectors geometrically, position them (without changing their lengths or directions) so that the initial point of one coincides with the terminal is formed by joining the initial point of the secpoint of the other. The sum ond vector with the terminal point of the first vector , as shown in Figure 6.20. This technique is called the parallelogram law for vector addition because the , often called the resultant of vector addition, is the diagonal of a vector parallelogram having and as its adjacent sides FIGURE 6.20 y u v + v u x x and u u1, u2 Definitions of Vector Addition and Scalar Multiplication v v1, v2 Let number). Then the sum of and u v2 k times . k be vectors and let be a scalar (a real v v1, u2 and the scalar multiple of u v u1 is the vector is the vector Scalar multiple ku ku1, u2 ku1, ku2 Sum u The negative of v 1v v v1, v2 is v1, v2 u and the difference of and u v u v u1 v1, u2 Negative v is Add v. See Figure 8.21. . Difference v2 u v To represent same initial point. The difference v geometrically, you can use directed line segments with the is the vector from the terminal point of as shown in Figure 6.21. to the terminal point of , which is equal to u v, u v u FIGURE 6.19 y −v u − v u v u + (−v) u v u v FIGURE 6.21 x 333202_0603.qxd 12/5/05 10:42 AM Page 450 450 Chapter 6 Additional Topics in Trigonometry The component definitions of vector addition and scalar multiplication are illustrated in Example 3. In this example, notice that each of the vector operations can be interpreted geometrically. Example 3 Vector Operations Let v 2, 5 w 3, 4, and w v and find each of the following vectors. v 2w c. a. 2 v b. Solution a. Because v 2, 5, you have 2v 22, 5 22, 25 4, 10. A sketch of 2 is shown in Figure 6.22. v b. The difference of w w v 3 2, 4 5 and is v 5, 1. w v A sketch of vector difference is shown in Figure 6.23. Note that the figure shows the w v as the sum w v. c. The sum of and 2 is w v 2w 2, 5 23, 4 v 2, 5 23, 24 2, 5 6, 8 2 6, 5 8 4, 13. v 2w A sketch of is shown in Figure 6.24. y y (3, 4) w −v 4 3 2 1 − ( 4, 10) − ( 2, 5) 10 8 2v 6 4 v −8 −6 −4 −2 2 x −1 FIGURE 6.22 FIGURE 6.23 x w − v 3 4 5 (5, −1) Now try Exercise 21. (4, 13) y 2w v + 2w 14 12 10 8 v (−2, 5) 2 4 6 8 x −6 −4 −2 FIGURE 6.24 333202_0603.qxd 12/5/05 10:42 AM Page 451 Section 6.3 Vectors in the Plane
451 Vector addition and scalar multiplication share many of the properties of ordinary arithmetic. c d w 1. vu , , and Properties of Vector Addition and Scalar Multiplication Let be vectors and let and be scalars. Then the following properties are true. u v v u u 0 u cdu cdu cu v cu cv du cu du 1u u, 0u 0 8. 6. 2. 4. 5. 3. 7. 9. cv c v Property 9 can be stated as follows: the magnitude of the vector cv is the absolute value of c times the magnitude of v. Unit Vectors In many applications of vectors, it is useful to find a unit vector that has the same direction as a given nonzero vector To do this, you can divide by its magnitude to obtain v. v u unit vector v v 1 vv. Unit vector in direction of v Note that same direction as The vector is a scalar multiple of The vector has a magnitude of 1 and the v. is called a unit vector in the direction of v. v. u u u Example 4 Finding a Unit Vector Find a unit vector in the direction of magnitude of 1. v 2, 5 and verify that the result has a Solution The unit vector in the direction of v is v v 2, 5 22 52 1 29 2 29 2, 5 . , 5 29 This vector has a magnitude of 1 because 5 29 2 29 4 2 2 29 25 29 29 29 1. Now try Exercise 31 Historical Note William Rowan Hamilton (1805–1865), an Irish mathematician, did some of the earliest work with vectors. Hamilton spent many years developing a system of vector-like quantities called quaternions. Although Hamilton was convinced of the benefits of quaternions, the operations he defined did not produce good models for physical phenomena. It wasn’t until the latter half of the nineteenth century that the Scottish physicist James Maxwell (1831–1879) restructured Hamilton’s quaternions in a form useful for representing physical quantities such as force, velocity, and acceleration. 333202_0603.qxd 12/5/05 10:42 AM Page 452 Chapter 6 Additional Topics in Trigonometry The unit vectors 1, 0 and 0, 1 are called the standard unit vectors and are denoted by i 1, 0 and j 0, 1 452 y 2 1 j = 〈0, 1〉 i = 〈1, 0〉 1 2 x FIGURE 6.25 (−1, 3) −8 −6 −4 −2 y 8 6 4 −2 −4 −6 4 6 2 u (2, −5) as shown in Figure 6.25. (Note that the lowercase letter is written in boldface to distinguish it from the imaginary number ) These vectors can be used to represent any vector i 1. as follows. , v v1, v2 i 0, 1 1, 0 v2 v v1, v2 v1 v1i v2 j v1 and The scalars respectively. The vector sum v2 are called the horizontal and vertical components of v, v1i v2 j is called a linear combination of the vectors and . Any vector in the plane can be written as a linear combination of the standard unit vectors and j. j i i Example 5 Writing a Linear Combination of Unit Vectors u Let be the vector with initial point as a linear combination of the standard unit vectors and and terminal point j. i 2, 5 1, 3. Write u Solution Begin by writing the component form of the vector u. x u 1 2, 3 5 3, 8 3i 8j FIGURE 6.26 This result is shown graphically in Figure 6.26. Now try Exercise 43. Example 6 Vector Operations Let u 3i 8j and let v 2i j. Find 2u 3v. Solution You could solve this problem by converting to component form. This, however, is not necessary. It is just as easy to perform the operations in unit vector form. and u v 2u 3v 23i 8j 32i j 6i 16j 6i 3j 12i 19j Now try Exercise 49. 333202_0603.qxd 12/8/05 9:26 AM Page 453 y 1 x y ( , ) y = sin θ u θ −1 x = cos θ 1 x −1 FIGURE 6.27 u 1 y 3 2 1 (3, 3) u x x θ = 45° 1 2 3 FIGURE 6.28 y 1 306.87° 1 2 3 4 v −1 −1 −2 −3 −4 (3, −4) FIGURE 6.29 Section 6.3 Vectors in the Plane 453 Direction Angles u is a unit vector such that If the positive -axis to u x , the terminal point of is the angle (measured counterclockwise) from lies on the unit circle and you have u u x, y cos , sin cos i sin j Suppose that as shown in Figure 6.27. The angle u u. is any vector that makes an angle with the positive -axis, it has the same direction as u is the direction angle of the vector v ai bj . is a unit vector with direction angle and you can write If x v v cos , sin v cos i v sin j. it follows that the direction is determined from v ai bj v cos i v sin j, Because v for angle tan sin cos v sin v cos Quotient identity Multiply numerator and denominator by v . b a . Simplify. Example 7 Finding Direction Angles of Vectors Find the direction angle of each vector. a. b. u 3i 3j v 3i 4j Solution a. The direction angle is tan b a 3 3 1. So, 45, as shown in Figure 6.28. b. The direction angle is 4 tan b a 3 v 3i 4j Moreover, because its reference angle is . arctan 4 53.13 53.13. 3 360 53.13 306.87, So, it follows that Now try Exercise 55. lies in Quadrant IV, lies in Quadrant IV and as shown in Figure 6.29. 333202_0603.qxd 12/5/05 10:42 AM Page 454 454 Chapter 6 Additional Topics in Trigonometry Applications of Vectors − 100 −75 −50 210° −50 −75 100 FIGURE 6.30 W 15° B C D 15° A FIGURE 6.31 y Example 8 Finding the Component Form of a Vector Find the component form of the vector that represents the velocity of an airplane descending at a speed of 100 miles per hour at an angle 30 below the horizontal, as shown in Figure 6.30. x Solution The velocity vector has a magnitude of 100 and a direction angle of v v v cos i v sin j 100cos 210i 100sin 210j 100 i 1001 2 3 2 j 210. 503 i 50j 503, 50 You can check that has a magnitude of 100, as follows. v v 5032 502 7500 2500 10,000 100 Now try Exercise 77. Example 9 Using Vectors to Determine Weight A force of 600 pounds is required to pull a boat and trailer up a ramp inclined at 15 from the horizontal. Find the combined weight of the boat and trailer. Solution Based on Figure 6.31, you can make the following observations. BA BC AC \ force of gravity combined weight of boat and trailer \ force against ramp \ force required to move boat up ramp 600 pounds BWD and ABC are similar. So, angle ABC , is 15 and By construction, triangles so in triangle sin 15 ABC you have AC \ 600 \ BA \ BA \ 600 sin 15 2318. BA Consequently, the combined weight is approximately 2318 pounds. (In Figure 6.31, note that is parallel to the ramp.) AC \ Now try Exercise 81. 333202_0603.qxd 12/5/05 10:42 AM Page 455 Recall from Section 4.8 that in air navigation, bearings are measured in degrees clockwise from north. Section 6.3 Vectors in the Plane 455 Example 10 Using Vectors to Find Speed and Direction 330 An airplane is traveling at a speed of 500 miles per hour with a bearing of at a fixed altitude with a negligible wind velocity as shown in Figure 6.32(a). When the airplane reaches a certain point, it encounters a wind with a velocity of 70 miles per hour in the direction as shown in Figure 6.32(b).What are the resultant speed and direction of the airplane? N 45 E, y y v1 120° (a) FIGURE 6.32 v2 d W i n v v1 θ x (b) xx Solution Using Figure 6.32, the velocity of the airplane (alone) is v1 500cos 120, sin 120 250, 2503 and the velocity of the wind is 70cos 45, sin 45 352, 352. v2 So, the velocity of the airplane (in the wind) is v2 v v1 250 352, 2503 352 200.5, 482.5 and the resultant speed of the airplane is v 200.52 482.52 miles per hour. 522.5 Finally, if is the direction angle of the flight path, you have tan 482.5 200.5 2.4065 which implies that 180 arctan2.4065 So, the true direction of the airplane is Now try Exercise 83. 180 67.4 337.4. 112.6. 333202_0603.qxd 12/5/05 10:42 AM Page 456 456 Chapter 6 Additional Topics in Trigonometry 6.3 Exercises VOCABULARY CHECK: Fill in the blanks. 1. A ________ ________ ________ can be used to represent a quantity that involves both magnitude and direction. 2. The directed line segment \ PQ has ________ point P 3. The ________ of the directed line segment \ PQ and ________ point PQ. is denoted by Q. 4. The set of all directed line segments that are equivalent to a given directed line segment \ PQ is a ________ v in the plane. 5. The directed line segment whose initial point is the origin is said to be in ________ ________ . 6. A vector that has a magnitude of 1 is called a ________ ________ . 7. The two basic vector operations are scalar ________ and vector ________ . 8. The vector u v 9. The vector sum v2 scalars and v1 is called the ________ of vector addition. v1i v2 j are called the ________ and ________ components of is called a ________ ________ of the vectors and i v, j, and the respectively. PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. In Exercises 1 and 2, show that u v. 1. y 2. 6 4 2 (0, 0) −2 −2 (6, 5) u (2, 4) (4, 1) 4 6 v 2 x y 4 (0, 4) u −2 −4 v (3, 3) x 2 4 −4 (−3, −4) (0, −5) In Exercises 3–14, find the component form and the magnitude of the vector v. 3. y 4 3 2 1 5. (3, 2) x 3 4 v 1 2 y (−1, 4) 5 v 3 2 1 (2, 2) −3 −2 −1 21 3 x 4. 6. y 1 −1 −2 −3 x −4 −3 −2 v − − ( 4, 2) (3, 5) y 6 4 2 v −4 −2 (−1, −1) x 2 4 7. y 8. y 4 3 2 1 (3, 3) v −2 −1 −2 −3 1 2 4 5 (3, −2) x −5 (−4, −1) 4 3 2 1 −2 −3 −4 −5 x 4 v (3, −1) Initial Point 1, 5 1, 11 3, 5 3, 11 1, 3 2, 7 9. 10. 11. 12. 13. 14. Terminal Point 15, 12 9, 3 5, 1 9, 40 8, 9 5, 17 In Exercises 15–20, use the figure to sketch a graph of the specified vector. To print an enlarged copy of the graph, go to the website, www.mathgraphs.com. y u v x v 16. 5 u v 18. 20. v 1 2u 15. 17. 19. v u v u 2v 333202_0603.qxd 12/5/05 10:42 AM Page 457 In Exercises 21–28, find (a) 2u 3v. , (b) Then sketch the resultant vector. u v Section 6.3 Vectors in the Plane 457 u v , and (c) In Exercises 53–56, find the magnitude and direction angle of the vector v. v 1, 3 v 4, 0 v 0, 0 v 2, 1 v 2i 3j 21. 22. 23. 24. 25. 26. 27. 28. u 2, 1, u 2, 3, u 5, 3, u 0, 0, u i j, u 2i j, u 2i, v j u 3j, v 2i v i 2j In Exercises 29–38, find a unit vector in the direction of the given vector. 29. 31. 33. 35. 37. u 3, 0 v 2, 2 v 6i 2j w 4j w i 2j 30. 32. 34. 36. 38. u 0, 2 v 5, 12 v i j w 6i w 7j 3i In Exercises 39– 42, find the vector v with the given magnitude and the same direction as u. Magnitude v 5 v 6 v 9 v 10 39. 40. 41. 42. Direction u 3, 3 u 3, 3 u 2, 5 u 10, 0 In Exercises 43–46, the initial and terminal points of a vector are g
iven. Write a linear combination of the standard unit vectors i and j. Initial Point 3, 1 0, 2 1, 5 6, 4 43. 44. 45. 46. Terminal Point 4, 5 3, 6 2, 3 0, 1 In Exercises 47–52, find the component form of and sketch the specified vector operations geometrically, w i 2j. where u 2i j and v 47. 48. 49. 50. 51. 52. v 3 2u v 3 4 w v u 2w v u w v 1 2 v u 2w 3u w 53. 54. 55. 56. v 3cos 60i sin 60j v 8cos 135i sin 135j v 6i 6j v 5i 4j In Exercises 57–64, find the component form of v given its magnitude and the angle it makes with the positive -axis. Sketch v. x Magnitude 32 v 43 v 2 v 3 57. 58. 59. 60. 61. 62. 63. 64. Angle 0 45 150 45 150 90 v in the direction v in the direction i 3j 3i 4j In Exercises 65–68, find the component form of the sum of u and v with direction angles and v . u 65. 66. 67. 68. Magnitude u 5 v 5 u 4 v 4 u 20 v 50 u 50 v 30 Angle 0 90 60 90 45 180 30 110 u v u v u v u v In Exercises 69 and 70, use the Law of Cosines to find the angle between the vectors. ( Assume 0 ≤ ≤ 180. ) 69. 70. v i j, v i 2j, w 2i 2j w 2i j Resultant Force In Exercises 71 and 72, find the angle between the forces given the magnitude of their resultant. (Hint: Write force 1 as a vector in the direction of the positive -axis and force 2 as a vector at an angle with the positive -axis.) x x Force 1 71. 45 pounds Force 2 60 pounds Resultant Force 90 pounds 72. 3000 pounds 1000 pounds 3750 pounds 333202_0603.qxd 12/5/05 10:42 AM Page 458 458 Chapter 6 Additional Topics in Trigonometry 73. Resultant Force Forces with magnitudes of 125 newtons and 300 newtons act on a hook (see figure). The angle between the two forces is Find the direction and magnitude of the resultant of these forces. 45. y 125 newtons 45° 300 newtons x 74. Resultant Force Forces with magnitudes of 2000 newtons and 900 newtons act on a machine part at angles of respectively, with the -axis (see figure). Find the direction and magnitude of the resultant of these forces. 45, and 30 x 2000 newtons 30° −45° x 900 newtons 75. Resultant Force Three forces with magnitudes of 75 pounds, 100 pounds, and 125 pounds act on an object at angles of respectively, with the positive x -axis. Find the direction and magnitude of the resultant of these forces. 120, 45, 30, and 76. Resultant Force Three forces with magnitudes of 70 pounds, 40 pounds, and 60 pounds act on an object at angles of respectively, with the x positive -axis. Find the direction and magnitude of the resultant of these forces. 30, 135, 45, 4 and 77. Velocity A ball is thrown with an initial velocity of 70 feet per second, at an angle of with the horizontal (see figure). Find the vertical and horizontal components of the velocity. 35 70 ft sec 35˚ 78. Velocity A gun with a muzzle velocity of 1200 feet per second is fired at an angle of with the horizontal. Find the vertical and horizontal components of the velocity. 6 Cable Tension In Exercises 79 and 80, use the figure to determine the tension in each cable supporting the load. 79. A 50° 30° B 80. 10 in. 20 in. C 2000 lb A 24 in. B C 5000 lb 81. Tow Line Tension A loaded barge is being towed by two tugboats, and the magnitude of the resultant is 6000 pounds directed along the axis of the barge (see figure). Find the tension in the tow lines if they each make an angle with the axis of the barge. 18 18° 18° 82. Rope Tension To carry a 100-pound cylindrical weight, two people lift on the ends of short ropes that are tied to an eyelet on the top center of the cylinder. Each rope makes a 20 angle with the vertical. Draw a figure that gives a visual representation of the problem, and find the tension in the ropes. 148, 83. Navigation An airplane is flying in the direction of with an airspeed of 875 kilometers per hour. Because of the wind, its groundspeed and direction are 140, respectively (see figure). 800 kilometers per hour and Find the direction and speed of the wind. y 140° 148° N S E W x d Win 800 kilometers per hour 875 kilometers per hour 333202_0603.qxd 12/5/05 10:42 AM Page 459 Model It 84. Navigation A commercial jet is flying from Miami to Seattle. The jet’s velocity with respect to the air is 580 The wind, at the miles per hour, and its bearing is altitude of the plane, is blowing from the southwest with a velocity of 60 miles per hour. 332. (a) Draw a figure that gives a visual representation of the problem. (b) Write the velocity of the wind as a vector in component form. (c) Write the velocity of the jet relative to the air in component form. (d) What is the speed of the jet with respect to the ground? (e) What is the true direction of the jet? 85. Work A heavy implement is pulled 30 feet across a floor, using a force of 100 pounds. The force is exerted at an above the horizontal (see figure). Find the angle of F work done. (Use the formula for work, is the component of the force in the direction of motion and D is the distance.) W FD, where 50 100 lb 50° 30 ft Tension 45° u 1 lb FIGURE FOR 85 FIGURE FOR 86 86. Rope Tension A tetherball weighing 1 pound is pulled outward from the pole by a horizontal force until the rope angle with the pole (see figure). Determine the makes a resulting tension in the rope and the magnitude of 45 u . u Synthesis True or False? statement is true or false. Justify your answer. In Exercises 87 and 88, decide whether the 87. If and v have the same magnitude and direction, then u u v. Section 6.3 Vectors in the Plane 459 (b) If the resultant of the forces is make a conjecture 0, about the angle between the forces. (c) Can the magnitude of the resultant be greater than the sum of the magnitudes of the two forces? Explain. 90. Graphical Reasoning Consider two forces F1 10, 0 and F2 F2 F1 (a) Find 5cos , sin . as a function of . (b) Use a graphing utility to graph the function in part (a) for 0 ≤ < 2. (c) Use the graph in part (b) to determine the range of the function. What is its maximum, and for what value of does it occur? What is its minimum, and for what value of does it occur? (d) Explain why the magnitude of the resultant is never 0. cos i sin j is a unit vector for 91. Proof Prove that any value of . 92. Technology Write a program for your graphing utility that graphs two vectors and their difference given the vectors in component form. In Exercises 93 and 94, use the program in Exercise 92 to find the difference of the vectors shown in the figure. 93. y 94. y 8 6 4 2 (1, 6) (4, 5) (9, 4) (5, 2) 2 4 6 8 125 (−20, 70) x (−100, 0) −50 (80, 80) (10, 60) x 50 Skills Review In Exercises 95–98, use the trigonometric substitution to write the algebraic expression as a trigonometric function of where 0 < < /2. , 95. 96. 97. 98. x 2 64, 64 x 2, x 2 36, x 2 253, x 8 sec x 8 sin x 6 tan x 5 sec 88. If u ai bj is a unit vector, then a 2 b2 1. In Exercises 99–102, solve the equation. 89. Think About It Consider two forces of equal magnitude acting on a point. 99. 100. (a) If the magnitude of the resultant is the sum of the magnitudes of the two forces, make a conjecture about the angle between the forces. cos xcos x 1 0 sin x2 sin x 2 0 3 sec x sin x 23 sin x 0 101. 102. cos x csc x cos x2 0 333202_0604.qxd 12/5/05 10:44 AM Page 460 460 Chapter 6 Additional Topics in Trigonometry 6.4 Vectors and Dot Products What you should learn • Find the dot product of two vectors and use the Properties of the Dot Product. • Find the angle between two vectors and determine whether two vectors are orthogonal. • Write a vector as the sum of two vector components. • Use vectors to find the work done by a force. Why you should learn it You can use the dot product of two vectors to solve real-life problems involving two vector quantities. For instance, in Exercise 68 on page 468, you can use the dot product to find the force necessary to keep a sport utility vehicle from rolling down a hill. Edward Ewert The Dot Product of Two Vectors So far you have studied two vector operations—vector addition and multiplication by a scalar—each of which yields another vector. In this section, you will study a third vector operation, the dot product. This product yields a scalar, rather than a vector. Definition of the Dot Product The dot product of and u v u1v1 u u1, u2 u2v2. v v1, v2 is be vectors in the plane or in space and let be a scalar. c Properties of the Dot Product Let 1. 2. 3. vu w , , and . 5. cu v cu v u cv For proofs of the properties of the dot product, see Proofs in Mathematics on page 492. Example 1 Finding Dot Products Find each dot product. 4, 5 2, 3 a. b. 2, 1 1, 2 c. 0, 3 4, 2 Solution a. 4, 5 2, 3 42 53 8 15 23 b. c. 2, 1 1, 2 21 12 2 2 0 0, 3 4, 2 04 32 0 6 6 Now try Exercise 1. In Example 1, be sure you see that the dot product of two vectors is a scalar (a real number), not a vector. Moreover, notice that the dot product can be positive, zero, or negative. 333202_0604.qxd 12/5/05 10:44 AM Page 461 Section 6.4 Vectors and Dot Products 461 Example 2 Using Properties of Dot Products Let a. u 1, 3, u vw v 2, 4, u 2v b. and w 1, 2. Find each dot product. Solution Begin by finding the dot product of and u v. u v 1, 3 2, 4 12 34 14 u vw 141, 2 14, 28 u 2v 2u v 214 28 a. b. Notice that the product in part (a) is a vector, whereas the product in part (b) is a scalar. Can you see why? Now try Exercise 11. Example 3 Dot Product and Magnitude u The dot product of with itself is 5. What is the magnitude of ? u Solution Because u 2 u u u u u 5. and u u 5, it follows that Now try Exercise 19. The Angle Between Two Vectors The angle between two nonzero vectors is the angle between their respective standard position vectors, as shown in Figure 6.33. This angle can be found using the dot product. (Note that the angle between the zero vector and another vector is not defined.) , 0 ≤ ≤ , Angle Between Two Vectors If is the angle between two nonzero vectors and u v, then cos u v u v . For a proof of the angle between two vectors, see Proofs in Mathematics on page 492. − v u θ u v Origin FIGURE 6.33 333202_0604.qxd 12/8/05 9:34 AM Page 462 462 Chapter 6 A
dditional Topics in Trigonometry Example 4 Finding the Angle Between Two Vectors Find the angle between u 4, 3 and v 3, 5. v = 〈3, 5〉 Solution cos u v u v 4, 3 3, 5 4, 3 3, 5 u = 〈4, 3〉 27 534 θ This implies that the angle between the two vectors is FIGURE 6.34 arccos 27 534 22.2 x as shown in Figure 6.34. Now try Exercise 29. Rewriting the expression for the angle between two vectors in the form u v u v cos Alternative form of dot product produces an alternative way to calculate the dot product. From this form, you can will always are always positive, see that because have the same sign. Figure 6.35 shows the five possible orientations of two vectors. u v cos u and and v θ u θ u u v v cos 1 Opposite Direction FIGURE 6.35 < < 2 1 < cos < 0 Obtuse Angle θ v 2 cos 0 90 Angle u θ v 0 < < 2 0 < cos < 1 Acute Angle v u 0 cos 1 Same Direction Definition of Orthogonal Vectors The vectors and are orthogonal if u v 0. u v The terms orthogonal and perpendicular mean essentially the same thing—meeting at right angles. Even though the angle between the zero vector and another vector is not defined, it is convenient to extend the definition of orthogonality to include the zero vector. In other words, the zero vector is orthogonal to every vector u, because 0 u 0. 333202_0604.qxd 12/5/05 10:44 AM Page 463 Te c h n o l o g y Example 5 Determining Orthogonal Vectors Section 6.4 Vectors and Dot Products 463 The graphing utility program Finding the Angle Between Two Vectors, found on our website college.hmco.com, graphs two v c, d vectors in standard position and finds the measure of the angle between them. Use the program to verify the solutions for Examples 4 and 5. u a, b and Are the vectors u 2, 3 and v 6, 4 orthogonal? Solution Begin by finding the dot product of the two vectors. 0 u v 2, 3 6, 4 26 34 Because the dot product is 0, the two vectors are orthogonal (see Figure 6.36). y 4 3 2 1 −1 −2 −3 v = 〈6, 42, −3〉 FIGURE 6.36 Now try Exercise 47. Finding Vector Components Consider a boat on an inclined ramp, as shown in Figure 6.37. The force You have already seen applications in which two vectors are added to produce a resultant vector. Many applications in physics and engineering pose the reverse problem—decomposing a given vector into the sum of two vector components. F due to gravity pulls the boat down the ramp and against the ramp. These two orthogonal forces, w2. F w1 are vector components of Vector components of . That is, and w2, w1 F F The negative of component rolling down the ramp, whereas withstand against the ramp. A procedure for finding following page. represents the force needed to keep the boat from represents the force that the tires must is shown on the and w2 w1 w2 w1 w1 w2 F FIGURE 6.37 333202_0604.qxd 12/5/05 10:44 AM Page 464 464 Chapter 6 Additional Topics in Trigonometry Definition of Vector Components Let and be nonzero vectors such that u v u w1 w1 w2 w2 w1 u projvu. w2 and are orthogonal and w1 where v , as shown in Figure 6.38. The vectors nents of is parallel to (or a scalar multiple of) w1 are called vector compo- w2 and is the projection of onto and is denoted by u . The vector v w1 The vector is given by w2 u w1. w2 u w1 vθ is acute. FIGURE 6.38 w2 θ v u w1 is obtuse. From the definition of vector components, you can see that it is easy to find v . To find the once you have found the projection of onto the component projection, you can use the dot product, as follows. w2 u w2 u w1 u v cv w2 cv w2 v cv v w2 v cv2 0 w1 is a scalar multiple of v. Take dot product of each side with v. w2 and are orthogonal. v So, and c u v v2 projv u cv u v v2 v. w1 Projection of u onto v Let and be nonzero vectors. The projection of onto u u v v is projvu u v v 2v. 333202_0604.qxd 12/5/05 10:44 AM Page 465 v = 〈6, 2〉 Example 6 Decomposing a Vector into Components Section 6.4 Vectors and Dot Products 465 Find the projection of onto two orthogonal vectors, one of which is u 3, 5 v 6, 2. projvu. Then write as the sum of u 1 2 3 4 5 6 w2 u = 〈3, −5〉 x Solution The projection of onto u v is projvu u v v2 v 8 40 6, 2 6 5 2 5 , w1 as shown in Figure 6.39. The other component, 9 5 3, 5 6 5 u w1 w2, is . , 27 5 w2 2 5 , y w1 −1 2 1 −1 −2 −3 −4 −5 FIGURE 6.39 So, u w1 w2 6 5 9 5 2 5 , 27 5 , 3, 5. Now try Exercise 53. Example 7 Finding a Force A 200-pound cart sits on a ramp inclined at 30 , as shown in Figure 6.40. What force is required to keep the cart from rolling down the ramp? Solution Because the force due to gravity is vertical and downward, you can represent the gravitational force by the vector F 200j. Force due to gravity To find the force required to keep the cart from rolling down the ramp, project in the direction of the ramp, as follows. onto a unit vector v F v cos 30i sin 30j 3 2 i 1 2 j Unit vector along ramp Therefore, the projection of onto F v is v 30° w1 F FIGURE 6.40 w1 projvF F v v2 v F vv 2001 v 2 1003 j. i 1 2 2 The magnitude of this force is 100, and so a force of 100 pounds is required to keep the cart from rolling down the ramp. Now try Exercise 67. 333202_0604.qxd 12/5/05 10:44 AM Page 466 466 Chapter 6 Additional Topics in Trigonometry Work The work object is given by W done by a constant force F acting along the line of motion of an W magnitude of forcedistance \ F PQ W proj PQ as shown in Figure 6.41. If the constant force W motion, as shown in Figure 6.42, the work \ F PQ cos F PQ \ F PQ proj PQ \ \ \ . Projection form for work F cos F Alternative form of dot product is not directed along the line of F done by the force is given by F P F θ F projPQ P Q Q Force acts along the line of motion. FIGURE 6.41 Force acts at angle with the line of motion. FIGURE 6.42 This notion of work is summarized in the following definition. done by a constant force F as its point of application moves is given by either of the following. Definition of Work The work W along the vector W projPQ W F PQ PQ \ \F PQ 1. 2. \ \ Projection form Dot product form Example 8 Finding Work To close a sliding door, a person pulls on a rope with a constant force of 50 pounds at a constant angle of 60 , as shown in Figure 6.43. Find the work done in moving the door 12 feet to its closed position. Solution Using a projection, you can calculate the work as follows. Projection form for work W proj PQ \ \F PQ cos 60F PQ \ 1 2 5012 300 foot-pounds So, the work done is 300 foot-pounds. You can verify this result by finding the vectors and calculating their dot product. and PQ F \ 12 ft P projPQF 60° Q F 12 ft FIGURE 6.43 Now try Exercise 69. 333202_0604.qxd 12/5/05 10:44 AM Page 467 Section 6.4 Vectors and Dot Products 467 6.4 Exercises VOCABULARY CHECK: Fill in the blanks. 1. The ________ ________ of two vectors yields a scalar, rather than a vector. 2. If is the angle between two nonzero vectors and cos ________ . 3. The vectors and are ________ if u v 4. The projection of onto u v is given by 5. The work W W ________ is given by done by a constant force or v, then u u v 0. projvu ________ . F W ________ . as its point of application moves along the vector \ PQ PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. In Exercises 1–8, find the dot product of and u v. 1. 3. 5. 7. u 6, 1 v 2, 3 u 4, 1 v 2, 3 u 4i 2j v i j u 3i 2j v 2i 3j 2. 4. 6. 8. u 5, 12 v 3, 2 u 2, 5 v 1, 2 u 3i 4j v 7i 2j u i 2j v 2i j 32. u 2i 3j v 4i 3j 31. 33. 34. u 5i 5j v 6i 6j i sin u cos j 3 3 i sin3 v cos3 j 4 4 i sin u cos j 4 4 i sin v cos j 2 2 In Exercises 35–38, graph the vectors and find the degree measure of the angle between the vectors. 35. 37. u 3i 4j v 7i 5j u 5i 5j v 8i 8j 36. 38. u 6i 3j v 4i 4j u 2i 3j v 8i 3j In Exercises 39–42, use vectors to find the interior angles of the triangle with the given vertices. 39. 41. 1, 2, 3, 4, 2, 5 3, 0, 2, 2, 0, 6) 40. 42. 3, 4, 3, 5, 1, 7, 1, 9, 8, 2 7, 9 where is the angle between In Exercises 9–18, use the vectors and whether the result is a vector or a scalar. u <2, 2>, v <3, 4>, to find the indicated quantity. State w <1, 2> u u u vv 3w vu w 1 u v u w 9. 11. 13. 15. 17. 10. 12. 14. 16. 18. 3u v v uw u 2vw 2 u v u w v In Exercises 19–24, use the dot product to find the magnitude of u. 19. 21. 23. u 5, 12 u 20i 25j u 6j 20. 22. 24. u 2, 4 u 12i 16j u 21i In Exercises 25 –34, find the angle between the vectors. 43. 25. 27. 29. u 1, 0 v 0, 2 u 3i 4j v 2j u 2i j v 6i 4j 26. 28. 30. u 3, 2 v 4, 0 u 2i 3j v i 2j u 6i 3j v 8i 4j u v, In Exercises 43–46, find u and v. u 4, v 10, 2 3 44. u 100, v 250, 6 45. u 9, v 36, 3 4 46. u 4, v 12, 3 333202_0604.qxd 12/5/05 10:44 AM Page 468 468 Chapter 6 Additional Topics in Trigonometry In Exercises 47–52, determine whether and are orthogonal, parallel, or neither. v u 47. 49. 51. u 12, 30 v 1 2, 5 4 3i j u 1 4 v 5i 6j u 2i 2j v i j 48. 50. 52. u 3, 15 v 1, 5 u i v 2i 2j u cos , sin v sin , cos In Exercises 53–56, find the projection of onto . Then write as the sum of two orthogonal vectors, one of which is u projv u. v u 53. 55. u 2, 2 v 6, 1 u 0, 3 v 2, 15 54. 56. u 4, 2 v 1, 2 u 3, 2 v 4, 1 In Exercises 57 and 58, use the graph to determine mentally the projection of onto . (The coordinates of the terminal points of the vectors in standard position are given.) Use the formula for the projection of onto to verify your result. u u v v 57. y 58. y 6 (−2, 3) u −2 −2 (6, 4) v x 2 4 6 (6, 4) v x 2 6 4 (2, −3) u 4 2 −2 −2 −4 In Exercises 59–62, find two vectors in opposite directions (There are many that are orthogonal to the vector correct answers.) u. 59. 60. 61. 62. u 3, 5 u 8 3j Work In Exercises 63 and 64, find the work done in if the magnitude and moving a particle from Q v. direction of the force are given by to P 63. 64. P 0, 0, Q 4, 7, v 1, 4 P 1, 3, Q 3, 5, v 2i 3j 65. Revenue The vector u 1650, 3200 gives the numbers of units of two types of baking pans produced by a company. The vector gives the prices (in dollars) of the two types of pans, respectively. v 15.25, 10.50 (a) Find the dot product u v and inter
pret the result in the context of the problem. (b) Identify the vector operation used to increase the prices by 5%. 66. Revenue The vector u 3240, 2450 gives the numbers of hamburgers and hot dogs, respectively, sold at a fast-food stand in one month. The vector gives the prices (in dollars) of the food items. v 1.75, 1.25 (a) Find the dot product u v and interpret the result in the context of the problem. (b) Identify the vector operation used to increase the prices by 2.5%. Model It 67. Braking Load A truck with a gross weight of 30,000 (see figure). Assume pounds is parked on a slope of that the only force to overcome is the force of gravity. d d° Weight = 30,000 lb (a) Find the force required to keep the truck from rolling down the hill in terms of the slope d. (b) Use a graphing utility to complete the table 10 d Force d Force (c) Find the force perpendicular to the hill when d 5. 68. Braking Load A sport utility vehicle with a gross weight of 5400 pounds is parked on a slope of Assume that the only force to overcome is the force of gravity. Find the force required to keep the vehicle from rolling down the hill. Find the force perpendicular to the hill. 10. 333202_0604.qxd 12/8/05 10:08 AM Page 469 69. Work Determine the work done by a person lifting a 25-kilogram (245-newton) bag of sugar. Synthesis Section 6.4 Vectors and Dot Products 469 70. Work Determine the work done by a crane lifting a 2400-pound car 5 feet. 71. Work A force of 45 pounds exerted at an angle of 30 above the horizontal is required to slide a table across a floor (see figure). The table is dragged 20 feet. Determine the work done in sliding the table. 45 lb 30° 20 ft 72. Work A tractor pulls a log 800 meters, and the tension in the cable connecting the tractor and log is approximately 1600 kilograms (15,691 newtons). The direction of the 35 above the horizontal. Approximate the work force is done in pulling the log. 73. Work One of the events in a local strongman contest is to pull a cement block 100 feet. One competitor pulls the block by exerting a force of 250 pounds on a rope attached to the block at an angle of with the horizontal (see figure). Find the work done in pulling the block. 30 30˚ 100 ft Not drawn to scale 74. Work A toy wagon is pulled by exerting a force of 25 pounds on a handle that makes a angle with the horizontal (see figure). Find the work done in pulling the wagon 50 feet. 20 20° True or False? the statement is true or false. Justify your answer. In Exercises 75 and 76, determine whether 75. The work done by a constant force line of motion of an object is represented by a vector. acting along the W F 76. A sliding door moves along the line of vector If a force is applied to the door along a vector that is orthogonal to then no work is done. PQ PQ . , \ \ 77. Think About It What is known about v between two nonzero vectors condition? and u , the angle , under each (a) u v 0 u v > 0 78. Think About It What can be said about the vectors and u v < 0 (b) (c) u v under each condition? (a) The projection of onto equals u v u . (b) The projection of onto equals u v 0 . 79. Proof Use vectors to prove that the diagonals of a rhombus are perpendicular. 80. Proof Prove the following. u v 2 u2 v 2 2u v Skills Review In Exercises 81–84, perform the operation and write the result in standard form. 81. 82. 83. 84. 42 24 18 112 3 8 12 96 In Exercises 85–88, find all solutions of the equation in the interval [0, 2. 85. 86. 87. 88. sin 2x 3 sin x 0 sin 2x 2 cos x 0 2 tan x tan 2x cos 2x 3 sin x 2 find the exact value of the and sin u 12 13 (Both and are in Quadrant IV.) v u 89. In Exercises 89–92, trigonometric function given that cos v 24 25. sinu v sinu v cosv u 91. 92. tanu v 90. 333202_0605.qxd 12/5/05 10:45 AM Page 470 470 Chapter 6 Additional Topics in Trigonometry 6.5 Trigonometric Form of a Complex Number What you should learn • Plot complex numbers in the complex plane and find absolute values of complex numbers. • Write the trigonometric forms of complex numbers. • Multiply and divide complex numbers written in trigonometric form. • Use DeMoivre’s Theorem to find powers of complex numbers. n numbers. • Find th roots of complex Why you should learn it You can use the trigonometric form of a complex number to perform operations with complex numbers. For instance, in Exercises 105–112 on page 480, you can use the trigonometric forms of complex numbers to help you solve polynomial equations. The Complex Plane Just as real numbers can be represented by points on the real number line, you can represent a complex number z a bi a, b in a coordinate plane (the complex plane). The horizontal axis as the point is called the real axis and the vertical axis is called the imaginary axis, as shown in Figure 6.44. Imaginary axis 3 2 1 (3, 1) or 3 + i 1 2 3 Real axis −3 −2 −1 (−2, −1) or −2 − i −1 −2 FIGURE 6.44 The absolute value of the complex number between the origin 0, 0 and the point a, b. a bi is defined as the distance Definition of the Absolute Value of a Complex Number The absolute value of the complex number z a bi is a bi a2 b2. If the complex number a bi is a real number (that is, if b 0 ), then this definition agrees with that given for the absolute value of a real number a 0i a2 02 a. Example 1 Finding the Absolute Value of a Complex Number Plot z 2 5i and find its absolute value. Solution The number is plotted in Figure 6.45. It has an absolute value of Imaginary axis 5 4 3 (−2, 5) 29 −4 −3 −2 −1 1 2 3 4 Real axis z 22 52 29. FIGURE 6.45 Now try Exercise 3. 333202_0605.qxd 12/5/05 10:45 AM Page 471 Section 6.5 Trigonometric Form of a Complex Number 471 Imaginary axis Trigonometric Form of a Complex Number (a , b) b r a θ Real axis In Section 2.4, you learned how to add, subtract, multiply, and divide complex numbers. To work effectively with powers and roots of complex numbers, it is helpful to write complex numbers in trigonometric form. In Figure 6.46, consider the nonzero complex number By letting be the angle from the positive real axis (measured counterclockwise) to the line segment connecting the origin and the point you can write a bi. a, b, a r cos and b r sin where r a2 b2. Consequently, you have a bi r cos r sin i FIGURE 6.46 from which you can obtain the trigonometric form of a complex number. Trigonometric Form of a Complex Number The trigonometric form of the complex number z a bi is z rcos i sin where number a r cos , b r sin , r a2 b2, tan ba. and is called an argument of z. The r is the modulus of z, and The trigonometric form of a complex number is also called the polar form. the trigonometric form of a is restricted to the interval Because there are infinitely many choices for complex number is not unique. Normally, 0 ≤ < 2, although on occasion it is convenient to use < 0. , Example 2 Writing a Complex Number in Trigonometric Form Write the complex number z 2 23i in trigonometric form. Solution The absolute value of z is r 2 23i 22 232 16 4 Imaginary axis and the reference angle is given by −3 −2 π 4 3 Real axis 1  z = 4 z = −2 − 2 3 i FIGURE 6.47 −2 −3 − 4 tan b a 23 2 tan3 3 Because choose to be 3. and because 3 43. z 2 23i So, the trigonometric form is lies in Quadrant III, you z rcos i sin 4cos i sin 4 3 4 . 3 See Figure 6.47. Now try Exercise 13. 333202_0605.qxd 12/5/05 10:45 AM Page 472 472 Chapter 6 Additional Topics in Trigonometry Example 3 Writing a Complex Number in Standard Form Write the complex number in standard form a bi. z 8cos i sin 3 3 Te c h n o l o g y A graphing utility can be used to convert a complex number in trigonometric (or polar) form to standard form. For specific keystrokes, see the user’s manual for your graphing utility. Solution Because cos3 1 2 sin3 32, you can write and i sin i 3 3 2 3 z 8cos 221 2 6i. 2 Now try Exercise 35. Multiplication and Division of Complex Numbers The trigonometric form adapts nicely to multiplication and division of complex numbers. Suppose you are given two complex numbers cos 1 i sin 1 and z2 r2 cos 2 i sin 2 . z1 r1 The product of r1r2 r1r2 z1z2 is given by z1 and cos 1 cos z 2 i sin 1 1 cos 2 cos i sin 2 isin sin 1 sin 2 2 1 cos 2 cos 1 sin 2 . Using the sum and difference formulas for cosine and sine, you can rewrite this equation as z1z2 r1r2 cos 1 2 i sin 1 2 . This establishes the first part of the following rule. The second part is left for you to verify (see Exercise 117). Product and Quotient of Two Complex Numbers z1 and Let numbers. i sin 1 i sin 2 cos 1 cos 2 r2 r1 z2 be complex z1z2 z1 z2 r1r2 r1 r2 cos 1 2 i sin 1 2 cos 1 2 i sin 1 2 , z 2 0 Product Quotient Note that this rule says that to multiply two complex numbers you multiply moduli and add arguments, whereas to divide two complex numbers you divide moduli and subtract arguments. 333202_0605.qxd 12/5/05 10:45 AM Page 473 Section 6.5 Trigonometric Form of a Complex Number 473 Example 4 Multiplying Complex Numbers Find the product 2cos z1 z1z2 2 3 Solution of the complex numbers. i sin 2 3 8cos z 2 11 6 i sin 11 6 Te c h n o l o g y z1z2 Some graphing utilities can multiply and divide complex numbers in trigonometric form. If you have access to such a graphing utility, use z1 it to find in Examples 4 and 5. and z1z2 z2 2cos 2 3 16cos i sin i sin 8cos 2 3 11 i sin 6 11 6 11 2 6 3 11 6 Multiply moduli and add arguments. 2 3 5 2 5 2 2 16cos i sin 16cos 2 i sin 160 i1 16i You can check this result by first converting the complex numbers to the standard z1 forms and then multiplying algebraically, as in Section 2.4. 1 3i 43 4i and z2 z1z2 1 3i43 4i 43 4i 12i 43 16i Now try Exercise 47. Example 5 Dividing Complex Numbers Find the quotient z 2 24cos 300 i sin 300 z1 of the complex numbers. 8cos 75 i sin 75 z 2 z1 Solution z1 z2 24cos 300 i sin 300 8cos 75 i sin 75 24 8 cos300 75 i sin300 75 Divide moduli and subtract arguments. 3cos 225 i sin 225 3 i 2 2 2 2 32 2 i 32 2 Now try Exercise 53. 333202_0605.qxd 12/5/05 10:45 AM Page 474 474 Chapter 6 Additional Topics in Trigonome
try Powers of Complex Numbers The trigonometric form of a complex number is used to raise a complex number to a power. To accomplish this, consider repeated use of the multiplication rule. z rcos i sin z 2 rcos i sin rcos i sin r 2cos 2 i sin 2 z3 r 2cos 2 i sin 2rcos i sin r 3cos 3 i sin 3 z4 r 4cos 4 i sin 4 z5 r5cos 5 i sin 5 . . . This pattern leads to DeMoivre’s Theorem, which is named after the French mathematician Abraham DeMoivre (1667–1754). DeMoivre’s Theorem z rcos i sin If then is a complex number and n is a positive integer, zn rcos i sin n rncos n i sin n. Example 6 Finding Powers of a Complex Number Use DeMoivre’s Theorem to find 1 3i12 . Solution First convert the complex number to trigonometric form using r 12 32 2 and arctan 3 1 2 . 3 So, the trigonometric form is z 1 3i 2cos 2 3 i sin 2 . 3 Then, by DeMoivre’s Theorem, you have Historical Note Abraham DeMoivre (1667–1754) is remembered for his work in probability theory and DeMoivre’s Theorem. His book The Doctrine of Chances (published in 1718) includes the theory of recurring series and the theory of partial fractions. 1 3i12 2cos 12 2 3 i sin 212cos 2 3 122 3 4096cos 8 i sin 8 40961 0 4096. i sin 122 3 Now try Exercise 75. 333202_0605.qxd 12/5/05 10:45 AM Page 475 Section 6.5 Trigonometric Form of a Complex Number 475 Roots of Complex Numbers Recall that a consequence of the Fundamental Theorem of Algebra is that a polynomial equation of degree has solutions in the complex number system. has six solutions, and in this particular case you can find So, the equation the six solutions by factoring and using the Quadratic Formula. x6 1 n n x 6 1 x3 1x3 1 x 1x 2 x 1x 1x 2 x 1 0 Consequently, the solutions are x ±1, x 1 ± 3i 2 , and x 1 ± 3 i 2 . Each of these numbers is a sixth root of 1. In general, the nth root of a complex number is defined as follows. Definition of the nth Root of a Complex Number The complex number is an nth root of the complex number u a bi z if z un a bin. n Exploration The th roots of a complex number are useful for solving some polynomial equations. For instance, explain how you can use DeMoivre’s Theorem to solve the polynomial equation x4 16 0. 16 [Hint: Write as 16cos i sin . ] To find a formula for an th root of a complex number, let be an th root n n u of where z, u scos i sin and z rcos i sin . By DeMoivre’s Theorem and the fact that un z, you have sn cos n i sin n rcos i sin . Taking the absolute value of each side of this equation, it follows that Substituting back into the previous equation and dividing by you get r, sn r. cos n i sin n cos i sin . So, it follows that cos n cos and sin n sin . Because both sine and cosine have a period of solutions if and only if the angles differ by a multiple of must exist an integer such that k 2, these last two equations have Consequently, there 2. n 2k 2k n . By substituting this value of stated on the following page. into the trigonometric form of you get the result u, 333202_0605.qxd 12/5/05 10:45 AM Page 476 476 Chapter 6 Additional Topics in Trigonometry Finding nth Roots of a Complex Number For a positive integer n exactly distinct th roots given by 2k n n nrcos 2k n the complex number i sin n, z rcos i sin has where k 0, 1, 2, . . . , n 1. When exceeds k n 1, the roots begin to repeat. For instance, if k n, the angle 2n n n 2 is coterminal with n, which is also obtained when n k 0. z nr, The formula for the th roots of a complex number has a nice geometrical n th roots of all interpretation, as shown in Figure 6.48. Note that because the nr have the same magnitude with center at they all lie on a circle of radius the origin. Furthermore, because successive th roots have arguments that differ by 2n, You have already found the sixth roots of 1 by factoring and by using the Quadratic Formula. Example 7 shows how you can solve the same problem with the formula for th roots. roots are equally spaced around the circle. the n n n z Example 7 Finding the nth Roots of a Real Number Find all the sixth roots of 1. Solution First write 1 in the trigonometric form root formula, with 61cos r 1, 0 2k 6 0 2k 6 i sin n 6 and 1 1cos 0 i sin 0. the roots have the form k 3 cos i sin k 3 Then, by the th n . So, for 1, 2, 3, 4, and 5, the sixth roots are as follows. (See Figure 6.49.) k 0, cos 0 i sin 0 1 cos i sin 3 1 2 3 2 i Increment by 2 n 2 6 3 3 2 3 cos 2 3 i sin 1 2 cos i sin sin i sin 3 2 i 3 2 i 3 2 i cos cos Now try Exercise 97. Imaginary axis rn π 2 n π 2 n Real axis FIGURE 6.48 Imaginary axis − 1 + 0i −1 1 + 0i 1 Real axis − − 1 2 3 2 i FIGURE 6.49 − 1 2 3 2 i 333202_0605.qxd 12/5/05 10:45 AM Page 477 Section 6.5 Trigonometric Form of a Complex Number 477 In Figure 8.49, notice that the roots obtained in Example 7 all have a magnitude of 1 and are equally spaced around the unit circle. Also notice that the complex roots occur in conjugate pairs, as discussed in Section 2.5. The distinct th roots of 1 are called the nth roots of unity. n n Example 8 Finding the nth Roots of a Complex Number Find the three cube roots of z 2 2i. Solution z Because lies in Quadrant II, the trigonometric form of z is z 2 2i 8 cos 135 i sin 135. arctan22 135 By the formula for th roots, the cube roots have the form 68 cos Finally, for 68cos n 135 360k 3 k 0, 1, and 2, 135 3600 3 68cos 135 3601 3 1 + i 68cos 135 3602 3 Real axis 1 2 See Figure 6.50. i sin 135º 360k 3 . you obtain the roots i sin 135 3600 3 2cos 45 i sin 45 i sin 135 3601 3 1 i 2cos 165 i sin 165 i sin 135 3602 3 1.3660 0.3660i 2cos 285 i sin 285 0.3660 1.3660i. 0.3660 − 1.3660i Now try Exercise 103. W RITING ABOUT MATHEMATICS A Famous Mathematical Formula The famous formula Imaginary axis −1.3660 + 0.3660i 1 −1 −2 −2 FIGURE 6.50 is Note in Example 8 that the z absolute value of r 2 2i 22 22 8 and the angle tan b a 2 2 1. is given by ea bi e acos b i sin b is called Euler’s Formula, after the Swiss mathematician Leonhard Euler (1707–1783). Although the interpretation of this formula is beyond the scope of this text, we decided to include it because it gives rise to one of the most wonderful equations in mathematics. ei 1 0 This elegant equation relates the five most famous numbers in mathematics—0, 1, e,, and —in a single equation. Show how Euler’s Formula can be used to derive this equation. i 333202_0605.qxd 12/8/05 10:09 AM Page 478 478 Chapter 6 Additional Topics in Trigonometry 6.5 Exercises VOCABULARY CHECK: Fill in the blanks. 1. The ________ ________ of a complex number and the point a, b. 2. The ________ ________ of a complex number is the ________ of and where z r 3. ________ Theorem states that if zn r ncos n i sin n. then a bi is the distance between the origin 0, 0 z a bi is given by z r cos i sin , is the ________ of z r cos i sin z. is a complex number and n is a positive integer, 4. The complex number u a bi is an ________ ________ of the complex number z if z un a bin. PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. In Exercises 1–6, plot the complex number and find its absolute value. 1. 3. 5. 7i 4 4i 6 7i 2. 4. 6. 7 5 12i 8 3i In Exercises 7–10, write the complex number in trigonometric form. 7. Imaginary axis 8. Imaginary axi s z = 3i 4 3 2 1 −2 −1 1 2 Real axis z = 2− 4 2 −6 −4 −2 2 −4 Real axis 9. Imaginary axis Real axis 3 = 3 − i z −3 10. Imaginary axis 3 i = 1 + 3 − z 25. 27. 29. 3 i 5 2i 8 53 i 26. 28. 30. 1 3i 8 3i 9 210 i In Exercises 31– 40, represent the complex number graphically, and find the standard form of the number. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 3cos 120 i sin 120 5cos 135 i sin 135 cos 300 i sin 300 3 2 cos 225 i sin 225 1 4 3 3 3.75cos 4 4 5 5 12 12 6cos i sin i sin 2 i sin 8cos 2 7cos 0 i sin 0 3cos18 45 i sin18 45 6cos230º 30 i sin230º 30 −3 −2 −1 Real axis In Exercises 41– 44, use a graphing utility to represent the complex number in standard form. In Exercises 11–30, represent the complex number graphically, and find the trigonometric form of the number. 11. 13. 15. 17. 19. 3 3i 3 i 21 3 i 5i 7 4i 21. 7 23. 3 3 i 12. 14. 16. 18. 20. 2 2i 4 43 i 3 i 5 2 4i 3 i 22. 4 24. 22 i 9 i sin 5cos 3cos 165.5 i sin 165.5 9cos 58º i sin 58º 9 41. 43. 44. 42. 10cos 2 5 i sin 2 5 In Exercises 45 and 46, represent the powers graphically. Describe the pattern. z, z2, z3, and z 4 45. z 2 2 1 i 46. z 1 2 1 3 i 333202_0605.qxd 12/8/05 10:09 AM Page 479 Section 6.5 Trigonometric Form of a Complex Number 479 In Exercises 47–58, perform the operation and leave the result in trigonometric form. 6cos 4 4cos i sin 12 3 i sin 4 cos 60 i sin 60 12 3 4 2cos 47. i sin 4 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 3 3 3 cos 3 i sin 4 cos 140 i sin 1402 5 3 0.5cos 100 i sin 100 0.8cos 300 i sin 300 0.45cos 310i sin 310 0.60cos 200 i sin 200 cos 5 i sin 5cos 20 i sin 20 cos 50 i sin 50 cos 20 i sin 20 2cos 120 i sin 120 4cos 40 i sin 40 cos53 i sin53 cos i sin 5cos 4.3 i sin 4.3 4cos 2.1 i sin 2.1 12cos 52 i sin 52 3cos 110 i sin 110 6cos 40 i sin 40 7cos 100 i sin 100 In Exercises 59–66, (a) write the trigonometric forms of the complex numbers, (b) perform the indicated operation using the trigonometric forms, and (c) perform the indicated operation using the standard forms, and check your result with that of part (b). 60. 62. 64. 3 i1 i 41 3 i 1 3 i 6 3i 59. 61. 63. 65. 66. 2 2i1 i 2i1 i 3 4i 1 3 i 5 2 3i 4i 4 2i In Exercises 67–70, sketch the graph of all complex numbers z satisfying the given condition. 67. 68. 69. 70. z 2 z 3 6 5 4 71. 73. In Exercises 71–88, use DeMoivre’s Theorem to find the indicated power of the complex number. Write the result in standard form. 1 i5 1 i10 23 i7 5cos 20 i sin 203 3cos 150 i sin 1504 cos 2 2i6 3 2i8 41 3 i3 i sin 77. 75. 79. 74. 76. 72. 78. 2 2 4 i sin 12 4 8 2cos 5cos 3.2 i sin 3.24 cos 0 i sin 020 3 2i5 5 4i3 3cos 15 i sin 154 2cos 10 i sin 108 5 2cos 10 6 2cos i sin i sin 10 8 8 80. 81. 82. 83. 84. 85. 86. 87. 88. In Exercises 89–104, (a) use the theorem on page 476 to find the indicated roots of the complex n
umber, (b) represent each of the roots graphically, and (c) write each of the roots in standard form. 89. Square roots of 90. Square roots of 91. Cube roots of 5cos 120 i sin 120 16cos 60 i sin 60 8cos i sin 2 3 5 6 2 3 5 6 92. Fifth roots of 93. Square roots of 94. Fourth roots of 95. Cube roots of 96. Cube roots of i sin 32cos 25i 625i 1 3 i 125 2 421 i 97. Fourth roots of 16 98. Fourth roots of i 99. Fifth roots of 1 100. Cube roots of 1000 125 4 102. Fourth roots of 101. Cube roots of 103. Fifth roots of 1281 i 104. Sixth roots of 64i 333202_0605.qxd 12/8/05 10:09 AM Page 480 480 Chapter 6 Additional Topics in Trigonometry In Exercises 105–112, use the theorem on page 476 to find all the solutions of the equation and represent the solutions graphically. 105. 106. 107. 108. 109. 110. 111. 112. x 4 i 0 x3 1 0 x5 243 0 x3 27 0 x 4 16i 0 x6 64i 0 x3 1 i 0 x 4 1 i 0 Synthesis True or False? the statement is true or false. Justify your answer. In Exercises 113–116, determine whether 113. Although the square of the complex number is given by the absolute value of the complex number bi bi2 b2, z a bi is defined as a bi a2 b2. 114. Geometrically, the th roots of any complex number are all equally spaced around the unit circle centered at the origin. n z Graphical Reasoning graph of the roots of a complex number. In Exercises 123 and 124, use the (a) Write each of the roots in trigonometric form. (b) Identify the complex number whose roots are given. (c) Use a graphing utility to verify the results of part (b). 123. Imaginary axis 30° 2 2 30° 2 1 −1 Real axis 124. Imaginary axis 45° 45° 3 3 3 45° 3 45° Real axis Skills Review In Exercises 125–130, solve the right triangle shown in the figure. Round your answers to two decimal places. 115. The product of two complex numbers z1 r1 cos 1 i sin 1 and z2 r2 cos 2 i sin 2 0 116. By DeMoivre’s Theorem, is zero only when r1 . and/or 0. r2 4 6 i8 cos32 i sin86. z1 117. Given two complex numbers 0, i sin 2 cos 2 z2 z2 , cos i sin r1 1 1 show that r2 r1 r2 z1 z 2 cos 1 2 i sin 1 2 . B a C 125. 127. 129. and a 8 b 112.6 A 22, A 30, A 42 15, c 11.2 c b 126. 128. 130. A a 33.5 b 211.2 B 66, B 6, B 81 30, c 6.8 Harmonic Motion In Exercises 131–134, for the simple harmonic motion described by the trigonometric function, find the maximum displacement and the least positive value of for which t d 0. 118. Show that conjugate of z r cos i sin z r cos i sin . is the complex 119. Use the trigonometric forms of and z z in Exercise 118 to find (a) zz and (b) zz, z 0. 120. Show that the negative of z r cos i sin . 1 3 i 121. Show that 1 2 2141 i is a sixth root of 1. 2. is a fourth root of 122. Show that z r cos i sin 131. d 16 cos t 4 133. d 1 16 sin t 5 4 132. 134. d 1 8 d 1 12 cos 12t sin 60t is In Exercises 135 and 136, write the product as a sum or difference. 135. 6 sin 8 cos 3 136. 2 cos 5 sin 2 333202_060R.qxd 12/5/05 10:48 AM Page 481 6 Chapter Summary What did you learn? Section 6.1 Use the Law of Sines to solve oblique triangles (AAS, ASA, or SSA) (p. 430, 432). Find areas of oblique triangles (p. 434). Use the Law of Sines to model and solve real-life problems (p. 435). Section 6.2 Use the Law of Cosines to solve oblique triangles (SSS or SAS) (p. 439). Use the Law of Cosines to model and solve real-life problems (p. 441). Use Heron's Area Formula to find areas of triangles (p. 442). Section 6.3 Represent vectors as directed line segments (p. 447). Write the component forms of vectors (p. 448). Perform basic vector operations and represent vectors graphically (p. 449). Write vectors as linear combinations of unit vectors (p. 451). Find the direction angles of vectors (p. 453). Use vectors to model and solve real-life problems (p. 454). Section 6.4 Find the dot product of two vectors and use the properties of the dot product (p. 460). Find the angle between two vectors and determine whether two vectors are orthogonal (p. 461). Write vectors as sums of two vector components (p. 463). Use vectors to find the work done by a force (p. 466). Section 6.5 Plot complex numbers in the complex plane and find absolute values of complex numbers (p. 470). Write the trigonometric forms of complex numbers (p. 471). Multiply and divide complex numbers written in trigonometric form (p. 472). Use DeMoivre’s Theorem to find powers of complex numbers (p. 474) Find nth roots of complex numbers (p. 475). Chapter Summary 481 Review Exercises 1–12 13–16 17–20 21–28 29–32 33–36 37, 38 39–44 45–56 57–62 63–68 69–72 73–80 81–88 89–92 93–96 97–100 101–104 105, 106 107–110 111–118 333202_060R.qxd 12/5/05 10:48 AM Page 482 482 Chapter 6 Additional Topics in Trigonometry 6 Review Exercises 6.1 In Exercises 1–12, use the Law of Sines to solve (if possible) the triangle. If two solutions exist, find both. Round your answers to two decimal places. 1. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. B 2. 71° a = 8 A C c 35° b A B c 22° 121° a = 17 C b B 72, B 10, A 16, A 95, A 24, B 64, B 150, B 150, A 75, B 25, C 82, C 20, B 98, B 45, C 48, C 36, b 30, a 10, a 51.2, a 6.2, b 54 c 33 c 8.4 c 104.8 b 27.5 a 367 c 10 b 3 b 33.7 b 4 In Exercises 13–16, find the area of the triangle having the indicated angle and sides. c 7 c 8 13. 14. 15. 16. A 27, B 80, C 123, A 11, b 5, a 4, a 16, b 22, b 5 c 21 17. Height From a certain distance, the angle of elevation to 17. At a point 50 meters closer to Approximate the the top of a building is the building, the angle of elevation is height of the building. 31. 18. Geometry Find the length of the side w of the parallelogram. w 140° 12 16 19. Height A tree stands on a hillside of slope from the horizontal. From a point 75 feet down the hill, the angle of elevation to the top of the tree is (see figure). Find the height of the tree. 45 28 5 ft 7 45° 28° FIGURE FOR 19 20. River Width A surveyor finds that a tree on the opposite bank of a river, flowing due east, has a bearing of N E from a certain point and a bearing of N W from a point 400 feet downstream. Find the width of the river. 22 30 15 In Exercises 21–28, use the Law of Cosines to solve 6.2 the triangle. Round your answers to two decimal places. 21. 22. 23. 24. 25. 26. 27. 28. c 10 a 5, a 80, a 2.5, a 16.4, B 110, B 150, C 43, A 62, b 8, b 60, b 5.0, b 8.8, a 4, a 10, a 22.5, b 11.34, c 100 c 4.5 c 12.2 c 4 c 20 b 31.4 c 19.52 29. Geometry The lengths of the diagonals of a parallelogram are 10 feet and 16 feet. Find the lengths of the sides of the 28. parallelogram if the diagonals intersect at an angle of 30. Geometry The lengths of the diagonals of a parallelogram are 30 meters and 40 meters. Find the lengths of the sides of the parallelogram if the diagonals intersect at an angle of 34. 31. Surveying To approximate the length of a marsh, a B. Then C and walks 300 meters to point (see surveyor walks 425 meters from point 65 the surveyor turns figure). Approximate the length of the marsh. to point AC A B 65° 300 m 425 m C A 333202_060R.qxd 12/5/05 10:48 AM Page 483 32. Navigation Two planes leave Raleigh-Durham Airport at approximately the same time. One is flying 425 miles per , and the other is flying 530 miles hour at a bearing of 67. per hour at a bearing of Draw a figure that gives a visual representation of the problem and determine the distance between the planes after they have flown for 2 hours. 355 In Exercises 33–36, use Heron’s Area Formula to find the area of the triangle. 33. 34. 35. 36. a 4, a 15, a 12.3, a 38.1, b 5, b 8, c 7 c 10 b 15.8, b 26.7, c 3.7 c 19.4 6.3 In Exercises 37 and 38, show that u v. 37. y 6 4 − ( 2, 1) (4, 6) u (6, 3) v −2 −2 (0, 2)− x 6 38. y (−3, 2) u 4 2 −4 (−1, −4) (1, 4) v x 2 4 (3, − 2) In Exercises 39– 44, find the component form of the vector v satisfying the conditions. 39. y 6 4 2 (−5, 4) v −4 −2 x (2, −1) 40. y 6 4 2 −2 ( 7 26, ) v (0, 1) 2 4 6 x 41. Initial point: terminal point: terminal point: 7, 3 15, 9 43. 42. Initial point: v 8, v 1 2, 44. 0, 10; 1, 5; 120 225 In Exercises 45–52, find (a) (d) 2v 5u. u v, (b) u v, (c) 3u , and 45. 46. 47. 48. 49. u 1, 3, v 3, 6 u 4, 5, v 0, 1 u 5, 2, v 4, 4 u 1, 8, v 3, 2 u 2i j, v 5i 3j Review Exercises 483 50. 51. 52. u 7i 3j, v 4i j u 4i, v i 6j u 6j, v i j and In Exercises 53–56, find the component form of sketch the specified vector operations geometrically, where v 1 i 3j. u 6i 5j and w 53. 54. 55. 56. w 2u v w 4u 5v w 3v w 1 2v In Exercises 57– 60, write vector as a linear combination of the standard unit vectors and u j. i 57. 58. 59. u 3, 4 u 6, 8 u has initial point 60. u has initial point 3, 4 2, 7 and terminal point and terminal point 9, 8. 5, 9. In Exercises 61 and 62, write the vector vcos i sin j. v in the form 61. 62. v 10i 10j v 4i j In Exercises 63–68, find the magnitude and the direction angle of the vector v. 63. 64. 65. 66. 67. 68. v 7cos 60i sin 60j v 3cos 150i sin 150j v 5i 4j v 4i 7j v 3i 3j v 8i j 69. Resultant Force Forces with magnitudes of 85 pounds and 50 pounds act on a single point. The angle between the forces is Describe the resultant force. 15. 70. Rope Tension A 180-pound weight is supported by two ropes, as shown in the figure. Find the tension in each rope. 30° 30° 180 lb 333202_060R.qxd 12/8/05 10:10 AM Page 484 484 Chapter 6 Additional Topics in Trigonometry 71. Navigation An airplane has an airspeed of 430 miles per The wind velocity is 35 miles per E. Find the resultant speed hour at a bearing of 30 hour in the direction of N and direction of the airplane. 135. 72. Navigation An airplane has an airspeed of 724 kilometers per hour at a bearing of The wind velocity is 32 kilometers per hour from the west. Find the resultant speed and direction of the airplane. 30. 6.4 In Exercises 73–76, find the dot product of u. and v. 73. 75. u 6, 7 v 3, 9 u 3i 7j v 11i 5j 74. 76. u 7, 12 v 4, 14 u 7i 2j v 16i 12j In Exercises 77– 80, use the vectors and to find the indicated quantity. State whether u <3, 4> the result is a vector or a scalar. 77. v <2, 1> 2u u v2 uu v 3u v 78. 80. 79. In Exercises 81– 84, find the angle between
the vectors. u cos i sin v cos i sin cos 45i sin 45j v cos 300i sin 300j u 22, 4, u 3, 3, v 4, 33 v 2, 1 81. 82. 83. 84. In Exercises 85–88, determine whether orthogonal, parallel, or neither. u and v are 85. 87. u 3, 8 v 8, 3 u i v i 2j 86. 88. u 1 4, 1 2 v 2, 4 u 2i j v 3i 6j In Exercises 89–92, find the projection of onto Then write as the sum of two orthogonal vectors, one of which is u projvu. v. u Work In Exercises 93 and 94, find the work done in Q if the magnitude and moving a particle from v. direction of the force are given by to P 93. 94. P 5, 3, Q 8, 9, v 2, 7 P 2, 9, Q 12, 8, v 3i 6j 95. Work Determine the work done by a crane lifting an 18,000-pound truck 48 inches. 96. Work A mover exerts a horizontal force of 25 pounds on a crate as it is pushed up a ramp that is 12 feet long and inclined at an angle of above the horizontal. Find the work done in pushing the crate. 20 In Exercises 97–100, plot the complex number and 6.5 find its absolute value. 97. 99. 100. 7i 5 3i 10 4i 98. 6i In Exercises 101–104, write the complex number in trigonometric form. 101. 103. 5 5i 33 3i 102. 104. 5 12i 7 In Exercises 105 and 106, (a) write the two complex (b) use the numbers 0. z2 where trigonometric forms to find in trigonometric form, and z1/ z2, and 105. 106. z1 z1 23 2i, 31 i, z2 z2 z1z2 10i 23 i In Exercises 107–110, use DeMoivre’s Theorem to find the indicated power of the complex number. Write the result in standard form. 5cos i sin 107. 4 12 4 15 5 i sin 12 4 2cos 15 2 3i 6 1 i 8 108. 109. 110. In Exercises 111–114, (a) use the theorem on page 476 to find the indicated roots of the complex number, (b) represent each of the roots graphically, and (c) write each of the roots in standard form. v 8, 2 111. Sixth roots of 89. 90. 91. 92. u 4, 3, u 5, 6, u 2, 7, u 3, 5, v 10, 0 v 1, 1 v 5, 2 729i 256i 112. Fourth roots of 113. Cube roots of 8 114. Fifth roots of 1024 333202_060R.qxd 12/8/05 10:10 AM Page 485 In Exercises 115–118, use the theorem on page 476 to find all solutions of the equation and represent the solutions graphically. 129. Give a geometric description of the scalar multiple ku of the vector u, for k > 0 and for k < 0. 130. Give a geometric description of the sum of the vectors u Review Exercises 485 115. 116. 117. 118. x4 81 0 x5 32 0 x3 8i 0 x3 1x2 1 0 Synthesis and v. Graphical Reasoning graph of the roots of a complex number. In Exercises 131 and 132, use the (a) Write each of the roots in trigonometric form. (b) Identify the complex number whose roots are given. (c) Use a graphing utility to verify the results of part (b). True or False? the statement is true or false. Justify your answer. In Exercises 119–123, determine whether 131. Imaginary axis 119. The Law of Sines is true if one of the angles in the triangle is a right angle. 120. When the Law of Sines is used, the solution is always unique. 121. If u is a unit vector in the direction of v, then v v u. 122. If v a i bj 0, then a b. 123. x 3 i is a solution of the equation x2 8i 0. 124. State the Law of Sines from memory. 125. State the Law of Cosines from memory. 126. What characterizes a vector in the plane? 127. Which vectors in the figure appear to be equivalent? y 2 4 −2 −2 4 60° 60° 4 132. Imaginary axis 3 30° 4 60° 4 4 60° 3 30° 4 Real axis Real axis C B A x D E 133. The figure shows z1 and Describe z2. z1z2 and z1 z2. Imaginary axis z2 1 θ z1 θ −1 1 Real axis u 128. The vectors and have the same magnitudes in the two figures. In which figure will the magnitude of the sum be greater? Give a reason for your answer. v (a) y (b) y v u v x u x 134. One of the fourth roots of a complex number z is shown in the figure. (a) How many roots are not shown? (b) Describe the other roots. Imaginary axis 1 −1 1 z 30° Real axis 333202_060R.qxd 12/8/05 10:11 AM Page 486 486 Chapter 6 Additional Topics in Trigonometry 6 Chapter Test Take this test as you would take a test in class. When you are finished, check your work against the answers given in the back of the book. In Exercises 1–6, use the information to solve the triangle. If two solutions exist, find both solutions. Round your answers to two decimal places. 1. 3. 5. A 24, A 24, B 100, B 68, a 11.2, a 15, a 12.2 b 13.4 b 23 2. 4. 6. B 104, a 4.0, C 123, C 33, a 18.1 b 7.3, a 41, c 12.4 b 57 7. A triangular parcel of land has borders of lengths 60 meters, 70 meters, and 82 meters. Find the area of the parcel of land. 240 mi C 37° B 8. An airplane flies 370 miles from point to point with a bearing of B 240 miles from point and bearing from point A B to point with a bearing of A C to point C. 24. It then flies (see figure). Find the distance 37 In Exercises 9 and 10, find the component form of the vector conditions. v satisfying the given 370 mi 10. Magnitude of v: 9. Initial point of v: 3, 7; v 12; terminal point of v: 11, 16 direction of v: u 3, 5 24° A FIGURE FOR 8 In Exercises 11–13, its graph. u <3, 5> and v <7, 1>. Find the resultant vector and sketch 11. u v 12. u v 13. 5u 3v 14. Find a unit vector in the direction of u 4, 3. 15. Forces with magnitudes of 250 pounds and 130 pounds act on an object at angles of , respectively, with the -axis. Find the direction and magnitude of the x and 45 60 resultant of these forces. 16. Find the angle between the vectors u 1, 5 and v 3, 2. 17. Are the vectors 18. Find the projection of orthogonal vectors. u 6, 10 and u 6, 7 v 2, 3 onto orthogonal? v 5, 1. Then write as the sum of two u 19. A 500-pound motorcycle is headed up a hill inclined at What force is required to 12. keep the motorcycle from rolling down the hill when stopped at a red light? 20. Write the complex number 21. Write the complex number z 5 5i z 6cos 120 i sin 120 in trigonometric form. in standard form. In Exercises 22 and 23, use DeMoivre’s Theorem to find the indicated power of the complex number. Write the result in standard form. 8 3cos 3 3i6 i sin 23. 22. 7 6 7 6 24. Find the fourth roots of 2561 3 i. 25. Find all solutions of the equation x3 27i 0 and represent the solutions graphically. 333202_060R.qxd 12/5/05 10:48 AM Page 487 6 Cumulative Test for Chapter 4–6 Cumulative Test for Chapters 4–6 487 y 4 −3 −4 FIGURE FOR 7 Take this test to review the material from earlier chapters. When you are finished, check your work against the answers given in the back of the book. 1. Consider the angle 120. (a) Sketch the angle in standard position. (b) Determine a coterminal angle in the interval 0, 360. (c) Convert the angle to radian measure. (d) Find the reference angle . (e) Find the exact values of the six trigonometric functions of . 2. Convert the angle 2.35 radians to degrees. Round the answer to one decimal place. 3. Find cos if tan 4 3 and sin < 0. x 1 3 In Exercises 4 –6, sketch the graph of the function. (Include two full periods.) 4. f x 3 2 sin x 5. gx 1 2 tanx 2 6. hx secx 7. Find a, b, graph in the figure. c and such that the graph of the function hx a cosbx c matches the 8. Sketch the graph of the function f x 1 2x sin x over the interval 3 ≤ x ≤ 3. In Exercises 9 and 10, find the exact value of the expression without using a calculator. 9. tanarctan 6.7 10. tanarcsin 3 5 11. Write an algebraic expression equivalent to 12. Use the fundamental identities to simplify: sinarccos 2x. cos 2 x csc x. 13. Subtract and simplify: sin 1 cos cos sin 1 . In Exercises 14 –16, verify the identity. 14. 15. 16. cot2 sec2 1 1 sinx y sinx y sin2 x sin2 y sin2 x cos2 x 1 8 1 cos 4x In Exercises 17 and 18, find all solutions of the equation in the interval [0, 2. 17. 18. 2 cos2 cos 0 3 tan cot 0 19. Use the Quadratic Formula to solve the equation in the interval 0, 2: sin2 x 2 sin x 1 0. sin u 12 13, cos v 3 5, and angles u and v are both in Quadrant I, find 20. Given that tanu v. tan 1 2, 21. If find the exact value of tan2. 333202_060R.qxd 12/5/05 10:48 AM Page 488 488 Chapter 6 Additional Topics in Trigonometry C b a A c B FIGURE FOR 25–28 22. If tan 4 3 , find the exact value of sin . 2 23. Write the product 5 sin 3 4 cos 7 4 as a sum or difference. 24. Write cos 8x cos 4x as a product. In Exercises 25–28, use the information to solve the triangle shown in the figure. Round your answers to two decimal places. 25. 26. 27. 28. A 30, a 9, b 8 A 30, b 8, c 10 A 30, a 4, C 90, b 8, c 9 b 10 29. Two sides of a triangle have lengths 7 inches and 12 inches. Their included angle measures 60. Find the area of the triangle. 30. Find the area of a triangle with sides of lengths 11 inches, 16 inches, and 17 inches. 31. Write the vector u 3, 5 as a linear combination of the standard unit vectors i and j. 32. Find a unit vector in the direction of v i j. u 3i 4j for 33. Find u v 34. Find the projection of orthogonal vectors. and u 8, 2 v i 2j. onto v 1, 5. Then write u as the sum of two 35. Write the complex number 36. Find the product of 2 2i in trigonometric form. 4cos 30 i sin 306cos 120 i sin 120. Write the answer in standard form. 37. Find the three cube roots of 1. 38. Find all the solutions of the equation x5 243 0. 39. A ceiling fan with 21-inch blades makes 63 revolutions per minute. Find the angular speed of the fan in radians per minute. Find the linear speed of the tips of the blades in inches per minute. 40. Find the area of the sector of a circle with a radius of 8 yards and a central angle of 5 feet 114. 41. From a point 200 feet from a flagpole, the angles of elevation to the bottom and top respectively. Approximate the height of the flag to the 16 45 18, and of the flag are nearest foot. 12 feet FIGURE FOR 42 42. To determine the angle of elevation of a star in the sky, you get the star in your line of vision with the backboard of a basketball hoop that is 5 feet higher than your eyes (see figure). Your horizontal distance from the backboard is 12 feet. What is the angle of elevation of the star? 43. Write a model for a particle in simple harmonic motion with a displacement of 4 inches and a period of 8 seconds. 44. An airplane’s velocity with respect to the air is 500 kilometers
per hour, with a The wind at the altitude of the plane has a velocity of 50 kilometers E. What is the true direction of the plane, and what bearing of 60 per hour with a bearing of N is its speed relative to the ground? 30. 45. A force of 85 pounds exerted at an angle of above the horizontal is required to slide an object across a floor. The object is dragged 10 feet. Determine the work done in sliding the object. 60 333202_060R.qxd 12/5/05 10:48 AM Page 489 Proofs in Mathematics Law of Tangents Besides the Law of Sines and the Law of Cosines, there is also a Law of Tangents, which was developed by Francois Vi`ete (1540–1603). The Law of Tangents follows from the Law of Sines and the sum-to-product formulas for sine and is defined as follows. a b a b tanA B2 tanA B2 The Law of Tangents can be used to solve a triangle when two sides and the included angle are given (SAS). Before calculators were invented, the Law of Tangents was used to solve the SAS case instead of the Law of Cosines, because computation with a table of tangent values was easier. C b a A c B A is acute. C b a A c B A is obtuse. Law of Sines If ABC (p. 430) is a triangle with sides a, b, and c, then a sin A b sin B c sin is acute. A is obtuse. Proof Let h be the altitude of either triangle found in the figure above. Then you have sin A h b sin B h a or or h b sin A h a sin B. Equating these two values of h, you have a sin B b sin A or a sin A b sin B . and sin B 0 sin A 0 0 or B (extended in the obtuse triangle), as shown at the left. Then you have In a similar manner, construct an altitude from vertex because no angle of a triangle can have a to 180. Note that measure of AC side sin A h c sin C h a or or h c sin A h a sin C. Equating these two values of h, you have a sin C c sin A or a sin A c sin C . By the Transitive Property of Equality you know that a sin A b sin B c sin C . So, the Law of Sines is established. 489 333202_060R.qxd 12/5/05 10:48 AM Page 490 Law of Cosines (p. 439) Standard Form a2 b2 c2 2bc cos A b2 a2 c2 2ac cos B c2 a2 b2 2ab cos C Alternative Form cos A b2 c2 a2 2bc cos B a2 c2 b2 2ac cos C a2 b2 c2 2ab Proof To prove the first formula, consider the top triangle at the left, which has three has acute angles. Note that vertex x, y, coordinates is the distance C from vertex has coordinates y b sin A. it follows that Furthermore, Because c, 0. and C B a x b cos A where B, to vertex a x c2 y 02 a2 x c2 y 02 a2 b cos A c2 b sin A2 a2 b2 cos2 A 2bc cos A c2 b2 sin2 A a2 b2sin2 A cos2 A c2 2bc cos A a2 b2 c2 2bc cos A. To prove the second formula, consider the bottom triangle at the left, which also A C has three acute angles. Note that vertex has coordinates is the has coordinates distance from vertex y a sin B. x a cos B x, y, C it follows that Furthermore, Because c, 0. and A, b where to vertex b x c2 y 02 b2 x c2 y 02 b2 a cos B c2 a sin B2 b2 a2 cos2 B 2ac cos B c2 a2 sin2 B b2 a2sin2 B cos2 B c2 2ac cos B b2 a2 c2 2ac cos B. A similar argument is used to establish the third formula. Distance Formula Square each side. Substitute for x and y. Expand. Factor out b2. sin2 A cos2 A 1 Distance Formula Square each side. Substitute for x and y. Expand. Factor out a2. sin2 B cos2 = ( , 0c, 0) 490 333202_060R.qxd 12/5/05 10:48 AM Page 491 a, b, and c, the area of the triangle is Heron’s Area Formula Given any triangle with sides of lengths Area ss as bs c (p. 442) where s a b c 2 . Proof From Section 6.1, you know that bc sin A b2c2 sin2 A Area 1 2 Area2 1 4 Area 1 1 1 4 4 b2c2 sin2 A b2c21 cos2 A Formula for the area of an oblique triangle Square each side. Take the square root of each side. Pythagorean Identity bc1 cos A1 2 bc1 cos A. Factor. 2 Using the Law of Cosines, you can show that bc1 cos and 1 2 bc1 cos A a b c 2 a b c 2 . Letting s a b c2, these two equations can be rewritten as and 1 2 1 2 bc1 cos A ss a bc1 cos A s bs c. By substituting into the last formula for area, you can conclude that Area ss as bs c. 491 333202_060R.qxd 12/8/05 9:34 AM Page 492 be vectors in the plane or in space and let be a scalar. c 1. Properties of the Dot Product Let v,u, w and . 5. cu v cu v u cv (p. 460) 2. 4. 0 v 0 v v v2 0 0, 0, and let c be a scalar. u2w2 u v u w w w1, w2 v2u2 , v u w2 u2w2 w2 v2 u2v2 u1w1 22 2 v2 v2 Proof Let 1. 2. 3. 4. 5. , u u1, u2 v v1, v2 u v u1v1 u2v2 0 v2 0 v 0 v1 u v w u v1 , v1u1 0 w1, v2 u2 v1 u1 u1v1 u1v1 w1 u1w1 u2v2 v v v1 cu v cu1, u2 2 v2 2 v1 v1, v2 u2v2 cu2 v2 v1, v2 cu1v1 cu1 v1 cu1, cu2 cu v Angle Between Two Vectors (p. 461) If is the angle between two nonzero vectors and u v, then cos u v u v. Proof Consider the triangle determined by vectors u, v, and figure. By the Law of Cosines, you can write v u, as shown in the v u2 u2 v2 2u vcos v u v u u2 v2 2u v cos v u v v u u u2 v2 2u v cos v v u v v u u u u2 v2 2u v cos v2 2u v u2 u2 v2 2u v cos cos u v u v. u − v u θ Origin v 492 333202_060R.qxd 12/5/05 10:48 AM Page 493 P.S. Problem Solving This collection of thought-provoking and challenging exercises further explores and expands upon concepts learned in this chapter. 1. In the figure, a beam of light is directed at the blue mirror, reflected to the red mirror, and then reflected back to the blue mirror. Find the distance that the light travels from the red mirror back to the blue mirror. PT Blue mirror 4 . 7 f t 25° O 6 ft 2. A triathlete sets a course to swim S 3 4 E from a point on shore to a buoy mile away. After swimming 300 yards through a strong current, the triathlete is off course at a bearing of S E. Find the bearing and distance the triathlete needs to swim to correct her course. 35 25 35° 25° 300 yd mi 3 4 Buoy W N S E 3. A hiking party is lost in a national park. Two ranger stations have received an emergency SOS signal from the party. Station B is 75 miles due east of station A. The bearing from E and the bearing from station station A to the signal is S B to the signal is S W. 60 75 (a) Draw a diagram that gives a visual representation of the problem. (b) Find the distance from each station to the SOS signal. (c) A rescue party is in the park 20 miles from station A at a bearing of S E. Find the distance and the bearing the rescue party must travel to reach the lost hiking party. 80 4. You are seeding a triangular courtyard. One side of the courtyard is 52 feet long and another side is 46 feet long. The angle opposite the 52-foot side is 65. (a) Draw a diagram that gives a visual representation of the problem. (b) How long is the third side of the courtyard? (c) One bag of grass covers an area of 50 square feet. How many bags of grass will you need to cover the courtyard? (i) (v) (ii) (iv) (iii) v v v 5. For each pair of vectors, find the following. u v u v u v u 0, 1 v 3, 3 u 2, 4 v 5, 5 u u u u 1, 1 v 1, 2 u 1, 1 2 v 2, 3 (vi) (b) (d) (c) (a) 6. A skydiver is falling at a constant downward velocity of 120 miles per hour. In the figure, vector u represents the skydiver’s velocity. A steady breeze pushes the skydiver to the east at 40 miles per hour. Vector v represents the wind velocity. Up 140 120 100 80 60 40 20 u v W −20 Down 20 40 60 E (a) Write the vectors and s u v. (b) Let u v in component form. Use the figure to sketch To print an enlarged copy of the graph, go to the website, www.mathgraphs.com. s. s. (c) Find the magnitude of What information does the magnitude give you about the skydiver’s fall? (d) If there were no wind, the skydiver would fall in a path perpendicular to the ground. At what angle to the ground is the path of the skydiver when the skydiver is affected by the 40 mile per hour wind from due west? (e) The skydiver is blown to the west at 30 miles per hour. Draw a new figure that gives a visual representation of the problem and find the skydiver’s new velocity. 493 333202_060R.qxd 12/5/05 10:48 AM Page 494 7. Write the vector terminal point of w w in terms of and bisects the line segment (see figure). given that the v, u v w u u is orthogonal to v and w, then u is 8. Prove that if orthogonal to cv dw for any scalars and c d (see figure). When taking off, a pilot must decide how much of the thrust to apply to each component. The more the thrust is applied to the horizontal component, the faster the airplane will gain speed. The more the thrust is applied to the vertical component, the quicker the airplane will climb. Thrust Lift Climb angle θ Velocity FIGURE FOR 10 θ Weight Drag v w u (a) Complete the table for an airplane that has a speed of v 100 miles per hour. 0.5 1.0 1.5 2.0 2.5 3.0 v sin v cos (b) Does an airplane’s speed equal the sum of the vertical and horizontal components of its velocity? If not, how could you find the speed of an airplane whose velocity components were known? (c) Use the result of part (b) to find the speed of an airplane with the given velocity components. (i) (ii) v sin 5.235 v cos 149.909 v sin 10.463 v cos 149.634 miles per hour miles per hour miles per hour miles per hour F2 F1 and and 9. Two forces of the same magnitude act at angles respectively. Use a diagram to compare the work 2, F1 in moving along the PQ if 2 60 and 1 done by with the work done by vector 1 1 30. (b) F2 2 (a) 10. Four basic forces are in action during flight: weight, lift, thrust, and drag. To fly through the air, an object must overcome its own weight. To do this, it must create an upward force called lift. To generate lift, a forward motion called thrust is needed. The thrust must be great enough to overcome air resistance, which is called drag. For a commercial jet aircraft, a quick climb is important to maximize efficiency, because the performance of an aircraft at high altitudes is enhanced. In addition, it is necessary to clear obstacles such as buildings and mountains and reduce noise in residential areas. In the diagram, the angle is called the climb angle. The velocity of the plane can be represented by a vector with a vertical (called climb speed) and a horizontal component is the speed of the plane. where component v sin v cos , v v 494 333202_0
700.qxd 12/5/05 9:38 AM Page 495 Systems of Equations and Inequalities 7.1 7.2 Linear and Nonlinear Systems of Equations Two-Variable Linear Systems 7.3 Multivariable Linear Systems 7.4 7.5 7.6 Partial Fractions Systems of Inequalities Linear Programming 77 Systems of equations can be used to determine the combinations of scoring plays for different sports, such as football AT I O N S Systems of equations and inequalities have many real-life applications. The applications listed below represent a small sample of the applications in this chapter. • Break-Even Analysis, Exercises 61–64, page 504 • Sports, • Data Analysis: Prescription Drugs, Exercise 51, page 529 Exercise 77, page 550 • Data Analysis: Renewable Energy, • Electrical Network, Exercise 71, page 505 Exercise 65, page 530 • Acid Mixture, Exercise 51, page 516 • Thermodynamics, Exercise 57, page 540 • Investment Portfolio, Exercises 47 and 48, page 561 • Supply and Demand, Exercises 75 and 76, page 565 495 333202_0701.qxd 12/5/05 9:39 AM Page 496 496 Chapter 7 Systems of Equations and Inequalities 7.1 Linear and Nonlinear Systems of Equations What you should learn • Use the method of substitution to solve systems of linear equations in two variables. • Use the method of substitution to solve systems of nonlinear equations in two variables. • Use a graphical approach to solve systems of equations in two variables. • Use systems of equations to model and solve real-life problems. Why you should learn it Graphs of systems of equations help you solve real-life problems. For instance, in Exercise 71 on page 505, you can use the graph of a system of equations to approximate when the consumption of wind energy exceeded the consumption of solar energy. The Method of Substitution Up to this point in the text, most problems have involved either a function of one variable or a single equation in two variables. However, many problems in science, business, and engineering involve two or more equations in two or more variables. To solve such problems, you need to find solutions of a system of equations. Here is an example of a system of two equations in two unknowns. 2x y 5 3x 2y 4 Equation 1 Equation 2 A solution of this system is an ordered pair that satisfies each equation in the system. Finding the set of all solutions is called solving the system of equations. is a solution of this system. To check this, you For instance, the ordered pair can substitute 2 for and 1 for in each equation. 2, 1 y x Check (2, 1) in Equation 1 and Equation 2: 2x y 5 22 1 ? 5 4 1 5 3x 2y 4 32 21 ? 4 6 2 4 Write Equation 1. x Substitute 2 for and 1 for Solution checks in Equation 1. ✓ y. Write Equation 2. x Substitute 2 for and 1 for Solution checks in Equation 2. ✓ y. In this chapter, you will study four ways to solve systems of equations, beginning with the method of substitution. Method Section Type of System 1. Substitution 2. Graphical method 3. Elimination 4. Gaussian elimination 7.1 7.1 7.2 7.3 Linear or nonlinear, two variables Linear or nonlinear, two variables Linear, two variables Linear, three or more variables © ML Sinibaldi /Corbis Method of Substitution 1. Solve one of the equations for one variable in terms of the other. 2. Substitute the expression found in Step 1 into the other equation to obtain an equation in one variable. 3. Solve the equation obtained in Step 2. 4. Back-substitute the value obtained in Step 3 into the expression obtained in Step 1 to find the value of the other variable. 5. Check that the solution satisfies each of the original equations. The HM mathSpace® CD-ROM and Eduspace® for this text contain additional resources related to the concepts discussed in this chapter. 333202_0701.qxd 12/5/05 9:39 AM Page 497 y2 x 2 4 x Exploration Use a graphing utility to graph y1 in and the same viewing window. Use the zoom and trace features to find the coordinates of the point of intersection. What is the relationship between the point of intersection and the solution found in Example 1? Section 7.1 Linear and Nonlinear Systems of Equations 497 Example 1 Solving a System of Equations by Substitution Solve the system of equations. x y 4 x y 2 Solution Begin by solving for y 4 x Equation 1 Equation 2 y in Equation 1. Solve for y in Equation 1. Next, substitute this expression for single-variable equation for x. y into Equation 2 and solve the resulting 2x 6 x 3 Write Equation 2. Substitute 4 x for y. Distributive Property Combine like terms. Divide each side by 2. Finally, you can solve for y 4 x, to obtain y by back-substituting x 3 into the equation The solution is the ordered pair You can check this solution as follows. Write revised Equation 1. Substitute 3 for x. Solve for y. 3, 1. into Equation 1: Write Equation 1. Substitute for x and y. Solution checks in Equation 1. ✓ into Equation 2: Write Equation 2. Substitute for x and y. Solution checks in Equation 2. Check Substitute Substitute 3, 1 Because of equations. satisfies both equations in the system, it is a solution of the system Now try Exercise 5. The term back-substitution implies that you work backwards. First you solve for one of the variables, and then you substitute that value back into one of the equations in the system to find the value of the other variable. Because many steps are required to solve a system of equations, it is very easy to make errors in arithmetic. So, you should always check your solution by substituting it into each equation in the original system. 333202_0701.qxd 12/5/05 9:39 AM Page 498 498 Chapter 7 Systems of Equations and Inequalities Example 2 Solving a System by Substitution A total of $12,000 is invested in two funds paying 5% and 3% simple interest. r where (Recall that the formula for simple interest is is the time.) The yearly interest is $500. How is the annual interest rate, and much is invested at each rate? is the principal, I Prt, P t Solution Verbal Model: 5% fund 3% fund Total investment 5% interest 3% interest Total interest x y When using the method of substitution, it does not matter which variable you choose to solve for first. Whether you solve for first, you first or will obtain the same solution. When making your choice, you should choose the variable and equation that are easier to work with. For instance, in Example x in Equation 1 2, solving for is easier than solving for in Equation 2. x Te c h n o l o g y One way to check the answers you obtain in this section is to use a graphing utility. For instance, enter the two equations in Example 2 y1 y2 12,000 x 500 0.05x 0.03 and find an appropriate viewing window that shows where the two lines intersect. Then use the intersect feature or the zoom and trace features to find the point of intersection. Does this point agree with the solution obtained at the right? Labels: System: x 0.05x y 0.03y 12,000 Amount in 5% fund Interest for 5% fund Amount in 3% fund Interest for 3% fund Total investment Total interest x 0.05x 500 y 0.03y 12,000 500 (dollars) (dollars) (dollars) (dollars) (dollars) (dollars) Equation 1 Equation 2 To begin, it is convenient to multiply each side of Equation 2 by 100. This eliminates the need to work with decimals. 1000.05x 0.03y 100500 Multiply each side by 100. 5x 3y 50,000 Revised Equation 2 To solve this system, you can solve for x in Equation 1. x 12,000 y Revised Equation 1 Then, substitute this expression for resulting equation for y. x into revised Equation 2 and solve the 5x 3y 50,000 512,000 y 3y 50,000 60,000 5y 3y 50,000 2y 10,000 y 5000 Write revised Equation 2. Substitute 12,000 y for x. Distributive Property Combine like terms. Divide each side by 2. Next, back-substitute the value y 5000 to solve for x. x 12,000 y x 12,000 5000 x 7000 Write revised Equation 1. Substitute 5000 for y. Simplify. The solution is at 3%. Check this in the original system. 7000, 5000. So, $7000 is invested at 5% and $5000 is invested Now try Exercise 19. 333202_0701.qxd 12/5/05 9:39 AM Page 499 Exploration Use a graphing utility to graph the two equations in Example 3 x2 4x 7 2x 1 y1 y2 in the same viewing window. How many solutions do you think this system has? Repeat this experiment for the equations in Example 4. How many solutions does this system have? Explain your reasoning. Section 7.1 Linear and Nonlinear Systems of Equations 499 Nonlinear Systems of Equations The equations in Examples 1 and 2 are linear. The method of substitution can also be used to solve systems in which one or both of the equations are nonlinear. Example 3 Substitution: Two-Solution Case Solve the system of equations. x2 4x y 2x y 7 1 Equation 1 Equation 2 Solution Begin by solving for expression for y y in Equation 2 to obtain into Equation 1 and solve for y 2x 1. x. Next, substitute this x 2 4x 2x 1 7 x 2 2x 1 7 x 2 2x 8 0 x 4x 2 0 x 4, 2 Substitute 2 x 1 for y into Equation 1. Simplify. Write in general form. Factor. Solve for x. Back-substituting these values of 4, 7 produces the solutions x and y to solve for the corresponding values of 2, 5. Check these in the original system. Now try Exercise 25. When using the method of substitution, you may encounter an equation that has no solution, as shown in Example 4. Example 4 Substitution: No-Real-Solution Case Solve the system of equations. x y 4 x2 y 3 Equation 1 Equation 2 Solution Begin by solving for expression for y y y x 4. in Equation 1 to obtain x. into Equation 2 and solve for Next, substitute this Substitute x 4 for y into Equation 2. Simplify. Use the Quadratic Formula. Because the discriminant is negative, the equation solution. So, the original system has no (real) solution. x2 x 1 0 has no (real) Now try Exercise 27. 333202_0701.qxd 12/5/05 9:39 AM Page 500 500 Chapter 7 Systems of Equations and Inequalities Te c h n o l o g y Most graphing utilities have builtin features that approximate the point(s) of intersection of two graphs. Typically, you must enter the equations of the graphs and visually locate a point of intersection before using t
he intersect feature. Use this feature to find the points of intersection of the graphs in Figures 7.1 to 7.3. Be sure to adjust your viewing window so that you see all the points of intersection. Graphical Approach to Finding Solutions From Examples 2, 3, and 4, you can see that a system of two equations in two unknowns can have exactly one solution, more than one solution, or no solution. By using a graphical method, you can gain insight about the number of solutions and the location(s) of the solution(s) of a system of equations by graphing each of the equations in the same coordinate plane. The solutions of the system correspond to the points of intersection of the graphs. For instance, the two equations in Figure 7.1 graph as two lines with a single point of intersection; the two equations in Figure 7.2 graph as a parabola and a line with two points of intersection; and the two equations in Figure 7.3 graph as a line and a parabola that have no points of intersection. y −1 −2 (2, 0) 1 2 x x 2 + 3y = 2 −1 y y = x2 − x − 1 y −x + y = 4 4 (2, 10, − 1) −3 1 −1 −2 x2 + y = 3 x 1 3 One intersection point FIGURE 7.1 Two intersection points FIGURE 7.2 No intersection points FIGURE 7.3 Example 5 Solving a System of Equations Graphically Solve the system of equations. y ln x Equation 1 x y 1 Equation 2 y 1 x + y = 1 y = ln x Solution Sketch the graphs of the two equations. From the graphs of these equations, it is is the solution clear that there is only one point of intersection and that x point (see Figure 7.4). You can confirm this by substituting 1 for and 0 for in both equations. 1, 0 y (1, 0) 1 2 x Check (1, 0) in Equation 1: y ln x 0 ln 1 Write Equation 1. Equation 1 checks. ✓ −1 FIGURE 7.4 Check (1, 0) in Equation 2: x y 1 1 0 1 Write Equation 2. Equation 2 checks. ✓ Now try Exercise 33. Example 5 shows the value of a graphical approach to solving systems of equations in two variables. Notice what would happen if you tried only the It substitution method in Example 5. You would obtain the equation would be difficult to solve this equation for using standard algebraic techniques. x ln x 1. x 333202_0701.qxd 12/5/05 9:39 AM Page 501 Section 7.1 Linear and Nonlinear Systems of Equations 501 Applications x C The total cost of producing units of a product typically has two components— the initial cost and the cost per unit. When enough units have been sold so that C, the sales are said to have reached the the total revenue break-even point. You will find that the break-even point corresponds to the point of intersection of the cost and revenue curves. equals the total cost R Example 6 Break-Even Analysis A shoe company invests $300,000 in equipment to produce a new line of athletic footwear. Each pair of shoes costs $5 to produce and is sold for $60. How many pairs of shoes must be sold before the business breaks even? Solution The total cost of producing units is x Total cost Cost per unit Number of units Initial cost C 5x 300,000. Equation 1 The revenue obtained by selling units is x Total revenue Price per unit Number of units R 60x. Equation 2 Because the break-even point occurs when system of equations to solve is R C, you have C 60x, and the C 5x 300,000 C 60x . Now you can solve by substitution. 60x 5x 300,000 55x 300,000 x 5455 Substitute 60x for C in Equation 1. Subtract 5x from each side. Divide each side by 55. Break-Even Analysis 600,000 500,000 Break-even point: 5455 units 400,000 R = 60x Profit 300,000 200,000 100,000 Loss C = 5x + 300,000 ) ,000 3,000 Number of units 9,000 x So, the company must sell about 5455 pairs of shoes to break even. Note in Figure 7.5 that revenue less than the break-even point corresponds to an overall loss, whereas revenue greater than the break-even point corresponds to a profit. FIGURE 7.5 Now try Exercise 63. Another way to view the solution in Example 6 is to consider the profit function P R C. The break-even point occurs when the profit is 0, which is the same as saying that R C. 333202_0701.qxd 12/5/05 9:39 AM Page 502 502 Chapter 7 Systems of Equations and Inequalities Example 7 Movie Ticket Sales The weekly ticket sales for a new comedy movie decreased each week. At the same time, the weekly ticket sales for a new drama movie increased each week. (in millions of dollars) for Models that approximate the weekly ticket sales each movie are S S 60 S 10 8x 4.5x Comedy Drama x represents the number of weeks each movie was in theaters, with x 0 where corresponding to the ticket sales during the opening weekend. After how many weeks will the ticket sales for the two movies be equal? Algebraic Solution Because the second equation has already been solved for in terms of x, solve for S substitute this value into the first equation and x, as follows. 10 4.5x 60 8x 4.5x 8x 60 10 12.5x 50 x 4 Substitute for S in Equation 1. Add 8x and 10 to each side. Combine like terms. Divide each side by 12.5. So, the weekly ticket sales for the two movies will be equal after 4 weeks. Numerical Solution You can create a table of values for each model to determine when the ticket sales for the two movies will be equal. Number of weeks, x Sales, S (comedy) Sales, S (drama) 0 1 2 3 4 5 6 60 52 44 36 28 20 12 10 14.5 19 23.5 28 32.5 37 Now try Exercise 65. So, from the table above, you can see that the weekly ticket sales for the two movies will be equal after 4 weeks. W RITING ABOUT MATHEMATICS Interpreting Points of Intersection You plan to rent a 14-foot truck for a two-day local move. At truck rental agency A, you can rent a truck for $29.95 per day plus $0.49 per mile. At agency B, you can rent a truck for $50 per day plus $0.25 per mile. a. Write a total cost equation in terms of and x y for the total cost of renting the truck from each agency. b. Use a graphing utility to graph the two equations in the same viewing window and find the point of intersection. Interpret the meaning of the point of intersection in the context of the problem. c. Which agency should you choose if you plan to travel a total of 100 miles during the two-day move? Why? d. How does the situation change if you plan to drive 200 miles during the two-day move? 333202_0701.qxd 12/5/05 9:39 AM Page 503 Section 7.1 Linear and Nonlinear Systems of Equations 503 7.1 Exercises The HM mathSpace® CD-ROM and Eduspace® for this text contain step-by-step solutions to all odd-numbered exercises. They also provide Tutorial Exercises for additional help. VOCABULARY CHECK: Fill in the blanks. 1. A set of two or more equations in two or more variables is called a ________ of ________. 2. A ________ of a system of equations is an ordered pair that satisfies each equation in the system. 3. Finding the set of all solutions to a system of equations is called ________ the system of equations. 4. The first step in solving a system of equations by the method of ________ is to solve one of the equations for one variable in terms of the other variable. 5. Graphically, the solution of a system of two equations is the ________ of ________ of the graphs of the two equations. 6. In business applications, the point at which the revenue equals costs is called the ________ point. PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. In Exercises 1– 4, determine whether each ordered pair is a solution of the system of equations. 9. 2x y x2 y2 5 25 10. x y 0 x3 5x y 0 y 1. 2. 3. 4. 4x y 1 6x y 6 4x2 y 3 x y 11 y 2e x 3x y 2 log x 3 y 9x y 28 1 9 (a) (c) (a) (c) (a) (c) (a) (c) 0, 3 3 2, 2 2, 13 3 2, 31 3 2, 0 0, 3 9, 37 9 1, 3 (b) (d) (b) (d) 1, 4 1 2, 3 2, 9 7 4, 37 4 0, 2 1, 2 10, 2 (b) (d) 2, 4 (d) (b) In Exercises 5–14, solve the system by the method of substitution. Check your solution graphically. 5. 2x y 6 x y 0 y 6. x y 4 x 2y 5 y 6 4 2 −2 −2 2 4 6 7. x y 4 x2 y 2 y 6 4 −2 2 4 x x 6 4 2 −2 2 8. 3x y 2 x3 2 y 0 y 8 6 −2 −2 −4 2 x x y 8 6 2 −6 −2 −6 x 6 8 −4 2 −2 −4 x 4 11. x2 y 0 x2 4x y 0 12. y 2x2 2 y 2x 4 2x2 1 y 2 −2 2 x −4 y 1 x 1 13. y x3 3x2 1 y x2 3x 1 14. y x3 3x2 4 y 2x 4 y 1 − 333202_0701.qxd 12/5/05 9:39 AM Page 504 504 Chapter 7 Systems of Equations and Inequalities In Exercises 15–28, solve the system by the method of substitution. In Exercises 49–60, solve the system graphically or algebraically. Explain your choice of method. 15. 17. 19. 21. 23. 25. 27. x y 0 5x 3y 10 2x y 2 0 4x y 5 0 1.5x 0.8y 2.3 0.3x 0.2y 0.1 1 1 8 2 y 20 y 6x 3 5y 5 x 7 6 y x2 y 0 2x y 0 x y 1 x2 y 4 5x x 16. 18. 20. 22. 24. 26. 28. x 2y 1 5x 4y 23 6x 3y 4 0 x 2y 4 0 0.5x 3.2y 9.0 0.2x 1.6y 3.6 1 2 x 3 4 y 10 4 x y 4 2 3x y 2 2x 3y 6 x 2y 0 3x y2 0 y x y x3 3x2 2x 3 In Exercises 29– 42, solve the system graphically. 29. 31. 33. 34. 35. 37. 39. 41. x 2y 2 3x y 15 x 3y 2 5x 3y 17 x y 4 x2 y2 4x 0 x y 3 x2 6x 27 y2 0 x y 3 0 x2 4x 7 y 7x 8y 24 x 8y 8 3x 2y 0 x2 y2 4 x2 25 0 y2 16y 3x2 30. 32. x y 0 3x 2y 10 x 2y 1 x y 2 36. 38. 40. 42. 0 1 2 1 y2 4x 11 2 x y x y 0 5x 2y 6 2x y 3 0 x2 y2 4x 0 x2 y2 25 x 82 y2 41 In Exercises 43– 48, use a graphing utility to solve the system of equations. Find the solution accurate to two decimal places. 49. 51. 53. 55. 57. 59. y 2x y x2 1 3x 7y 6 0 x2 y2 4 x 2y 4 x2 y 0 y ex 1 y ln x 3 y x 4 2x2 1 y 1 x2 xy 1 0 2x 4y 7 0 50. 52. 54. 56. 58. 60. x y 4 x2 y 2 x2 y2 25 2x y 10 y x 13 y x 1 x2 y 4 ex y 0 y x 3 2x2 x 1 y x2 3x 1 x 2y 1 y x 1 Break-Even Analysis necessary to break even producing x R C units and the revenue In Exercises 61 and 62, find the sales for the cost of obtained by selling units. (Round to the nearest whole unit.) C R x 61. 62. C 8650x 250,000, C 5.5x 10,000, R 9950x R 3.29x 63. Break-Even Analysis A small software company invests $16,000 to produce a software package that will sell for $55.95. Each unit can be produced for $35.45. (a) How many units must be sold to break even? (b) How many units must be sold to make a profit of $60,000? 64. Break-Even Analysis A small fast-food restaurant invests $5000 to produce a new food item that will sell for $3.49.
Each item can be produced for $2.16. (a) How many items must be sold to break even? (b) How many items must be sold to make a profit of $8500? 65. DVD Rentals The weekly rentals for a newly released DVD of an animated film at a local video store decreased each week. At the same time, the weekly rentals for a newly released DVD of a horror film increased each week. Models that approximate the weekly rentals for each DVD are R 43. 45. 46. 47. y e x x y 1 0 x 2y 8 y log2 x y 2 lnx 1 3y 2x 9 x2 y2 169 x2 8y 104 48. 44. y 4ex y 3x 8 0 R 360 24x R 24 18x Animated film Horror film x where the store, with represents the number of weeks each DVD was in x 1 corresponding to the first week. x2 y2 4 2x2 y 2 (a) After how many weeks will the rentals for the two movies be equal? (b) Use a table to solve the system of equations numeri- cally. Compare your result with that of part (a). 333202_0701.qxd 12/5/05 9:39 AM Page 505 Section 7.1 Linear and Nonlinear Systems of Equations 505 66. CD Sales The total weekly sales for a newly released rock CD increased each week. At the same time, the total weekly sales for a newly released rap CD decreased each S week. Models that approximate the total weekly sales (in thousands of units) for each CD are 70. Log Volume You are offered two different rules for estimating the number of board feet in a 16-foot log. (A board foot is a unit of measure for lumber equal to a board 1 foot square and 1 inch thick.) The first rule is the Doyle Log Rule and is modeled by S 25x 100 S 50x 475 Rock CD Rap CD x represents the number of weeks each CD was in corresponding to the CD sales on the where stores, with day each CD was first released in stores. x 0 (a) After how many weeks will the sales for the two CDs D 42, V1 5 ≤ D ≤ 40 and the other is the Scribner Log Rule and is modeled by V2 0.79D 2 2D 4, 5 ≤ D ≤ 40 D where volume (in board feet). is the diameter (in inches) of the log and V is its be equal? (a) Use a graphing utility to graph the two log rules in the (b) Use a table to solve the system of equations numeri- same viewing window. cally. Compare your result with that of part (a). (b) For what diameter do the two scales agree? 67. Choice of Two Jobs You are offered two jobs selling dental supplies. One company offers a straight commission of 6% of sales. The other company offers a salary of $350 per week plus 3% of sales. How much would you have to sell in a week in order to make the straight commission offer better? 68. Supply and Demand The supply and demand curves for a business dealing with wheat are Supply: p 1.45 0.00014x 2 Demand: p 2.388 0.007x 2 (c) You are selling large logs by the board foot. Which scale would you use? Explain your reasoning. Model It 71. Data Analysis: Renewable Energy The table shows (in trillions of Btus) of solar energy the consumption and wind energy in the United States from 1998 to (Source: Energy Information Administration) 2003. C p where is the is the price in dollars per bushel and quantity in bushels per day. Use a graphing utility to graph the supply and demand equations and find the market equilibrium. (The market equilibrium is the point of intersection of the graphs for x > 0. ) x 69. Investment Portfolio A total of $25,000 is invested in two funds paying 6% and 8.5% simple interest. (The 6% investment has a lower risk.) The investor wants a yearly interest income of $2000 from the two investments. (a) Write a system of equations in which one equation represents the total amount invested and the other equation represents the $2000 required in interest. Let x y and represent the amounts invested at 6% and 8.5%, respectively. (b) Use a graphing utility to graph the two equations in the same viewing window. As the amount invested at 6% increases, how does the amount invested at 8.5% change? How does the amount of interest income change? Explain. (c) What amount should be invested at 6% to meet the requirement of $2000 per year in interest? Year Solar, C Wind, C 1998 1999 2000 2001 2002 2003 70 69 66 65 64 63 31 46 57 68 105 108 (a) Use the regression feature of a graphing utility to find a quadratic model for the solar energy consumption data and a linear model for the wind represent the year, energy consumption data. Let with corresponding to 1998. t 8 t (b) Use a graphing utility to graph the data and the two models in the same viewing window. (c) Use the graph from part (b) to approximate the point of intersection of the graphs of the models. Interpret your answer in the context of the problem. (d) Approximate the point of intersection of the graphs of the models algebraically. (e) Compare your results from parts (c) and (d). (f) Use your school’s library, the Internet, or some other reference source to research the advantages and disadvantages of using renewable energy. 333202_0701.qxd 12/5/05 9:39 AM Page 506 506 Chapter 7 Systems of Equations and Inequalities 81. Writing List and explain the steps used to solve a system of equations by the method of substitution. 82. Think About It When solving a system of equations by substitution, how do you recognize that the system has no solution? 83. Exploration Find an equation of a line whose graph intersects the graph of the parabola at (a) two points, (b) one point, and (c) no points. (There is more than one correct answer.) y x 2 84. Conjecture Consider the system of equations y b x y x b . (a) Use a graphing utility to graph the system for b 1, 2, 3, and 4. (b) For a fixed even value of make a conjecture about the number of points of intersection of the graphs in part (a). b > 1, Skills Review In Exercises 85–90, find the general form of the equation of the line passing through the two points. 85. 86. 87. 88. 89. 90. 2, 7, 5, 5 3.5, 4, 10, 6 6, 3, 10, 3 4, 2, 4, 5 5, 0, 4, 6 3 7 3, 8, 5 2, 1 2 In Exercises 91–94, find the domain of the function and identify any horizontal or vertical asymptotes. 91. 92. 93. 94. f x 5 x 6 f x 2x 7 3x 2 f x x2 2 x2 16 f x 3 2 x2 72. Data Analysis: Population The table shows the popula(in thousands) of Alabama and Colorado from 1999 P tions to 2003. (Source: U.S. Census Bureau) Year Alabama, P Colorado, P 1999 2000 2001 2002 2003 4430 4447 4466 4479 4501 4226 4302 4429 4501 4551 (a) Use the regression feature of a graphing utility to find linear models for each set of data. Graph the models in the same viewing window. Let represent the year, with t corresponding to 1999. t 9 (b) Use your graph from part (a) to approximate when the population of Colorado exceeded the population of Alabama. (c) Verify your answer from part (b) algebraically. Geometry rectangle meeting the specified conditions. In Exercises 73–76, find the dimensions of the 73. The perimeter is 30 meters and the length is 3 meters greater than the width. 74. The perimeter is 280 centimeters and the width is 20 centimeters less than the length. 75. The perimeter is 42 inches and the width is three-fourths the length. 76. The perimeter is 210 feet and the length is 11 2 times the width. 77. Geometry What are the dimensions of a rectangular tract of land if its perimeter is 40 kilometers and its area is 96 square kilometers? 78. Geometry What are the dimensions of an isosceles right triangle with a two-inch hypotenuse and an area of 1 square inch? Synthesis True or False? the statement is true or false. Justify your answer. In Exercises 79 and 80, determine whether 79. In order to solve a system of equations by substitution, you in one of the two equations and y must always solve for then back-substitute. 80. If a system consists of a parabola and a circle, then the system can have at most two solutions. 333202_0702.qxd 12/5/05 9:41 AM Page 507 7.2 Two-Variable Linear Systems Section 7.2 Two-Variable Linear Systems 507 What you should learn • Use the method of elimination to solve systems of linear equations in two variables. • Interpret graphically the numbers of solutions of systems of linear equations in two variables. • Use systems of linear equations in two variables to model and solve real-life problems. Why you should learn it You can use systems of equations in two variables to model and solve real-life problems. For instance, in Exercise 63 on page 517, you will solve a system of equations to find a linear model that represents the relationship between wheat yield and amount of fertilizer applied. So, © Bill Stormont /Corbis The Method of Elimination In Section 7.1, you studied two methods for solving a system of equations: substitution and graphing. Now you will study the method of elimination. The key step in this method is to obtain, for one of the variables, coefficients that differ only in sign so that adding the equations eliminates the variable. 3x 5y 7 3x 2y 1 3y 6 Equation 1 Equation 2 Add equations. Note that by adding the two equations, you eliminate the -terms and obtain a which you can single equation in Solving this equation for produces then back-substitute into one of the original equations to solve for x y 2, y. x. y Example 1 Solving a System of Equations by Elimination Solve the system of linear equations. 3x 2y 4 5x 2y 8 Equation 1 Equation 2 y Solution Because the coefficients of differ only in sign, you can eliminate the -terms by adding the two equations. 3x 2y 4 5x 2y 8 12 Write Equation 1. Write Equation 2. Add equations. y By back-substituting this value into Equation 1, you can solve for y. Write Equation 1. Substitute 3 2 for x. Simplify. Solve for y. . Check this in the original system, as follows. 8x x 3 2. 3x 2y 4 2y 4 33 9 2y 4 2 2 y 1 4 3 2, 1 4 The solution is Check 33 2 21 4 1 9 2 2 21 4 15 1 2 2 ? 4 4 ? 8 8 Exploration Use the method of substitution to solve the system in Example 1. Which method is easier? 53 2 Substitute into Equation 1. Equation 1 checks. ✓ Substitute into Equation 2. Equation 2 checks. ✓ Now try Exercise 11. 333202_0702.qxd 12/5/05 9:41 AM Page 508 508 Chapter 7 Systems of Equations and Inequalities Method of Elimination To use the method of el
imination to solve a system of two linear equations x in and perform the following steps. y, 1. Obtain coefficients for y terms of one or both equations by suitably chosen constants. (or ) that differ only in sign by multiplying all x 2. Add the equations to eliminate one variable, and solve the resulting equation. 3. Back-substitute the value obtained in Step 2 into either of the original equations and solve for the other variable. 4. Check your solution in both of the original equations. Example 2 Solving a System of Equations by Elimination Solve the system of linear equations. 2x 3y 7 3x y 5 Equation 1 Equation 2 Solution For this system, you can obtain coefficients that differ only in sign by multiplying Equation 2 by 3. 2x 3y 7 3x y 5 So, you can see that y. you can solve for 2x 3y 7 9x 3y 15 22 11x Write Equation 1. Multiply Equation 2 by 3. Add equations. x 2. By back-substituting this value of x into Equation 1, 2x 3y 7 22 3y 7 3y 3 y 1 2, 1. The solution is Check Write Equation 1. Substitute 2 for x. Combine like terms. Solve for y. Check this in the original system, as follows. 2x 3y 7 22 31 ? 7 4 3 7 3x y 5 32 1 ? 5 6 1 5 Now try Exercise 13. Write original Equation 1. Substitute into Equation 1. Equation 1 checks. ✓ Write original Equation 2. Substitute into Equation 2. Equation 2 checks. ✓ Exploration Rewrite each system of equations in slope-intercept form and sketch the graph of each system. What is the relationship between the slopes of the two lines and the number of points of intersection? a. b. 5x y 1 x y 5 4x 3y 1 8x 6y 2 c. x 2y 3 x 2y 8 333202_0702.qxd 12/5/05 9:41 AM Page 509 Section 7.2 Two-Variable Linear Systems 509 In Example 2, the two systems of linear equations (the original system and the system obtained by multiplying by constants) 2x 3y 7 3x y 5 and 2x 3y 7 9x 3y 15 are called equivalent systems because they have precisely the same solution set. The operations that can be performed on a system of linear equations to produce an equivalent system are (1) interchanging any two equations, (2) multiplying an equation by a nonzero constant, and (3) adding a multiple of one equation to any other equation in the system. Example 3 Solving the System of Equations by Elimination Solve the system of linear equations. 5x 3y 9 2x 4y 14 Equation 1 Equation 2 Algebraic Solution You can obtain coefficients that differ only in sign by multiplying Equation 1 by 4 and multiplying Equation 2 by 3. 5x 3y 9 2x 4y 14 20x 12y 36 6x 12y 42 78 26x Multiply Equation 1 by 4. Multiply Equation 2 by 3. Add equations. From this equation, you can see that of into Equation 2, you can solve for x x 3. y. 2x 4y 14 23 4y 14 4y 8 y 2 By back-substituting this value Write Equation 2. Substitute 3 for x. Combine like terms. Solve for y. The solution is 3, 2. Check this in the original system. Now try Exercise 15. 2 y2 1 y. y1 Graphical Solution Then use a Solve each equation for 5 3x 3 graphing utility to graph 2x 7 in the same viewing and window. Use the intersect feature or the zoom and trace features to approximate the point of intersection of the graphs. From the graph in Figure 7.6, you can see that the point of intersection is You can determine that this is the exact solution by checking in both equations. 3, 2. 3, 2 5 y1 = − x + 3 3 3 7 1 y2 = x − 2 7 2 −5 −5 FIGURE 7.6 You can check the solution from Example 3 as follows. 53 32 ? 15 23 42 ? 9 6 9 14 8 14 6 Substitute 3 for x and Equation 1 checks. ✓ 2 Substitute 3 for x and Equation 2 checks. ✓ 2 for y in Equation 1. for y in Equation 2. Keep in mind that the terminology and methods discussed in this section apply only to systems of linear equations. 333202_0702.qxd 12/5/05 9:41 AM Page 510 510 Chapter 7 Systems of Equations and Inequalities Graphical Interpretation of Solutions It is possible for a general system of equations to have exactly one solution, two or more solutions, or no solution. If a system of linear equations has two different solutions, it must have an infinite number of solutions. Graphical Interpretations of Solutions For a system of two linear equations in two variables, the number of solutions is one of the following. Number of Solutions Graphical Interpretation Slopes of Lines 1. Exactly one solution The two lines intersect at one point. The slopes of the two lines are not equal. 2. Infinitely many solutions The two lines coincide (are identical). The slopes of the two lines are equal. 3. No solution The two lines are parallel. The slopes of the two lines are equal. A system of linear equations is consistent if it has at least one solution. A consistent system with exactly one solution is independent, whereas a consistent system with infinitely many solutions is dependent. A system is inconsistent if it has no solution. Example 4 Recognizing Graphs of Linear Systems Match each system of linear equations with its graph in Figure 7.7. Describe the number of solutions and state whether the system is consistent or inconsistent. 2x 3y 3 4x 6y 6 a. i. b. ii. y 4 2 −2 2 4 x −2 −4 FIGURE 7.7 2x 3y 3 x 2y 5 c. 2x 3y 3 4x 6y 6 iii. y 4 2 x 2 4 −2 2 4 x −2 −4 y 4 2 −2 −4 A comparison of the slopes of two lines gives useful information about the number of solutions of the corresponding system of equations. To solve a system of equations graphically, it helps to begin by writing the equations in slope-intercept form. Try doing this for the systems in Example 4. Solution a. The graph of system (a) is a pair of parallel lines (ii). The lines have no point of intersection, so the system has no solution. The system is inconsistent. b. The graph of system (b) is a pair of intersecting lines (iii). The lines have one point of intersection, so the system has exactly one solution. The system is consistent. c. The graph of system (c) is a pair of lines that coincide (i). The lines have infinitely many points of intersection, so the system has infinitely many solutions. The system is consistent. Now try Exercises 31–34. 333202_0702.qxd 12/5/05 9:41 AM Page 511 Section 7.2 Two-Variable Linear Systems 511 In Examples 5 and 6, note how you can use the method of elimination to determine that a system of linear equations has no solution or infinitely many solutions. Example 5 No-Solution Case: Method of Elimination Solve the system of linear equations. x 2y 3 2x 4y 1 Equation 1 Equation 2 Solution To obtain coefficients that differ only in sign, multiply Equation 1 by 2. −2x + 4y = 1 1 3 x x − 2y = 3 x 2y 3 2x 4y 1 2x 4y 6 2x 4y 1 0 7 Multiply Equation 1 by 2. Write Equation 2. False statement Because there are no values of and you can conclude that the system is inconsistent and has no solution. The lines corresponding to the two equations in this system are shown in Figure 7.8. Note that the two lines are parallel and therefore have no point of intersection. for which y x 0 7, y 2 1 −1 −2 FIGURE 7.8 Now try Exercise 19. 0 7, In Example 5, note that the occurrence of a false statement, such as indicates that the system has no solution. In the next example, note that the occurrence of a statement that is true for all values of the variables, such as 0 0, indicates that the system has infinitely many solutions. Example 6 Many-Solution Case: Method of Elimination Solve the system of linear equations. 2x y 1 4x 2y 2 Equation 1 Equation 2 Solution To obtain coefficients that differ only in sign, multiply Equation 2 by 1 2. 2x y 1 4x 2y 2 2x y 1 2x y 1 0 0 Write Equation 1. Multiply Equation 2 by 1 2. Add equations. Because the two equations turn out to be equivalent (have the same solution set), you can conclude that the system has infinitely many solutions. The solution set x, y as shown in Figure 7.9. consists of all points x a, a where Letting is any real number, you can see that the solutions to the a, 2a 1. system are lying on the line 2x y 1, Now try Exercise 23. y 3 2 1 (2, 3) 2x − y = 1 (1, 1) −1 1 2 3 x −1 FIGURE 7.9 333202_0702.qxd 12/5/05 9:41 AM Page 512 512 Chapter 7 Systems of Equations and Inequalities Te c h n o l o g y The general solution of the linear system ax by c dx ey f and x ce bf ae bd is y af cdae bd. If ae bd 0, the system does not have a unique solution. A graphing utility program (called Systems of Linear Equations) for solving such a system can be found at our website college.hmco.com. Try using the program for your graphing utility to solve the system in Example 7. Example 7 illustrates a strategy for solving a system of linear equations that has decimal coefficients. Example 7 A Linear System Having Decimal Coefficients Solve the system of linear equations. 0.02x 0.05y 0.38 0.03x 0.04y 1.04 Equation 1 Equation 2 Solution Because the coefficients in this system have two decimal places, you can begin by multiplying each equation by 100. This produces a system in which the coefficients are all integers. 2x 5y 38 3x 4y 104 Revised Equation 1 Revised Equation 2 Now, to obtain coefficients that differ only in sign, multiply Equation 1 by 3 and multiply Equation 2 by 2x 5y 38 3x 4y 104 Multiply Equation 1 by 3. Multiply Equation 2 by 2. 2. 6x 15y 114 6x 8y 208 23y 322 Add equations. So, you can conclude that 322 23 y 14. Back-substituting this value into revised Equation 2 produces the following. 3x 4y 104 3x 414 104 3x 48 x 16 16, 14. The solution is Check Write revised Equation 2. Substitute 14 for y. Combine like terms. Solve for x. Check this in the original system, as follows. 0.02x 0.05y 0.38 0.0216 0.0514 ? 0.38 0.32 0.70 0.38 0.03x 0.04y 1.04 0.0316 0.0414 ? 1.04 0.48 0.56 1.04 Now try Exercise 25. Write original Equation 1. Substitute into Equation 1. Equation 1 checks. ✓ Write original Equation 2. Substitute into Equation 2. Equation 2 checks. ✓ 333202_0702.qxd 12/5/05 9:41 AM Page 513 Section 7.2 Two-Variable Linear Systems 513 Applications At this point, you may be asking the question “How can I tell which application problems can be solved using a system of linear equations?” The answer comes from the following considerations. 1
. Does the problem involve more than one unknown quantity? 2. Are there two (or more) equations or conditions to be satisfied? If one or both of these situations occur, the appropriate mathematical model for the problem may be a system of linear equations. Example 8 An Application of a Linear System An airplane flying into a headwind travels the 2000-mile flying distance between Chicopee, Massachusetts and Salt Lake City, Utah in 4 hours and 24 minutes. On the return flight, the same distance is traveled in 4 hours. Find the airspeed of the plane and the speed of the wind, assuming that both remain constant. Solution The two unknown quantities are the speeds of the wind and the plane. If speed of the plane and is the speed of the wind, then r2 r1 is the Original flight WIND r1 − r2 Return flight WIND r1 + r2 FIGURE 7.10 r1 r1 r2 r2 speed of the plane against the wind speed of the plane with the wind as shown in Figure 7.10. Using the formula speeds, you obtain the following equations. distance ratetime for these two 2000 r1 r2 4 24 60 2000 r1 r2 4 These two equations simplify as follows. 5000 11r1 500 r1 11r2 r2 Equation 1 Equation 2 To solve this system by elimination, multiply Equation 2 by 11. 5000 11r1 500 r1 11r2 11r2 11r2 r2 Write Equation 1. Multiply Equation 2 by 11. 5000 11r1 5500 11r1 10,500 22r1 Add equations. So, r1 r2 10,500 5250 11 22 500 5250 11 250 11 477.27 miles per hour Speed of plane 22.73 miles per hour. Speed of wind Check this solution in the original statement of the problem. Now try Exercise 43. 333202_0702.qxd 12/5/05 9:41 AM Page 514 514 Chapter 7 Systems of Equations and Inequalities In a free market, the demands for many products are related to the prices of the products. As the prices decrease, the demands by consumers increase and the amounts that producers are able or willing to supply decrease. Example 9 Finding the Equilibrium Point The demand and supply functions for a new type of personal digital assistant are p 150 0.00001x p 60 0.00002x Demand equation Supply equation Equilibrium p (3,000,000, 120) Demand p is the price in dollars and x where equilibrium point for this market. The equilibrium point is the price number of units represents the number of units. Find the and that satisfy both the demand and supply equations. p x Solution p Because is written in terms of the supply equation into the demand equation. x, begin by substituting the value of given in p Supply p 150 0.00001x 60 0.00002x 150 0.00001x Write demand equation. Substitute 60 0.00002x for p. 0.00003x 90 Combine like terms. x 3,000,000 Solve for x. 150 125 100 75 50 25 ) ,000,000 3,000,000 Number of units x FIGURE 7.11 So, the equilibrium point occurs when the demand and supply are each 3 million units. (See Figure 7.11.) The price that corresponds to this -value is obtained by back-substituting into either of the original equations. For instance, back-substituting into the demand equation produces x 3,000,000 x p 150 0.000013,000,000 150 30 $120. The solution is 3,000,000, 120. You can check this as follows. into the demand equation. Check Substitute 3,000,000, 120 p 150 0.00001x 150 0.000013,000,000 120 ? 120 120 Substitute 3,000,000, 120 p 60 0.00002x into the supply equation. 60 0.000023,000,000 120 ? 120 120 Now try Exercise 45. Write demand equation. Substitute 120 for p and 3,000,000 for x. Solution checks in demand equation. ✓ Write supply equation. Substitute 120 for p and 3,000,000 for x. Solution checks in supply equation. ✓ 333202_0702.qxd 12/5/05 9:41 AM Page 515 Section 7.2 Two-Variable Linear Systems 515 7.2 Exercises VOCABULARY CHECK: Fill in the blanks. 1. The first step in solving a system of equations by the method of ________ is to obtain coefficients for x (or ) that differ only in sign. y 2. Two systems of equations that have the same solution set are called ________ systems. 3. A system of linear equations that has at least one solution is called ________, whereas a system of linear equations that has no solution is called ________. 4. In business applications, the ________ ________ is defined as the price and the number of units p x that satisfy both the demand and supply equations. PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. In Exercises 1–10, solve the system by the method of elimination. Label each line with its equation. To print an enlarged copy of the graph, go to the website www.mathgraphs.com. 7. 3x 2y 5 6x 4y 10 8. 9x 3y 15 3x y 5 1. 2x y 5 x y 1 y 4 2 −2 2 4 6 x −4 3. x y 0 3x 2y 1 y 4 x 2 4 −4 −2 −2 −4 5. x y 2 2x 2y 5 y 4 x 2 4 −2 −2 2. x 3y 1 x 2y 2 −2 −4 −2 −2 x 2 9. 9x 3y 1 3x 6y 5 y 4. 2x y 3 4x 3y 21 y 6 4 −4 x 2 10. 5x 3y 18 2x 6y 1 y 4 2 −2 x 2 x 2 4 −6 −4 −2 −2 −4 2 4 x In Exercises 11–30, solve the system by the method of elimination and check any solutions algebraically. 6. 3x 2y 3 6x 4y 14 y x 4 −2 −2 −4 11. 13. 15. 17. 19. x 2y 4 x 2y 1 2x 3y 18 5x y 11 3x 2y 10 2x 5y 3 5u 6v 24 3u 5v 18 9 6 5x 5y 4 6y 3 9x 12. 14. 16. 18. 3x 5y 2 2x 5y 13 x 7y 12 3x 5y 10 2r 4s 5 16r 50s 55 3x 11y 4 2x 5y 9 y 1 4x 8 3y 3 4x 8 9 20. 3 y 6 4 2 −2 333202_0702.qxd 12/5/05 9:41 AM Page 516 516 Chapter 7 Systems of Equations and Inequalities 21. 23. 25. 27. 29. 4 x y 6 y 3 12 1 3 x 5x 6y 24y 20x 0.05x 0.03y 0.21 0.07x 0.02y 0.16 4b 3m 3 3b 11m 13 x 3 y 1 3 2x y 12 1 4 22. 24. 26. 28. 30 8y 7x 6 12 16y 14x 0.2x 0.5y 27.8 0.3x 0.4y 68.7 2x 5y 8 5x 8y 10 x 1 y 2 3 x 2y 5 4 2 In Exercises 31–34, match the system of linear equations with its graph. Describe the number of solutions and state whether the system is consistent or inconsistent. [The graphs are labeled (a), (b), (c) and (d).] (b) y 4 2 (d) (a) y 4 2 −2 2 4 −4 (c) y −6 2 −2 −4 2 4 31. 33. 2x 5y 0 x y 3 2x 5y 0 2x 3y 4 32. 34. 7x 6y 4 14x 12y 8 7x 6y 6 7x 6y 4 In Exercises 35–42, use any method to solve the system. 35. 37. 39. 3x 5y 7 2x y 9 y 2x 5 y 5x 11 x 5y 21 6x 5y 21 36. 38. 40. x 3y 17 4x 3y 7 7x 3y 16 y x 2 y 3x 8 y 15 2x 41. 2x 8y 19 y x 3 42. 4x 3y 6 5x 7y 1 43. Airplane Speed An airplane flying into a headwind travels the 1800-mile flying distance between Pittsburgh, Pennsylvania and Phoenix, Arizona in 3 hours and 36 minutes. On the return flight, the distance is traveled in 3 hours. Find the airspeed of the plane and the speed of the wind, assuming that both remain constant. 44. Airplane Speed Two planes start from Los Angeles International Airport and fly in opposite directions. The second plane starts hour after the first plane, but its speed is 80 kilometers per hour faster. Find the airspeed of each plane if 2 hours after the first plane departs the planes are 3200 kilometers apart. 1 2 In Exercises 45– 48, Supply and Demand the equilibrium point of the demand and supply equations. The equilibrium point is the price p and number of units x that satisfy both the demand and supply equations. find Demand p 50 0.5x p 100 0.05x p 140 0.00002x p 400 0.0002x 45. 46. 47. 48. Supply p 0.125x p 25 0.1x p 80 0.00001x p 225 0.0005x 49. Nutrition Two cheeseburgers and one small order of French fries from a fast-food restaurant contain a total of 850 calories. Three cheeseburgers and two small orders of French fries contain a total of 1390 calories. Find the caloric content of each item. 50. Nutrition One eight-ounce glass of apple juice and one eight-ounce glass of orange juice contain a total of 185 milligrams of vitamin C. Two eight-ounce glasses of apple juice and three eight-ounce glasses of orange juice contain a total of 452 milligrams of vitamin C. How much vitamin C is in an eight-ounce glass of each type of juice? 51. Acid Mixture Ten liters of a 30% acid solution is obtained by mixing a 20% solution with a 50% solution. (a) Write a system of equations in which one equation represents the amount of final mixture required and the other represents the percent of acid in the final mixture. x y Let and represent the amounts of the 20% and 50% solutions, respectively. (b) Use a graphing utility to graph the two equations in part (a) in the same viewing window. As the amount of the 20% solution increases, how does the amount of the 50% solution change? (c) How much of each solution is required to obtain the specified concentration of the final mixture? 333202_0702.qxd 12/5/05 9:41 AM Page 517 52. Fuel Mixture Five hundred gallons of 89 octane gasoline is obtained by mixing 87 octane gasoline with 92 octane gasoline. (a) Write a system of equations in which one equation represents the amount of final mixture required and the other represents the amounts of 87 and 92 octane gasolines in the final mixture. Let and represent the numbers of gallons of 87 octane and 92 octane gasolines, respectively. x y (b) Use a graphing utility to graph the two equations in part (a) in the same viewing window. As the amount of 87 octane gasoline increases, how does the amount of 92 octane gasoline change? (c) How much of each type of gasoline is required to obtain the 500 gallons of 89 octane gasoline? 53. Investment Portfolio A total of $12,000 is invested in two corporate bonds that pay 7.5% and 9% simple interest. The investor wants an annual interest income of $990 from the investments. What amount should be invested in the 7.5% bond? 54. Investment Portfolio A total of $32,000 is invested in two municipal bonds that pay 5.75% and 6.25% simple interest. The investor wants an annual interest income of $1900 from the investments. What amount should be invested in the 5.75% bond? 55. Ticket Sales At a local high school city championship basketball game, 1435 tickets were sold. A student admission ticket cost $1.50 and an adult admission ticket cost $5.00. The sum of all the total ticket receipts for the basketball game were $3552.50. How many of each type of ticket were sold? 56. Consumer Awareness A department store held a sale to sell all of the 214 winter jackets that remained after the season ended. Until noon, each jacket in the store was priced at $31.95. At noon, the ja
ckets was further reduced to $18.95. After the last jacket was sold, total receipts for the clearance sale were $5108.30. How many jackets were sold before noon and how many were sold after noon? the price of Fitting a Line to Data squares regression line In Exercises 57–62, find the least y ax b for the points x1, y1 , x2, y2 , . . . , xn, yn by solving the system for a and b. nb n i1 xia n i1 yi n i1 xib n i1 i a n x2 i1 xi yi Section 7.2 Two-Variable Linear Systems 517 Then use a graphing utility to confirm the result. (If you are unfamiliar with summation notation, look at the discussion in Section 9.1 or in Appendix B at the website for this text at college.hmco.com.) 57. 5b 10a 20.2 10b 30a 50.1 58. 5b 10a 11.7 10b 30a 25.6 y 6 5 4 3 2 1 (4, 5.8) (3, 5.2) (2, 4.2) (1, 2.9) (0, 2.1) −1 1 2 3 4 5 59. 7b 21a 35.1 21b 91a 114.2 y 8 6 2 (5, 5.6) (6, 6) (3, 5) (4, 5.4) (2, 4.6) (1, 4.4) (0, 4.11 −2 (4, 2.8) (2, 2.4) (3, 2.5) (1, 2.1) 2 3 4 5 (0, 1.9) x 60. 6b 15a 23.6 15b 55a 48.8 y 8 (0, 5.4) (1, 4.8) 4 2 (3, 3.5) (5, 2.5) (2, 4.3) (4, 3.1) 2 4 6 x 61. 62. 0, 4, 1, 3, 1, 1, 2, 0 1, 0, 2, 0, 3, 0, 3, 1, 4, 1, 4, 2, 5, 2, 6, 2 63. Data Analysis A farmer used four test plots to determine the relationship between wheat yield (in bushels per acre) x (in hundreds of pounds per and the amount of fertilizer acre). The results are shown in the table. y Fertilizer, x Yield, y 1.0 1.5 2.0 2.5 32 41 48 53 (a) Use the technique demonstrated in Exercises 57–62 to set up a system of equations for the data and to find the least squares regression line y ax b. (b) Use the linear model to predict the yield for a fertilizer application of 160 pounds per acre. 333202_0702.qxd 12/5/05 9:41 AM Page 518 518 Chapter 7 Systems of Equations and Inequalities Model It 64. Data Analysis The table shows the average room for a hotel room in the United States for the (Source: American Hotel rates years 1995 through 2001. & Motel Association) y Year Average room rate, y 1995 1996 1997 1998 1999 2000 2001 $66.65 $70.93 $75.31 $78.62 $81.33 $85.89 $88.27 (a) Use the technique demonstrated in Exercises 57–62 to set up a system of equations for the data and to find the least squares regression line y at b. t 5 corresponding to 1995. represent the year, with Let t (b) Use the regression feature of a graphing utility to find a linear model for the data. How does this model compare with the model obtained in part (a)? (c) Use the linear model to create a table of estimated y. values of Compare the estimated values with the actual data. (d) Use the linear model to predict the average room rate in 2002. The actual average room rate in 2002 was $83.54. How does this value compare with your prediction? (e) Use the linear model to predict when the average room rate will be $100.00. Using your result from part (d), do you think this prediction is accurate? Synthesis True or False? the statement is true or false. Justify your answer. In Exercises 65 and 66, determine whether 65. If two lines do not have exactly one point of intersection, then they must be parallel. 66. Solving a system of equations graphically will always give an exact solution. 67. Writing Briefly explain whether or not it is possible for a consistent system of linear equations to have exactly two solutions. 68. Think About It Give examples of a system of linear equations that has (a) no solution and (b) an infinite number of solutions. Think About It In Exercises 69 and 70, the graphs of the two equations appear to be parallel. Yet, when the system is solved algebraically, you find that the system does have a solution. Find the solution and explain why it does not appear on the portion of the graph that is shown. 69. 100y x 200 99y x 198 y 70. 21x 20y 0 13x 12y 120 y 4 −4 −2 2 4 x −4 10 −10 −10 x 10 In Exercises 71 and 72, find the value of k such that the system of linear equations is inconsistent. 71. 4x 8y 3 2x ky 16 72. 15x 3y 6 10x ky 9 Skills Review In Exercises 73–80, solve the inequality and graph the solution on the real number line. 73. 75. 77. 79. 11 6x ≥ 33 8x 15 ≤ 42x 1 x 8 < 10 2x2 3x 35 < 0 74. 76. 78. 80. 2x 3 > 5x 1 6 ≤ 3x 10 < 6 x 10 ≥ 3 3x2 12x > 0 In Exercises 81–84, write the expression as the logarithm of a single quantity. 81. 83. ln x ln 6 log9 12 log9 x 82. 84. ln x 5 lnx 3 1 4 log6 3x In Exercises 85 and 86, solve the system by the method of substitution. 85. 2x y 4 4x 2y 12 86. 30x 40y 33 0 10x 20y 21 0 87. Make a Decision To work an extended application analyzing the average undergraduate tuition, room, and board charges at private colleges in the United States from 1985 to 2003, visit this text’s website at college.hmco.com. (Data Source: U.S. Dept. of Education) 333202_0703.qxd 12/5/05 9:42 AM Page 519 Section 7.3 Multivariable Linear Systems 519 7.3 Multivariable Linear Systems What you should learn • Use back-substitution to solve linear systems in row-echelon form. • Use Gaussian elimination to solve systems of linear equations. • Solve nonsquare systems of linear equations. • Use systems of linear equations in three or more variables to model and solve real-life problems. Why you should learn it Systems of linear equations in three or more variables can be used to model and solve real-life problems. For instance, in Exercise 71 on page 531, a system of linear equations can be used to analyze the reproduction rates of deer in a wildlife preserve. Row-Echelon Form and Back-Substitution The method of elimination can be applied to a system of linear equations in more than two variables. In fact, this method easily adapts to computer use for solving linear systems with dozens of variables. When elimination is used to solve a system of linear equations, the goal is to rewrite the system in a form to which back-substitution can be applied. To see how this works, consider the following two systems of linear equations. System of Three Linear Equations in Three Variables: (See Example 3.) x 2y 3z 9 x 3y 4 2x 5y 5z 17 Equivalent System in Row-Echelon Form: (See Example 1.) x 2y 3z 9 y 3z 5 z 2 The second system is said to be in row-echelon form, which means that it has a “stair-step” pattern with leading coefficients of 1. After comparing the two systems, it should be clear that it is easier to solve the system in row-echelon form, using back-substitution. Example 1 Using Back-Substitution in Row-Echelon Form Solve the system of linear equations. x 2y 3z 9 y 3z 5 z 2 Equation 1 Equation 2 Equation 3 Jeanne Drake/Tony Stone Images Solution From Equation 3, you know the value of To solve for Equation 2 to obtain y 32 5 Substitute 2 for z. z. y 1. Solve for y. y, substitute z 2 into Finally, substitute y 1 x 21 32 9 x 1. and The solution is triple 1, 1, 2. x 1, y 1, z 2 into Equation 1 to obtain Substitute 1 for y and 2 for z. Solve for x. and z 2, which can be written as the ordered Check this in the original system of equations. Now try Exercise 5. 333202_0703.qxd 12/5/05 9:42 AM Page 520 520 Chapter 7 Systems of Equations and Inequalities Historical Note One of the most influential Chinese mathematics books was the Chui-chang suan-shu or Nine Chapters on the Mathematical Art (written in approximately 250 B.C.). Chapter Eight of the Nine Chapters contained solutions of systems of linear equations using positive and negative numbers. One such system was as follows. 3x 2y z 39 2x 3y z 34 x 2y 3z 26 This system was solved using column operations on a matrix. Matrices (plural for matrix) will be discussed in the next chapter. As demonstrated in the first step in the solution of Example 2, interchanging rows is an easy way of obtaining a leading coefficient of 1. Gaussian Elimination Two systems of equations are equivalent if they have the same solution set. To solve a system that is not in row-echelon form, first convert it to an equivalent system that is in row-echelon form by using the following operations. Operations That Produce Equivalent Systems Each of the following row operations on a system of linear equations produces an equivalent system of linear equations. 1. Interchange two equations. 2. Multiply one of the equations by a nonzero constant. 3. Add a multiple of one of the equations to another equation to replace the latter equation. To see how this is done, take another look at the method of elimination, as applied to a system of two linear equations. Example 2 Using Gaussian Elimination to Solve a System Solve the system of linear equations. 1 0 3x 2y x y Equation 1 Equation 2 Solution There are two strategies that seem reasonable: eliminate the variable or eliminate the variable The following steps show how to use the first strategy. y. x x y 0 1 3x 2y 3x 3y 3x 2y 3x 3y 0 3x 2y Interchange the two equations in the system. Multiply the first equation by 3. Add the multiple of the first equation to the second equation to obtain a new second equation. New system in row-echelon form Now, using back-substitution, you can determine that the solution is x 1, in the original system of equations. which can be written as the ordered pair 1, 1. y 1 and Check this solution Now try Exercise 13. 333202_0703.qxd 12/5/05 9:42 AM Page 521 Arithmetic errors are often made when performing elementary row operations. You should note the operation performed in each step so that you can go back and check your work. Section 7.3 Multivariable Linear Systems 521 As shown in Example 2, rewriting a system of linear equations in row-echelon form usually involves a chain of equivalent systems, each of which is obtained by using one of the three basic row operations listed on the previous page. This process is called Gaussian elimination, after the German mathematician Carl Friedrich Gauss (1777–1855). Example 3 Using Gaussian Elimination to Solve a System Solve the system of linear equations. x 2y 3z 9 x 3y 4 2x 5y 5z 17 Equation 1 Equation 2 Equation 3 Solution Because the leading coefficient of the first equation is 1, you can begin by saving the at the upper left and eliminating
the other -terms from the first column. x x x 3y x 2y 3z 9 4 y 3z 5 x 2y 3z 9 y 3z 5 2x 5y 5z 17 2x 4y 6z 18 2x 5y 5z 17 y z 1 x 2y 3z 9 y 3z 5 y z 1 Write Equation 1. Write Equation 2. Add Equation 1 to Equation 2. Adding the first equation to the second equation produces a new second equation. Multiply Equation 1 by 2. Write Equation 3. Add revised Equation 1 to Equation 3. 2 Adding times the first equation to the third equation produces a new third equation. Now that all but the first have been eliminated from the first column, go to work on the second column. (You need to eliminate from the third equation.) y x x 2y 3z 9 y 3z 5 2z 4 Adding the second equation to the third equation produces a new third equation. Finally, you need a coefficient of 1 for z in the third equation. x 2y 3z 9 y 3z 5 z 2 Multiplying the third equation 1 by produces a new third 2 equation. This is the same system that was solved in Example 1, and, as in that example, you can conclude that the solution is x 1, y 1, and z 2. Now try Exercise 15. 333202_0703.qxd 12/5/05 9:42 AM Page 522 522 Chapter 7 Systems of Equations and Inequalities The next example involves an inconsistent system—one that has no solution. The key to recognizing an inconsistent system is that at some stage in the elimination process you obtain a false statement such as 0 2. Example 4 An Inconsistent System Solve the system of linear equations. x 3y z 1 2x y 2z 2 x 2y 3z 1 Solution Equation 1 Equation 2 Equation 3 FIGURE 7.12 Solution: one point FIGURE 7.13 Solution: one line 5y 4z 0 x 2y 3z 1 x 3y z 1 x 3y z 1 x 3y z 1 5y 4z 0 5y 4z 2 5y 4z 0 0 2 2 times the first Adding equation to the second equation produces a new second equation. 1 times the first Adding equation to the third equation produces a new third equation. 1 times the second Adding equation to the third equation produces a new third equation. 0 2 is a false statement, you can conclude that this system is Because inconsistent and so has no solution. Moreover, because this system is equivalent to the original system, you can conclude that the original system also has no solution. Now try Exercise 19. FIGURE 7.14 Solution: one plane As with a system of linear equations in two variables, the solution(s) of a system of linear equations in more than two variables must fall into one of three categories. The Number of Solutions of a Linear System For a system of linear equations, exactly one of the following is true. FIGURE 7.15 Solution: none 1. There is exactly one solution. 2. There are infinitely many solutions. 3. There is no solution. In Section 7.2, you learned that a system of two linear equations in two variables can be represented graphically as a pair of lines that are intersecting, coincident, or parallel. A system of three linear equations in three variables has a similar graphical representation—it can be represented as three planes in space that intersect in one point (exactly one solution) [see Figure 7.12], intersect in a line or a plane (infinitely many solutions) [see Figures 7.13 and 7.14], or have no points common to all three planes (no solution) [see Figures 7.15 and 7.16]. FIGURE 7.16 Solution: none 333202_0703.qxd 12/7/05 4:22 PM Page 523 Section 7.3 Multivariable Linear Systems 523 Example 5 A System with Infinitely Many Solutions Solve the system of linear equations. x y 3z 1 y z 0 x 2y 1 Solution Equation 1 Equation 2 Equation 3 y z 0 3y 3z 0 x y 3z 1 x y 3z 1 y z 0 0 0 Adding the first equation to the third equation produces a new third equation. 3 times the second Adding equation to the third equation produces a new third equation. This result means that Equation 3 depends on Equations 1 and 2 in the sense that it gives no additional information about the variables. Because is a true statement, you can conclude that this system will have infinitely many solutions. However, it is incorrect to say simply that the solution is “infinite.” You must also specify the correct form of the solution. So, the original system is equivalent to the system 0 0 x y 3z 1 . y z 0 y in the first equation produces In the last equation, solve for for a real number, the solutions to the given system are all of the form y a, So, every ordered triple of the form y z. Finally, letting x 2z 1. in terms of to obtain y z Back-substituting z a, where is x 2a 1, a and z a. 2a 1, a, a, a is a real number is a solution of the system. Now try Exercise 23. In Example 5, there are other ways to write the same infinite set of solutions. For instance, letting b, 1 2 b 1, 1 2 x b, b 1, the solutions could have been written as b is a real number. To convince yourself that this description produces the same set of solutions, consider the following. y and are solved x In Example 5, z. in terms of the third variable To write the correct form of the solution to the system that does not use any of the three variables a represent any of the system, let z a. real number and let Then x y. solve for and The solution can then be written in terms of a, variables of the system. which is not one of the When comparing descriptions of an infinite solution set, keep in mind that there is more than one way to describe the set. Substitution Solution 1 1 1, 0, 0 2 1 1, 1 2 20 1, 0, 0 1, 0, 0 1, 1 21 1, 1, 1 1, 1, 1 1, 1 1 1, 1 2 22 1, 2, 2 3, 2, 2 3, 1 3 1, 1 2 2 2 1 1 1, 1, 1 3 1 3, 2, 2 Same solution Same solution Same solution 333202_0703.qxd 12/5/05 9:42 AM Page 524 524 Chapter 7 Systems of Equations and Inequalities Nonsquare Systems So far, each system of linear equations you have looked at has been square, which means that the number of equations is equal to the number of variables. In a nonsquare system, the number of equations differs from the number of variables. A system of linear equations cannot have a unique solution unless there are at least as many equations as there are variables in the system. Example 6 A System with Fewer Equations than Variables Solve the system of linear equations. x 2y z 2 2x y z 1 Equation 1 Equation 2 Solution Begin by rewriting the system in row-echelon form. x 2y z 2 3y 3z 3 x 2y z 2 y z 1 2 times the first Adding equation to the second equation produces a new second equation. Multiplying the second equation 1 by produces a new second 3 equation. Solve for in terms of z, to obtain y y z 1. By back-substituting into Equation 1, you can solve for x, as follows. x 2y z 2 x 2z 1 z 2 x 2z 2 z 2 x z Write Equation 1. Substitute for y in Equation 1. Distributive Property Solve for x. Finally, by letting x a, z a, y a 1, where a is a real number, you have the solution and z a. So, every ordered triple of the form a, a 1, a, a is a real number is a solution of the system. Because there were originally three variables and only two equations, the system cannot have a unique solution. Now try Exercise 27. In Example 6, try choosing some values of to obtain different solutions of the system, such as Then check each of the solutions in the original system to verify that they are solutions of the original system. 1, 0, 1, 2, 1, 2, a 3, 2, 3. and 333202_0703.qxd 12/5/05 9:42 AM Page 525 Section 7.3 Multivariable Linear Systems 525 Applications t = 1 t = 2 Example 7 Vertical Motion t The height at time of an object that is moving in a (vertical) line with constant acceleration s 1 is given by the position equation 2 at 2 v0 t s0. a t = 3 t = 0 a is measured in feet, the acceleration is the initial velocity (at t s The height is measured in seconds, squared, a, initial height. Find the values of t 3, and s 20 at v0 v0, t 1, s0 and interpret the result. (See Figure 7.17.) s 52 and if is measured in feet per second t 0), is the t 2, s 52 s0 at and at s 60 55 50 45 40 35 30 25 20 15 10 5 FIGURE 7.17 Solution s By substituting the three values of and s0 obtain three linear equations in and . 2 a1 2 v0 52 2 a22 v0 52 2 a32 v0 20 t v0, a, 1 s0 2 s0 3 s0 t 1: t 2: t 3: When When When 1 1 1 into the position equation, you can 2a 2v0 2a 2v0 9a 6v0 2s0 2s0 2s0 104 152 140 This produces the following system of linear equations. a 2a 9a 2v0 2v0 6v0 2s0 s0 2s0 104 52 40 Now solve the system using Gaussian elimination. 9a 2v0 2v0 6v0 2v0 2v0 12v0 a a a a 2v0 2v0 2v0 v0 2s0 3s0 2s0 2s0 3s0 16s0 2s0 3s0 2s0 2s0 3 2s0 s0 104 156 40 104 156 896 104 156 40 104 78 20 2 Adding times the first equation to the second equation produces a new second equation. 9 Adding times the first equation to the third equation produces a new third equation. 6 Adding times the second equation to the third equation produces a new third equation. 1 2 Multiplying the second equation by produces a new second equation and multiplying the third equation by produces a new third equation. 1 2 48, 20. s0 and This solution So, the solution of this system is 16t 2 48t 20 and implies that the results in a position equation of object was thrown upward at a velocity of 48 feet per second from a height of 20 feet. a 32, s v0 Now try Exercise 39. 333202_0703.qxd 12/5/05 9:42 AM Page 526 526 Chapter 7 Systems of Equations and Inequalities Example 8 Data Analysis: Curve-Fitting Find a quadratic equation y ax 2 bx c whose graph passes through the points 1, 3, 1, 1, and 2, 6. (2, 6) y = 2x2 − x y 6 5 4 3 2 (−1, 3) (1, 1) − 3 − 2 −1 1 2 3 x FIGURE 7.18 y ax 2 bx c you can write the following. passes through the points 1, 3, Solution Because the graph of 1, 1, and 2, 6, x 1, y 3: x 1, y 1: x 2, y 6: When When When a12 b1 c 3 a1 2 b1 c 1 a2 2 b2 c 6 This produces the following system of linear equations. a b c 3 a b c 1 4a 2b c 6 Equation 1 Equation 2 Equation 3 The solution of this system is parabola is y 2x 2 x, a 2, b 1, and as shown in Figure 7.18. c 0. So, the equation of the Now try Exercise 43. Example 9 Investment Analysis An inheritance of $12,000 was invested among three funds: a money-market fund that paid 5% annually, municipal bonds that paid 6% annually, and mutual funds that paid 12% annually. The amount invested in mutual funds was $4000 more than the amou
nt invested in municipal bonds. The total interest earned during the first year was $1120. How much was invested in each type of fund? Solution x, y, Let represent the amounts invested in the money-market fund, municipal bonds, and mutual funds, respectively. From the given information, you can write the following equations. and z x y z 12,000 z y 4000 0.05x 0.06y 0.12z 1120 Equation 1 Equation 2 Equation 3 Rewriting this system in standard form without decimals produces the following. x 5x y y 6y z z 12z 12,000 4,000 112,000 Equation 1 Equation 2 Equation 3 Using Gaussian elimination to solve this system yields z 7000. ed in municipal bonds, and $7000 was invested in mutual funds. and So, $2000 was invested in the money-market fund, $3000 was invest- x 2000, y 3000, Now try Exercise 53. 333202_0703.qxd 12/7/05 4:23 PM Page 527 Section 7.3 Multivariable Linear Systems 527 7.3 Exercises VOCABULARY CHECK: Fill in the blanks. 1. A system of equations that is in ________ form has a “stair-step” pattern with leading coefficients of 1. 2. A solution to a system of three linear equations in three unknowns can be written as an ________ ________, which has the form x, y, z. 3. The process used to write a system of linear equations in row-echelon form is called ________ elimination. 4. Interchanging two equations of a system of linear equations is a ________ ________ that produces an equivalent system. 5. A system of equations is called ________ if the number of equations differs from the number of variables in the system. s 1 2 at2 v0t s0 6. The equation height of an object at time s t is called the ________ equation, and it models the that is moving in a vertical line with a constant acceleration a. PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. In Exercises 1–4, determine whether each ordered triple is a solution of the system of equations. 1. 3x y z 1 2x 3z 14 5y 2z 8 (a) (c) 2, 0, 3 0, 1, 3 2. 3x 4y z 17 5x y 2z 2 2x 3y 7z 21 (b) (d) 2, 0, 8 1, 0, 4 (a) (c) 3, 1, 2 4, 1, 3 (b) (d) 1, 3, 2 1, 2, 2 7. 8. 9. 10. y z 12 z 2 3y 8z 9 z 3 2x y 3z 10 x y 2z 22 4x 2y z 8 5x y z 4 z 2 8z 5z z 22 10 4 3y 3. 4. z z 4x y 8x 6y 3x y 1 2, 3 1 2, 3 (c) (a) 4 4, 7 4, 5 4 0 7 4 9 4 (b) (d) 3 1 2, 5 2, 1 4, 5 6, 3 4 4 4x y 8z 6 y z 0 4x 7y 6 2, 2, 2 1 2, 1 8, 1 (a) (c) 2 (b) (d) 33 11 2 , 10, 10 2 , 4, 4 In Exercises 5–10, use back-substitution to solve the system of linear equations. 5. 2x y 5z 24 y 2z 6 z 4 6. 4x 3y 2z 21 6y 5z 8 z 2 In Exercises 11 and 12, perform the row operation and write the equivalent system. 11. Add Equation 1 to Equation 2. x x 2x 2y 3y 3z 5 5z 4 3z 0 Equation 1 Equation 2 Equation 3 What did this operation accomplish? 12. Add 2 times Equation 1 to Equation 3. x x 2x 2y 3y 3z 5z 3z 5 4 0 Equation 1 Equation 2 Equation 3 What did this operation accomplish? 333202_0703.qxd 12/5/05 9:42 AM Page 528 528 Chapter 7 Systems of Equations and Inequalities In Exercises 13–38, solve the system of linear equations and check any solution algebraically. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 1 2 1 12 9 10 7 6 0 7 9 5 x 4z 3z 2z 4z 2x y y x x 6y 3y 2x 3x z 3z z y 2y 3y 4y 2y y 5x 3y 3y x 2y 2z 3x y z 2x 4y 2z x 4y 4z 5 4z 2 4 4 x x 2x 2x 3x 2x 4y z 2x y z 5x 3y 2z 3 3x 5y 5z 1 2x y 3z 1 x 2y 7z 2x 3x 3y 6z 6 x 2z 5 3x y z 1 6x y 5z 16 x 2 0 2x y z 3x 9y 36z y 2x 6y 8z 3 6x 8y 18z 5 x 2y z 5 5x 8y 13z 7 2x 4y z 7 x 11y 4z 3 5x 2y 3z 0 7x y 3z 0 2y 4x 2x 3z 2z 13z 5z z 3y 4x 4 13 33 4 10 8 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38 3y 2z 18 5x 13y 12z 80 2x 2 z 3y 4x 9y 7 2x 3y 3z 7 4x 18y 15z 44 z y y x 2x 2x 4z z 3w w 2w w w 2w w 1 10 5 z z 4z 10z 2z 2y 3y y x y 2x 3y 3x 4y x 2y x x 2x 2y 6z 2x 3y 4x 3y 17z 0 5x 4y 22z 0 4x 2y 19z 0 12x 5y z 0 23x 4y z 0 2x y z 0 2x 6y 4z 2 3x 2y 6z x y 5z 0 0 0 4x 3y 8x 3y z 3z 4 1 3 Vertical Motion In Exercises 39– 42, an object moving vertically is at the given heights at the specified times. Find for the object. the position equation 2 at 2 v0t s0 s 1 39. At 40. At 41. At 42. At feet feet feet feet At At At At At At At At second, seconds, seconds, second, seconds, seconds, second, seconds, seconds, second, seconds, seconds, s 128 s 80 s 0 s 48 s 64 s 48 s 452 s 372 s 260 s 132 s 100 s 36 feet feet feet feet feet feet feet feet 333202_0703.qxd 12/5/05 9:42 AM Page 529 In Exercises 43– 46, find the equation of the parabola y ax 2 bx c that passes through the points. To verify your result, use a graphing utility to plot the points and graph the parabola. 43. 45. 0, 0, 2, 2, 4, 0 2, 0, 3, 1, 4, 0 44. 46. 0, 3, 1, 4, 2, 3 1, 3, 2, 2, 3, 3 In Exercises 47–50, find the equation of the circle x 2 y 2 Dx Ey F 0 that passes through the points. To verify your result, use a graphing utility to plot the points and graph the circle. 47. 48. 49. 50. 0, 0, 2, 2, 4, 0 0, 0, 0, 6, 3, 3 3, 1, 2, 4, 6, 8 0, 0, 0, 2, 3, 0 51. Sports In Super Bowl I, on January 15, 1967, the Green Bay Packers defeated the Kansas City Chiefs by a score of 35 to 10. The total points scored came from 13 different scoring plays, which were a combination of touchdowns, extra-point kicks, and field goals, worth 6, 1, and 3 points respectively. The same number of touchdowns and extra point kicks were scored. There were six times as many touchdowns as field goals. How many touchdowns, extra-point kicks, and field goals were scored during the game? (Source: SuperBowl.com) 52. Sports In the 2004 Women’s NCAA Final Four the University of Connecticut Championship game, Huskies defeated the University of Tennessee Lady Volunteers by a score of 70 to 61. The Huskies won by scoring a combination of two-point baskets, three-point baskets, and one-point free throws. The number of two-point baskets was two more than the number of free throws. The number of free throws was one more than two times the number of three-point baskets. What combination of scoring accounted for the Huskies’ 70 points? (Source: National Collegiate Athletic Association) 53. Finance A small corporation borrowed $775,000 to expand its clothing line. Some of the money was borrowed at 8%, some at 9%, and some at 10%. How much was borrowed at each rate if the annual interest owed was $67,500 and the amount borrowed at 8% was four times the amount borrowed at 10%? 54. Finance A small corporation borrowed $800,000 to expand its line of toys. Some of the money was borrowed at 8%, some at 9%, and some at 10%. How much was borrowed at each rate if the annual interest owed was $67,000 and the amount borrowed at 8% was five times the amount borrowed at 10%? Section 7.3 Multivariable Linear Systems 529 Investment Portfolio In Exercises 55 and 56, consider an investor with a portfolio totaling $500,000 that is invested in certificates of deposit, municipal bonds, blue-chip stocks, and growth or speculative stocks. How much is invested in each type of investment? 55. The certificates of deposit pay 10% annually, and the municipal bonds pay 8% annually. Over a five-year period, the investor expects the blue-chip stocks to return 12% annually and the growth stocks to return 13% annually. The investor wants a combined annual return of 10% and also wants to have only one-fourth of the portfolio invested in stocks. 56. The certificates of deposit pay 9% annually, and the municipal bonds pay 5% annually. Over a five-year period, the investor expects the blue-chip stocks to return 12% annually and the growth stocks to return 14% annually. The investor wants a combined annual return of 10% and also wants to have only one-fourth of the portfolio invested in stocks. 57. Agriculture A mixture of 5 pounds of fertilizer A, 13 pounds of fertilizer B, and 4 pounds of fertilizer C provides the optimal nutrients for a plant. Commercial brand X contains equal parts of fertilizer B and fertilizer C. Commercial brand Y contains one part of fertilizer A and two parts of fertilizer B. Commercial brand Z contains two parts of fertilizer A, five parts of fertilizer B, and two parts of fertilizer C. How much of each fertilizer brand is needed to obtain the desired mixture? 58. Agriculture A mixture of 12 liters of chemical A, 16 liters of chemical B, and 26 liters of chemical C is required to kill a destructive crop insect. Commercial spray X contains 1, 2, and 2 parts, respectively, of these chemicals. Commercial spray Y contains only chemical C. Commercial spray Z contains only chemicals A and B in equal amounts. How much of each type of commercial spray is needed to get the desired mixture? 59. Coffee Mixture A coffee manufacturer sells a 10-pound package of coffee that consists of three flavors of coffee. Vanilla-flavored coffee costs $2 per pound, hazelnutflavored coffee costs $2.50 per pound, and mocha-flavored coffee costs $3 per pound. The package contains the same amount of hazelnut coffee as mocha coffee. The cost of the 10-pound package is $26. How many pounds of each type of coffee are in the package? 60. Floral Arrangements A florist is creating 10 centerpieces for a wedding. The florist can use roses that cost $2.50 each, lilies that cost $4 each, and irises that cost $2 each to make the bouquets. The customer has a budget of $300 and wants each bouquet to contain 12 flowers, with twice as many roses used as the other two types of flowers combined. How many of each type of flower should be in each centerpiece? 333202_0703.qxd 12/5/05 9:42 AM Page 530 530 Chapter 7 Systems of Equations and Inequalities 61. Advertising A health insurance company advertises on television, radio, and in the local newspaper. The marketing department has an advertising budget of $42,000 per month. A television ad costs $1000, a radio ad costs $200, and a newspaper ad costs $500. The department wants to run 60 ads per month, and have as many television ads as radio and newspaper ads combined. How many of each type of ad can the department run each month? 62. Radio You work as a disc jockey at your college radio station. You are suppo
sed to play 32 songs within two hours. You are to choose the songs from the latest rock, dance, and pop albums. You want to play twice as many rock songs as pop songs and four more pop songs than dance songs. How many of each type of song will you play? 63. Acid Mixture A chemist needs 10 liters of a 25% acid solution. The solution is to be mixed from three solutions whose concentrations are 10%, 20%, and 50%. How many liters of each solution will satisfy each condition? (a) Use 2 liters of the 50% solution. (b) Use as little as possible of the 50% solution. (c) Use as much as possible of the 50% solution. 64. Acid Mixture A chemist needs 12 gallons of a 20% acid solution. The solution is to be mixed from three solutions whose concentrations are 10%, 15%, and 25%. How many gallons of each solution will satisfy each condition? (a) Use 4 gallons of the 25% solution. (b) Use as little as possible of the 25% solution. (c) Use as much as possible of the 25% solution. 65. Electrical Network Applying Kirchhoff’s Laws to the I3 and are electrical network in the figure, the currents the solution of the system I1, I2, I1 3I1 I2 2I2 2I2 I3 4I3 0 7 8 find the currents. 3Ω I1 I3 4Ω 2Ω I2 7 volts 8 volts 66. Pulley System A system of pulleys is loaded with 128t1 in the ropes and the acceleration of the 32-pound pound and 32-pound weights (see figure). The tensions and weight are found by solving the system of equations t2 a t1 t1 2t2 2a a t2 0 128 32 t1 t2 where in feet per second squared. and are measured in pounds and a is measured t2 32 lb t1 128 lb (a) Solve this system. (b) The 32-pound weight in the pulley system is replaced by a 64-pound weight. The new pulley system will be modeled by the following system of equations. t1 t1 2t2 2a a t2 0 128 64 Solve this system and use your answer for the acceleration to describe what (if anything) is happening in the pulley system. . . , x1, y1 Fitting a Parabola In Exercises 67–70, find the least y ax 2 bx c for the squares regression parabola , x2, y2 xn, yn by solving the followpoints ing system of linear equations for Then use a, c. the regression feature of a graphing utility to confirm the result. (If you are unfamiliar with summation notation, look at the discussion in Section 9.1 or in Appendix B at the website for this text at college.hmco.com.) and b, . , nc n i1 xib n i1 i a n x2 i1 yi n i1 xic n i1 i b n x2 i1 n i1 i c n x2 i1 i b n x3 i1 i a n x 3 i1 i a n x 4 i1 xi yi x2 i yi 333202_0703.qxd 12/5/05 9:42 AM Page 531 67. (−2, 6) (−4, 5) y 8 6 4 2 (2, 6) (4, 2) −4 −2 2 4 69. y 12 10 8 6 (0, 0) (4, 12) (3, 6) (2, 2) −8 −6 −4 −2 8642 x x 68. y 4 2 (−1, 0) (2, 5) (1, 2) (−2, 0) −4 −2 (0, 1) 2 70. y 12 10 4 2 (0, 10) (1, 9) (2, 6) (3, 0) −8 −6 −4 8642 x x Model It 71. Data Analysis: Wildlife A wildlife management team studied the reproduction rates of deer in three tracts of a wildlife preserve. Each tract contained 5 acres. In each tract, the number of females and the that had offspring the following percent of females year, were recorded. The results are shown in the table. x, y Number, x Percent, y 100 120 140 75 68 55 (a) Use the technique demonstrated in Exercises 67–70 to set up a system of equations for the data and to find a least squares regression parabola that models the data. (b) Use a graphing utility to graph the parabola and the data in the same viewing window. (c) Use the model to create a table of estimated values y. of Compare the estimated values with the actual data. (d) Use the model to estimate the percent of females that had offspring when there were 170 females. (e) Use the model to estimate the number of females when 40% of the females had offspring. Section 7.3 Multivariable Linear Systems 531 72. Data Analysis: Stopping Distance automobile braking system, the speed and the stopping distance table. In testing a new (in miles per hour) (in feet) were recorded in the y x Speed, x Stopping distance, y 30 40 50 55 105 188 (a) Use the technique demonstrated in Exercises 67–70 to set up a system of equations for the data and to find a least squares regression parabola that models the data. (b) Graph the parabola and the data on the same set of axes. (c) Use the model to estimate the stopping distance when the speed is 70 miles per hour. 73. Sports In Super Bowl XXXVIII, on February 1, 2004, the New England Patriots beat the Carolina Panthers by a score of 32 to 29. The total points scored came from 16 different scoring plays, which were a combination of touchdowns, extra-point kicks, two-point conversions, and field goals, worth 6, 1, 2, and 3 points, respectively. There were four times as many touchdowns as field goals and two times as many field goals as two-point conversions. How many touchdowns, extra-point kicks, two-point conversions, and field goals were scored during the game? (Source: SuperBowl.com) 74. Sports In the 2005 Orange Bowl, the University of Southern California won the National Championship by defeating the University of Oklahoma by a score of 55 to 19. The total points scored came from 22 different scoring plays, which were a combination of touchdowns, extrapoint kicks, field goals and safeties, worth 6, 1, 3, and 2 points respectively. The same number of touchdowns and extra-point kicks were scored, and there were three times as many field goals as safeties. How many touchdowns, extra-point kicks, field goals, and safeties were scored? (Source: ESPN.com) 333202_0703.qxd 12/5/05 9:42 AM Page 532 532 Chapter 7 Systems of Equations and Inequalities and In Exercises 75–78, find values of Advanced Applications y, certain optimization problems in calculus, and a Lagrange multiplier. x, that satisfy the system. These systems arise in is called 75. 76. 77. 78. y 0 x 0 x y 10 0 2x 0 2y 0 x y 4 0 2x 2x 0 2 2y 2 0 2y 0 y x2 0 2x 1 0 2x y 100 0 Skills Review In Exercises 87–90, solve the percent problem. 87. What is 71 2% 88. 225 is what percent of 150? of 85? 89. 0.5% of what number is 400? 90. 48% of what number is 132? 91. 92. In Exercises 91–96, perform the operation and write the result in standard form. 7 i 4 2i 6 3i 1 6i 4 i5 2i 1 2i3 4i 6 94. 93. 95. i 1 i i 4 i 1 i 2i 8 3i Synthesis 96. True or False? the statement is true or false. Justify your answer. In Exercises 79 and 80, determine whether 79. The system x 3y 6z 2y z z 16 1 3 is in row-echelon form. 80. If a system of three linear equations is inconsistent, then its graph has no points common to all three equations. 81. Think About It Are the following two systems of equations equivalent? Give reasons for your answer. x 3y z 6 2x y 2z 1 3x 2y z 2 x 3y z 7y 4z 7y 4z 6 1 16 82. Writing When using Gaussian elimination to solve a system of linear equations, explain how you can recognize that the system has no solution. Give an example that illustrates your answer. In Exercises 83–86, find two systems of linear equations that have the ordered triple as a solution. (There are many correct answers.) 4, 1, 2 3, 1 2, 7 5, 2, 1 3 2, 4, 7 83. 85. 86. 84. 4 In Exercises 97–100, (a) determine the real zeros of and (b) sketch the graph of f f. f x x 3 x2 12x f x 8x 4 32x2 f x 2x 3 5x2 21x 36 f x 6x 3 29x2 6x 5 97. 98. 99. 100. In Exercises 101–104, use a graphing utility to construct a table of values for the equation. Then sketch the graph of the equation by hand. 101. 102. 103. 104. y 4x4 5 y 5 x1 4 y 1.90.8x 3 y 3.5x2 6 2 In Exercises 105 and 106, solve the system by elimination. 105. 106. 2x y 120 x 2y 120 6x 5y 3 10x 12y 5 107. Make a Decision To work an extended application analyzing the earnings per share for Wal-Mart Stores, Inc. from 1988 text’s website at (Data Source: Wal-Mart Stores, Inc.) college.hmco.com. to 2003, visit this 333202_0704.qxd 12/5/05 9:43 AM Page 533 7.4 Partial Fractions Section 7.4 Partial Fractions 533 What you should learn • Recognize partial fraction decompositions of rational expressions. • Find partial fraction decompositions of rational expressions. Why you should learn it Partial fractions can help you analyze the behavior of a rational function. For instance, in Exercise 57 on page 540, you can analyze the exhaust temperatures of a diesel engine using partial fractions. © Michael Rosenfeld/Getty Images Section A.4, shows you how to combine expressions such as 1 x 2 1 x 3 5 x 2x 3. The method of partial fractions shows you how to reverse this process. 5 x 2x 3 ? x 2 ? x 3 Introduction In this section, you will learn to write a rational expression as the sum of two or more simpler rational expressions. For example, the rational expression x 7 x2 x 6 can be written as the sum of two fractions with first-degree denominators. That is, Partial fraction decomposition x 7 x2 x 6 of . Partial fraction Partial fraction Each fraction on the right side of the equation is a partial fraction, and together they make up the partial fraction decomposition of the left side. Decomposition of 1. Divide if improper: If Nx ≥ degree of Dx, obtain Nx/Dx NxDx into Partial Fractions is an improper fraction degree of divide the denominator into the numerator to Nx Dx polynomial N1 x Dx and apply Steps 2, 3, and 4 below to the proper rational expression N1 is the remainder from the division of by xDx. Dx. Note that x N1 Nx 2. Factor the denominator: Completely factor the denominator into factors of the form px qm and ax 2 bx c where ax 2 bx cn is irreducible. 3. Linear factors: For each factor of the form px qm, the partial fracm tion decomposition must include the following sum of fractions. A1 px q A2 px q2 . . . Am px qm 4. Quadratic factors: For each factor of the form ax 2 bx cn, n tial fraction decomposition must include the following sum of Bnx Cn B2x C2 ax 2 bx cn ax 2 bx c2 B1x C1 ax 2 bx c . . . the parfractions. 333202_0704.qxd 12/5/05 9:43 AM Page 534 534 Chapter 7 Systems of Equations and Inequalities Partial Fraction Decomposition Algebraic techniques for determining the constants in the numerators of partial fractions are demonstrated in the examples that follow. No
te that the techniques vary slightly, depending on the type of factors of the denominator: linear or quadratic, distinct or repeated. Example 1 Distinct Linear Factors Write the partial fraction decomposition of x 7 x 2 x 6 . Solution The expression is proper, so be sure to factor the denominator. Because x 2 x 6 x 3x 2, you should include one partial fraction with a constant numerator for each linear factor of the denominator. Write the form of the decomposition as follows Write form of decomposition. Multiplying each side of this equation by the least common denominator, x 3x 2, leads to the basic equation x 7 Ax 2 Bx 3. Basic equation Because this equation is true for all you can substitute any convenient values of x A B. and Values of that are especially that will help determine the constants x 2 convenient are ones that make the factors and equal to zero. For instance, let x x 3 x 2. x, Then 2 7 A2 2 B2 3 Te c h n o l o g y You can use a graphing utility to check graphically the decomposition found in Example 1. To do this, graph x 7 x2 x 6 y1 and y2 2 x 3 1 x 2 in the same viewing window. The graphs should be identical, as shown below. 5 A0 B5 5 5B 1 B. To solve for x 3 A, let and obtain 3 7 A3 2 B3 3 10 A5 B0 10 5A 2 A. Substitute 2 for x. Substitute 3 for x. −9 6 −6 So, the partial fraction decomposition is Check this result by combining the two partial fractions on the right side of the equation, or by using your graphing utility. Now try Exercise 15. 333202_0704.qxd 12/5/05 9:43 AM Page 535 Section 7.4 Partial Fractions 535 The next example shows how to find the partial fraction decomposition of a rational expression whose denominator has a repeated linear factor. Example 2 Repeated Linear Factors Write the partial fraction decomposition of x 4 2x3 6x2 20x 6 x3 2x2 x . Solution This rational expression is improper, so you should begin by dividing the numerator by the denominator to obtain x 5x2 20x 6 x3 2x2 x . Because the denominator of the remainder factors as x 3 2x 2 x xx 2 2x 1 xx 12 you should include one partial fraction with a constant numerator for each power x of and and write the form of the decomposition as follows. x 1 5x 2 20x 6 xx 12 Multiplying by the LCD, A x B x 1 xx 12, C x 12 Write form of decomposition. leads to the basic equation 5x 2 20x 6 Ax 12 Bxx 1 Cx. Basic equation Letting x 1 eliminates the B 512 201 6 A1 12 B11 1 C1 - and -terms and yields A 5 20 6 0 0 C C 9. Letting x 0 eliminates the - and B C -terms and yields 502 200 6 A0 12 B00 1 C0 6 A1 0 0 6 A. use any other value for along with the known values of x and A 6, C 9, B, x 1, At this point, you have exhausted the most convenient choices for the value of C. So, using 512 201 6 61 12 B11 1 91 31 64 2B 9 2 2B 1 B. x, so to find A and So, the partial fraction decomposition is x 6 x x 4 2x3 6x2 20x 6 x3 2x2 x 1 x 1 9 x 12 . Now try Exercise 27. 333202_0704.qxd 12/5/05 9:43 AM Page 536 536 Chapter 7 Systems of Equations and Inequalities Historical Note John Bernoulli (1667–1748), a Swiss mathematician, introduced the method of partial fractions and was instrumental in the early development of calculus. Bernoulli was a professor at the University of Basel and taught many outstanding students, the most famous of whom was Leonhard Euler. The procedure used to solve for the constants in Examples 1 and 2 works well when the factors of the denominator are linear. However, when the denominator contains irreducible quadratic factors, you should use a different procedure, which involves writing the right side of the basic equation in polynomial form and equating the coefficients of like terms. Then you can use a system of equations to solve for the coefficients. Example 3 Distinct Linear and Quadratic Factors Write the partial fraction decomposition of 3x 2 4x 4 x 3 4x . Solution This expression is proper, so factor the denominator. Because the denominator factors as x 3 4x xx 2 4 you should include one partial fraction with a constant numerator and one partial fraction with a linear numerator and write the form of the decomposition as follows. 3x 2 4x 4 x 3 4x A x Multiplying by the LCD, Bx C x 2 4 xx 2 4, Write form of decomposition. yields the basic equation 3x 2 4x 4 Ax 2 4 Bx Cx. Basic equation Expanding this basic equation and collecting like terms produces 3x 2 4x 4 Ax 2 4A Bx 2 Cx A Bx 2 Cx 4A. Polynomial form Finally, because two polynomials are equal if and only if the coefficients of like terms are equal, you can equate the coefficients of like terms on opposite sides of the equation. 3x 2 4x 4 A Bx 2 Cx 4A Equate coefficients of like terms. You can now write the following system of linear equations. B A 4A C 3 4 4 From this system you can see that A 1 into Equation 1 yields 1 B 3 ⇒ B 2. Equation 1 Equation 2 Equation 3 A 1 and C 4. Moreover, substituting So, the partial fraction decomposition is 3x 2 4x 4 x 3 4x 1 x 2x 4 x 2 4 . Now try Exercise 29. 333202_0704.qxd 12/5/05 9:43 AM Page 537 Section 7.4 Partial Fractions 537 The next example shows how to find the partial fraction decomposition of a rational expression whose denominator has a repeated quadratic factor. Example 4 Repeated Quadratic Factors Write the partial fraction decomposition of 8x 3 13x x 2 22 . Solution You need to include one partial fraction with a linear numerator for each power of x 2 2. 8x 3 13x x 2 22 Ax B x 2 2 Cx D x 2 22 Write form of decomposition. Multiplying by the LCD, x 2 22, yields the basic equation 8x 3 13x Ax Bx 2 2 Cx D Basic equation Ax 3 2Ax Bx 2 2B Cx D Ax 3 Bx 2 2A Cx 2B D. Polynomial form Equating coefficients of like terms on opposite sides of the equation 8x 3 0x 2 13x 0 Ax 3 Bx 2 2A Cx 2B D produces the following system of linear equations. A 2A B C 2B 8 0 13 0 A 8 D and Finally, use the values 28 C 13 C 3 20 D 0 D 0 Equation 1 Equation 2 Equation 3 Equation 4 B 0 to obtain the following. Substitute 8 for A in Equation 3. Substitute 0 for B in Equation 4. A 8, B 0, C 3, and D 0, the partial fraction decomposition So, using is 8x 3 13x x 2 22 8x x2 2 3x x 2 22 . Check this result by combining the two partial fractions on the right side of the equation, or by using your graphing utility. Now try Exercise 49. 333202_0704.qxd 12/5/05 9:43 AM Page 538 538 Chapter 7 Systems of Equations and Inequalities Guidelines for Solving the Basic Equation Linear Factors 1. Substitute the zeros of the distinct linear factors into the basic equation. 2. For repeated linear factors, use the coefficients determined in Step 1 to x rewrite the basic equation. Then substitute other convenient values of and solve for the remaining coefficients. Quadratic Factors 1. Expand the basic equation. 2. Collect terms according to powers of x. 3. Equate the coefficients of like terms to obtain equations involving A, B, C, and so on. 4. Use a system of linear equations to solve for A, B, C, . . . . Keep in mind that for improper rational expressions such as Nx Dx 2x3 x2 7x 7 x2 x 2 you must first divide before applying partial fraction decomposition. W RITING ABOUT MATHEMATICS Error Analysis You are tutoring a student in algebra. In trying to find a partial fraction decomposition, the student writes the following. x 2 1 xx 1 x 2 1 xx 1 B A x x 1 Ax 1 xx 1 Bx xx 1 x 2 1 Ax 1 Bx Basic equation By substituting A 1 that the following. x 0 and and B 2. x 1 into the basic equation, the student concludes However, in checking this solution, the student obtains 1 x 2 x 1 1x 1 2x xx 1 x 1 xx 1 x2 1 xx 1 What has gone wrong? 333202_0704.qxd 12/5/05 9:43 AM Page 539 Section 7.4 Partial Fractions 539 7.4 Exercises VOCABULARY CHECK: Fill in the blanks. 1. The process of writing a rational expression as the sum or difference of two or more simpler rational expressions is called ________ ________ ________. 2. If the degree of the numerator of a rational expression is greater than or equal to the degree of the denominator, then the fraction is called ________. 3. In order to find the partial fraction decomposition of a rational expression, the denominator must be and ________ factors of the form px qm completely factored into ________ factors of the form ax2 bx cn, which are ________ over the rationals. 4. The ________ ________ is derived after multiplying each side of the partial fraction decomposition form by the least common denominator. PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. In Exercises 1– 4, match the rational expression with the form of its decomposition. [The decompositions are labeled (a), (b), (c), and (d).] (a) (c) 1. 3 x2 x 3x 1 xx 4 3x 1 xx2 4 (b) (d) 2. 4. A x B x 4 Bx C x2 4 A x 3x 1 x2x 4 3x 1 xx2 4 In Exercises 5–14, write the form of the partial fraction decomposition of the rational expression. Do not solve for the constants. 5. 7. 9. 11. 13. 7 x 2 14x 12 x 3 10x 2 4x2 3 x 53 2x 3 x 3 10x x 1 xx2 12 6. 8. 10. 12. 14. x 2 x 2 4x 3 x2 3x 2 4x 3 11x2 6x 5 x 24 x 6 2x3 8x x 4 x23x 12 In Exercises 15–38, write the partial fraction decomposition of the rational expression. Check your result algebraically. 15. 17. 1 x 2 1 1 x 2 x 16. 18. 1 4x 2 9 3 x 2 3x 19. 21. 23. 25. 27. 29. 31. 33. 35. 37. 1 2x 2 x 3 x 2 x 2 x 2 12x 12 x 3 4x 4x 2 2x 1 x 2x 1 3x x 32 x 2 1 xx 2 1 x x 3 x2 2x 2 x 2 x 4 2x 2 8 x 16x 4 1 x 2 5 x 1x 2 2x 3 20. 22. 24. 26. 28. 30. 32. 34. 36. 38 4x 3 x 2 xx 4 2x 3 x 12 6x 2 1 x 2x 12 x x 1x 2 x 1 x 6 x 3 3x2 4x 12 2x 2 x 8 x 2 42 x 1 x 3 x x 2 4x 7 x 1x 2 2x 3 In Exercises 39– 44, write the partial fraction decomposition of the improper rational expression. 39. 41. 43. x2 x x2 x 1 2x 3 x2 x 5 x2 3x 2 x 4 x 13 40. 42. 44. x2 4x x2 x 6 x 3 2x2 x 1 x2 3x 4 16x 4 2x 13 333202_0704.qxd 12/5/05 9:43 AM Page 540 540 Chapter 7 Systems of Equations and Inequalities In Exercises 45–52, write the partial fraction decomposition of the rational expression. Use a graphing utility to check your result graphically. 45. 47. 49. 51. 5 x 2x 22 2x 3 4x 2 15x 5 x 2 2x 8 46. 48. 50. 52. 3x 2 7x 2 x 3 x
4x 2 1 2xx 12 x 3 x 22x 22 x 3 x 3 x 2 x 2 Graphical Analysis In Exercises 53–56, (a) write the partial fraction decomposition of the rational function, (b) identify the graph of the rational function and the graph of each term of its decomposition, and (c) state any relationship between the vertical asymptotes of the graph of the rational function and the vertical asymptotes of the graphs of the terms of the decomposition. To print an enlarged copy of the graph, go to the website www.mathgraphs.com. 53. y x 12 xx 4 y 54. y 2(x 1)2 xx2 55. y 24x 3 x2 9 y x 4 8 −4 −8 4 −4 x 2 4 56. y 24x2 15x 39 x2x2 10x 26 y 12 8 4 x 4 8 –4 –4 Model It 57. Thermodynamics The magnitude of the range of exhaust temperatures (in degrees Fahrenheit) in an experimental diesel engine is approximated by the model R 20004 3x 0 < x ≤ 1 R 11 7x7 4x, Model It (co n t i n u e d ) where x is the relative load (in foot-pounds). (a) Write the partial fraction decomposition of the equation. (b) The decomposition in part (a) is the difference of two fractions. The absolute values of the terms give the expected maximum and minimum temperatures of the exhaust gases for different loads. Ymax 1st term Ymin 2nd term Write the equations for Ymax and Ymin. (c) Use a graphing utility to graph each equation from part (b) in the same viewing window. (d) Determine the expected maximum and minimum temperatures for a relative load of 0.5. Synthesis 58. Writing Describe two ways of solving for the constants in a partial fraction decomposition. True or False? the statement is true or false. Justify your answer. In Exercises 59 and 60, determine whether 59. For the rational expression x x 10x 102 the partial fraction decomposition is of the form B x 102 . 60. When writing the partial fraction decomposition of the A x 10 expression x 3 x 2 x2 5x 14 denominator. the first step is to factor the In Exercises 61– 64, write the partial fraction decomposition of the rational expression. Check your result algebraically. Then assign a value to the constant a to check the result graphically. 61. 63. 1 a 2 x 2 1 ya y Skills Review 62. 64. 1 xx a 1 x 1a x In Exercises 65–70, sketch the graph of the function. 65. 67. 69. f x x2 9x 18 f x x2x 3 f x x2 x 6 x 5 66. 68. f x 2x2 9x 5 f x 1 2x 3 1 70. f x 3x 1 x2 4x 12 333202_0705.qxd 12/5/05 9:45 AM Page 541 7.5 Systems of Inequalities Section 7.5 Systems of Inequalities 541 What you should learn • Sketch the graphs of inequali- ties in two variables. • Solve systems of inequalities. • Use systems of inequalities in two variables to model and solve real-life problems. Why you should learn it You can use systems of inequalities in two variables to model and solve real-life problems. For instance, in Exercise 77 on page 550, you will use a system of inequalities to analyze the retail sales of prescription drugs. The Graph of an Inequality a 2x 2 3y 2 ≥ 6 3x 2y < 6 a, b b and is a solution of an inequality in and are inequalities in two The statements x variables. An ordered pair if the respectively. The graph inequality is true when and are substituted for and of an inequality is the collection of all solutions of the inequality. To sketch the graph of an inequality, begin by sketching the graph of the corresponding equation. The graph of the equation will normally separate the plane into two or more regions. In each such region, one of the following must be true. 1. All points in the region are solutions of the inequality. 2. No point in the region is a solution of the inequality. y, y x So, you can determine whether the points in an entire region satisfy the inequality by simply testing one point in the region. Sketching the Graph of an Inequality in Two Variables 1. Replace the inequality sign by an equal sign, and sketch the graph of the resulting equation. (Use a dashed line for < or > and a solid line for ≤ ≥ or .) 2. Test one point in each of the regions formed by the graph in Step 1. If the point satisfies the inequality, shade the entire region to denote that every point in the region satisfies the inequality. Example 1 Sketching the Graph of an Inequality y ≥ x2 1, To sketch the graph of y x 2 1, above the parabola the points that satisfy the inequality are those lying above (or on) the parabola. begin by graphing the corresponding equation which is a parabola, as shown in Figure 7.19. By testing a point you can see that and a point below the parabola 0, 2, 0, 0 Jon Feingersh/Masterfile y ≥ x2 − 1 y y = x2 − 1 Note that when sketching the graph of an inequality in two variables, a dashed line means all points on the line or curve are not solutions of the inequality. A solid line means all points on the line or curve are solutions of the inequality. 2 1 (0, 0) −2 x 2 Test point above parabola −2 Test point below parabola (0, −2) FIGURE 7.19 Now try Exercise 1. 333202_0705.qxd 12/5/05 9:45 AM Page 542 542 Chapter 7 Systems of Equations and Inequalities The inequality in Example 1 is a nonlinear inequality in two variables. Most of the following examples involve linear inequalities such as a ( and are not both zero). The graph of a linear inequality is a half-plane lying on one side of the line ax by c. ax by < c b Example 2 Sketching the Graph of a Linear Inequality Sketch the graph of each linear inequality. a. x > 2 b. y ≤ 3 Te c h n o l o g y and A graphing utility can be used to graph an inequality or a system of inequalities. For instance, to graph y ≥ x 2, y x 2 enter use the shade feature of the graphing utility to shade the correct part of the graph. You should obtain the graph below. Consult the user’s guide for your graphing utility for specific keystrokes. −10 10 −10 10 y x − y < 2 1 2 x Solution a. The graph of the corresponding equation that satisfy the inequality shown in Figure 7.20. x 2 is a vertical line. The points are those lying to the right of this line, as x > 2 b. The graph of the corresponding equation y 3 is a horizontal line. The points are those lying below (or on) this line, as that satisfy the inequality shown in Figure 7.21. y ≤ 3 x > −2 y x = −2 −4 −3 −1 2 1 −1 −2 x FIGURE 7.20 Now try Exercise 32 −1 FIGURE 7.21 x 1 2 Example 3 Sketching the Graph of a Linear Inequality Sketch the graph of x y < 2. Solution is a line, as shown in Figure The graph of the corresponding equation 7.22. Because the origin satisfies the inequality, the graph consists of the half-plane lying above the line. (Try checking a point below the line. Regardless of which point you choose, you will see that it does not satisfy the inequality.) x y 2 0, 0 x − y = 2 To graph a linear inequality, Now try Exercise 9. slope-intercept form. For instance, by writing y > x 2 it can help to write the inequality in x y < 2 in the form you can see that the solution points lie above the line as shown in Figure 7.22. x y 2 or y x 2, (0, 0) −1 −2 FIGURE 7.22 333202_0705.qxd 12/5/05 9:45 AM Page 543 Section 7.5 Systems of Inequalities 543 Systems of Inequalities Many practical problems in business, science, and engineering involve systems of linear inequalities. A solution of a system of inequalities in and is a point x, y that satisfies each inequality in the system. y x To sketch the graph of a system of inequalities in two variables, first sketch the graph of each individual inequality (on the same coordinate system) and then find the region that is common to every graph in the system. This region represents the solution set of the system. For systems of linear inequalities, it is helpful to find the vertices of the solution region. Example 4 Solving a System of Inequalities Sketch the graph (and label the vertices) of the solution set of the system Inequality 1 Inequality 2 Inequality 3 Solution The graphs of these inequalities are shown in Figures 7.22, 7.20, and 7.21, respectively, on page 542. The triangular region common to all three graphs can be found by superimposing the graphs on the same coordinate system, as shown in Figure 7.23. To find the vertices of the region, solve the three systems of corresponding equations obtained by taking pairs of equations representing the boundaries of the individual regions. Using different colored pencils to shade the solution of each inequality in a system will make identifying the solution of the system of inequalities easier. Vertex A: 2, 4 2 2 x y x Vertex B: 5, 3 2 x y y 3 Vertex C: x y 2, 3 2 3 y y = 3 C = (− 2, 3) y B = (5, 3 Solution set A = (−2, −4) −2 −3 −4 x = −2 −1 1 −2 −3 −4 FIGURE 7.23 Note in Figure 7.23 that the vertices of the region are represented by open dots. This means that the vertices are not solutions of the system of inequalities. Now try Exercise 35. 333202_0705.qxd 12/5/05 9:45 AM Page 544 544 Chapter 7 Systems of Equations and Inequalities For the triangular region shown in Figure 7.23, each point of intersection of a pair of boundary lines corresponds to a vertex. With more complicated regions, two border lines can sometimes intersect at a point that is not a vertex of the region, as shown in Figure 7.24. To keep track of which points of intersection are actually vertices of the region, you should sketch the region and refer to your sketch as you find each point of intersection. y Not a vertex x FIGURE 7.24 Example 5 Solving a System of Inequalities Sketch the region containing all points that satisfy the system of inequalities. x2 y ≤ 1 x y ≤ 1 Inequality 1 Inequality 2 Solution As shown in Figure 7.25, the points that satisfy the inequality x 2 y ≤ 1 Inequality 1 are the points lying above (or on) the parabola given by y x 2 1. Parabola The points satisfying the inequality x y ≤ 1 Inequality 2 y = x2 + 2 (−1, 0) FIGURE 7.25 (2, 3) are the points lying below (or on) the line given by y x 1. Line To find the points of intersection of the parabola and the line, solve the system of corresponding equations. x 2 x2 y 1 x y 1 and So, the region containing all points that satisfy the system is indicated by Using the method of substitution, you can find the solutions
to be 2, 3. the shaded region in Figure 7.25. 1, 0 Now try Exercise 37. 333202_0705.qxd 12/5/05 9:45 AM Page 545 Section 7.5 Systems of Inequalities 545 When solving a system of inequalities, you should be aware that the system might have no solution or it might be represented by an unbounded region in the plane. These two possibilities are shown in Examples 6 and 7. Example 6 A System with No Solution Sketch the solution set of the system of inequalities. x y > x y < 3 1 Inequality 1 Inequality 2 Solution From the way the system is written, it is clear that the system has no solution, x y because the quantity and greater than 3. Graphically, the inequality is represented by the half-plane lying x y < 1 above the line is represented by the as shown in Figure 7.26. These two half-plane lying below the line half-planes have no points in common. So, the system of inequalities has no solution. and the inequality x y 1, cannot be both less than x y 32 −1 1 2 3 x −1 −2 x + y < −1 FIGURE 7.26 Now try Exercise 39. Example 7 An Unbounded Solution Set Sketch the solution set of the system of inequalities. x y < 3 x 2y > 3 Inequality 1 Inequality 2 Solution The graph of the inequality x y 3, x 2y 3. the half-plane that lies above the line half-planes is an infinite wedge that has a vertex at the system of inequalities is unbounded. x y < 3 is the half-plane that lies below the line is The intersection of these two 3, 0. So, the solution set of x 2y > 3 as shown in Figure 7.27. The graph of the inequality + 2y = 3 (3, 0) −1 1 2 3 x FIGURE 7.27 Now try Exercise 41. 333202_0705.qxd 12/5/05 9:45 AM Page 546 546 p Chapter 7 Systems of Equations and Inequalities Applications Consumer surplus Demand curve Equilibrium point e c i r P Supply curve Producer surplus Number of units x FIGURE 7.28 Example 9 in Section 7.2 discussed the equilibrium point for a system of demand and supply functions. The next example discusses two related concepts that economists call consumer surplus and producer surplus. As shown in Figure 7.28, the consumer surplus is defined as the area of the region that lies below the demand curve, above the horizontal line passing through the equilibrium point, and to the p -axis. Similarly, the producer surplus is defined as the area of the right of the region that lies above the supply curve, below the horizontal line passing through p the equilibrium point, and to the right of the -axis. The consumer surplus is a measure of the amount that consumers would have been willing to pay above what they actually paid, whereas the producer surplus is a measure of the amount that producers would have been willing to receive below what they actually received. Example 8 Consumer Surplus and Producer Surplus The demand and supply functions for a new type of personal digital assistant are given by p 150 0.00001x p 60 0.00002x Demand equation Supply equation 175 150 125 100 75 50 25 ) Supply vs. Demand p p = 150 − 0.00001x Consumer surplus p = 120 Producer surplus p where consumer surplus and producer surplus for these two equations. is the price (in dollars) and represents the number of units. Find the x Solution Begin by finding the equilibrium point (when supply and demand are equal) by solving the equation 60 0.00002x 150 0.00001x. In Example 9 in Section 7.2, you saw that the solution is p $120. which corresponds to an equilibrium price of surplus and producer surplus are the areas of the following triangular regions. units, So, the consumer x 3,000,000 p = 60 + 0.00002x 1,000,000 3,000,000 Number of units x Consumer Surplus p ≤ 150 0.00001x p ≥ 120 x ≥ 0 Producer Surplus p ≥ 60 0.00002x p ≤ 120 x ≥ 0 FIGURE 7.29 In Figure 7.29, you can see that the consumer and producer surpluses are defined as the areas of the shaded triangles. Consumer surplus Producer surplus 1 2 1 2 1 2 1 2 (base)(height) 3,000,00030 $45,000,000 (base)(height) 3,000,00060 $90,000,000 Now try Exercise 65. 333202_0705.qxd 12/5/05 9:45 AM Page 547 Section 7.5 Systems of Inequalities 547 Example 9 Nutrition The liquid portion of a diet is to provide at least 300 calories, 36 units of vitamin A, and 90 units of vitamin C. A cup of dietary drink X provides 60 calories, 12 units of vitamin A, and 10 units of vitamin C. A cup of dietary drink Y provides 60 calories, 6 units of vitamin A, and 30 units of vitamin C. Set up a system of linear inequalities that describes how many cups of each drink should be consumed each day to meet or exceed the minimum daily requirements for calories and vitamins. Solution Begin by letting and x y represent the following. x y number of cups of dietary drink X number of cups of dietary drink Y To meet or exceed the minimum daily requirements, the following inequalities must be satisfied. 60x 60y ≥ 12x 6y ≥ 10x 30y ≥ x ≥ y ≥ 300 36 90 0 0 Calories Vitamin A Vitamin C The last two inequalities are included because and cannot be negative. The graph of this system of inequalities is shown in Figure 7.30. (More is said about this application in Example 6 in Section 7.6.) y x y 8 6 4 2 (0, 6) (1, 4) (3, 2) (9, 0) x 2 4 6 8 10 FIGURE 7.30 Now try Exercise 69. W RITING ABOUT MATHEMATICS Creating a System of Inequalities Plot the points coordinate plane. Draw the quadrilateral that has these four points as its vertices. Write a system of linear inequalities that has the quadrilateral as its solution. Explain how you found the system of inequalities. and in a 4, 0, 3, 2, 0, 0, 0, 2 333202_0705.qxd 12/5/05 9:45 AM Page 548 548 Chapter 7 Systems of Equations and Inequalities 7.5 Exercises VOCABULARY CHECK: Fill in the blanks. a, b 1. An ordered pair a when and are substituted for and y, b x respectively. is a ________ of an inequality in and x y if the inequality is true 2. The ________ of an inequality is the collection of all solutions of the inequality. 3. The graph of a ________ inequality is a half-plane lying on one side of the line ax by c. 4. A ________ of a system of inequalities in and x y is a point x, y that satisfies each inequality in the system. 5. The area of the region that lies below the demand curve, above the horizontal line passing through the equilibrium point, to the right of the -axis is called the ________ _________. p PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. In Exercises 1–14, sketch the graph of the inequality. 29. y 30. 1. 3. 5. 7. 9. 11. 12. y < 2 x2 2y x ≥ 4 x 12 y 22 < 9 x 12 y 42 > 9 2. 4. 6. 8. 10 > 2x 4 5x 3y ≥ 15 13. y ≤ 1 1 x 2 14. y > 15 x2 x 4 In Exercises 15–26, use a graphing utility to graph the inequality. Shade the region representing the solution. 15. 17. 19. 21. 23. 25. y < ln x y < 3x4 y ≥ 2 3x 1 y < 3.8x 1.1 x 2 5y 10 ≤ 0 5 2 y 3x2 6 ≥ 0 16. 18. 20. 22. 24. 26. y ≥ 6 lnx 5 y ≤ 22x0.5 7 y ≤ 6 3 2x y ≥ 20.74 2.66x 2x 10 x2 3 4 In Exercises 27–30, write an inequality for the shaded region shown in the figure. 6 4 2 −4 −2 −2 2 −4 x y 4 2 −2 −4 x 2 4 In Exercises 31–34, determine whether each ordered pair is a solution of the system of linear inequalities. 31. 32. 33. 34. y > 3 y ≤ 8x 3 x ≥ 4 2x 5y ≥ 3 x2 y2 ≥ 36 y < 4 4x 2y < 7 3x y > y 1 2 x2 ≤ 15x 4y > 3x y ≤ 10 a) (c) 0, 0 4, 0 1, 3 (b) (d) 3, 11 (a) (c) 0, 2 8, 2 6, 4 (b) (d) 3, 2 (a) (c) 0, 10 2, 9 (a) (c) 1, 7 6, 0 0, 1 (b) (d) 1, 6 5, 1 (b) (d) 4, 8 27. y 4 28. y 4 2 −4 −2 −2 x 2 −4 In Exercises 35–48, sketch the graph and label the vertices of the solution set of the system of inequalities. x 4 35. 37. x y ≤ 1 x2 36. 2y 3x 38. 2x2 333202_0705.qxd 12/5/05 9:45 AM Page 549 39. 41. 43. 45. 47. x 2x 2y < 4y > y < 2x y > 2 6x 3y < 2 3x x > y2 x < y 2 x2 y2 ≤ 9 x2 y2 ≥ 1 3x 4 ≥ y2 x y < 0 6 2 3 40. 42. 44. 46. 48. 36 5 6 x 7y > 5x 2y > 6x 5y > x 2y < 6 5x 3y > 9 x y2 > 0 x y > 2 x2 ≤ 25 y2 ≤ 0 3y x < 2y y2 0 < x y 4x Section 7.5 Systems of Inequalities 549 59. y 60, 8 ) 1 2 3 4 x 61. Rectangle: vertices at 2, 1, 5, 1, 5, 7, 2, 7 62. Parallelogram: vertices at 0, 0, 4, 0, 1, 4, 5, 4 63. Triangle: vertices at 64. Triangle: vertices at 0, 0, 5, 0, 2, 3 1, 0, 1, 0, 0, 1 In Exercises 49–54, use a graphing utility to graph the inequalities. Shade the region representing the solution set of the system. Supply and Demand In Exercises 65–68, (a) graph the systems representing the consumer surplus and producer surplus for the supply and demand equations and (b) find the consumer surplus and producer surplus. 49. 51. 53. y ≤ 3x 1 y ≥ x2 1 y < x 3 2x 1 x2y ≥ 1 y > 2x 50. 52. 54. y > y < x2 2x 3 x2 4x 3 y ≥ x4 2x2 1 y ≤ 1 x2 y ≤ ex 22 0 x ≤ 2 y ≥ 2 ≤ In Exercises 55–64, derive a set of inequalities to describe the region. 55. y 56. y 4 3 2 1 572 2 4 8 6 4 2 −2 −2 58. 6 y 3 1 −3 −1 1 3 −3 x x x x Demand p 50 0.5x p 100 0.05x p 140 0.00002x p 400 0.0002x 65. 66. 67. 68. Supply p 0.125x p 25 0.1x p 80 0.00001x p 225 0.0005x 11 3 69. Production A furniture company can sell all the tables and chairs it produces. Each table requires 1 hour in the hours in the finishing center. Each assembly center and chair requires hours in the finishing center. The company’s assembly center is available 12 hours per day, and its finishing center is available 15 hours per day. Find and graph a system of inequalities describing all possible production levels. hours in the assembly center and 11 2 11 2 70. Inventory A store sells two models of computers. Because of the demand, the store stocks at least twice as many units of model A as of model B. The costs to the store for the two models are $800 and $1200, respectively. The management does not want more than $20,000 in computer inventory at any one time, and it wants at least four model A computers and two model B computers in inventory at all times. Find and graph a system of inequalities describing all possible inventory levels. 71. Investment Analysis A person plans to invest up to $20,000 in two different interest-bearing accounts. Each account is to contain at least $5000. Moreover, the amount in one account should be at least twice the amount in the oth
er account. Find and graph a system of inequalities to describe the various amounts that can be deposited in each account. 333202_0705.qxd 12/5/05 9:45 AM Page 550 550 Chapter 7 Systems of Equations and Inequalities 72. Ticket Sales For a concert event, there are $30 reserved seat tickets and $20 general admission tickets. There are 2000 reserved seats available, and fire regulations limit the number of paid ticket holders to 3000. The promoter must take in at least $75,000 in ticket sales. Find and graph a system of inequalities describing the different numbers of tickets that can be sold. 73. Shipping A warehouse supervisor is told to ship at least 50 packages of gravel that weigh 55 pounds each and at least 40 bags of stone that weigh 70 pounds each. The maximum weight capacity in the truck he is loading is 7500 pounds. Find and graph a system of inequalities describing the numbers of bags of stone and gravel that he can send. 74. Truck Scheduling A small company that manufactures two models of exercise machines has an order for 15 units of the standard model and 16 units of the deluxe model. The company has trucks of two different sizes that can haul the products, as shown in the table. Truck Standard Deluxe Large Medium 6 4 3 6 Find and graph a system of inequalities describing the numbers of trucks of each size that are needed to deliver the order. 75. Nutrition A dietitian is asked to design a special dietary supplement using two different foods. Each ounce of food X contains 20 units of calcium, 15 units of iron, and 10 units of vitamin B. Each ounce of food Y contains 10 units of calcium, 10 units of iron, and 20 units of vitamin B. The minimum daily requirements of the diet are 300 units of calcium, 150 units of iron, and 200 units of vitamin B. (a) Write a system of inequalities describing the different amounts of food X and food Y that can be used. (b) Sketch a graph of the region corresponding to the system in part (a). (c) Find two solutions of the system and interpret their meanings in the context of the problem. x 76. Health A person’s maximum heart rate is is the person’s age in years for 220 x, 20 ≤ x ≤ 70. where it is recommended that the When a person exercises, person strive for a heart rate that is at least 50% of the maximum and at most 75% of the maximum. (Source: American Heart Association) (a) Write a system of inequalities that describes the exercise target heart rate region. (b) Sketch a graph of the region in part (a). (c) Find two solutions to the system and interpret their meanings in the context of the problem. Model It 77. Data Analysis: Prescription Drugs The table shows the retail sales (in billions of dollars) of prescription drugs in the United States from 1999 to 2003. (Source: National Association of Chain Drug Stores) y Year Retail sales, y 1999 2000 2001 2002 2003 125.8 145.6 164.1 182.7 203.1 (a) Use the regression feature of a graphing utility to find a linear model for the data. Let represent the year, with corresponding to 1999. t 9 t (b) The total retail sales of prescription drugs in the United States during this five-year period can be approximated by finding the area of the trapezoid bounded by the linear model you found in part (a) y 0, t 8.5, and the lines Use a graphing utility to graph this region. t 13.5. and (c) Use the formula for the area of a trapezoid to approximate the total retail sales of prescription drugs. 78. Physical Fitness Facility An indoor running track is to be constructed with a space for body-building equipment inside the track (see figure). The track must be at least 125 meters long, and the body-building space must have an area of at least 500 square meters. y Body-building equipment x (a) Find a system of inequalities describing the require- ments of the facility. (b) Graph the system from part (a). 333202_0705.qxd 12/5/05 9:45 AM Page 551 Synthesis True or False? the statement is true or false. Justify your answer. In Exercises 79 and 80, determine whether 79. The area of the figure defined by the system is 99 square units. 80. The graph below shows the solution of the system y ≤ 6 4x 9y > 6 3x y2 ≥ 2 . y 10 8 4 −4 −6 −8 −4 81. Writing Explain the difference between the graphs of the on the real number line and on the x ≤ 4 inequality rectangular coordinate system. 82. Think About It After graphing the boundary of an inequality in and how do you decide on which side of the boundary the solution set of the inequality lies? y, x 83. Graphical Reasoning Two concentric circles have radii The area between the circles must be y, and where x at least 10 square units. y > x. (a) Find a system of inequalities describing the constraints on the circles. (b) Use a graphing utility to graph the system of inequaliin the same y x ties in part (a). Graph the line viewing window. (c) Identify the graph of the line in relation to the boundary of the inequality. Explain its meaning in the context of the problem. 84. The graph of the solution of the inequality is shown in the figure. Describe how the solution set would change for each of the following. x 2y < 6 (a) x 2y ≤ 6 (b) x 2y > 6 y 6 2 −2 −4 2 4 6 x x 6 −6 −6 Section 7.5 Systems of Inequalities 551 In Exercises 85–88, match the system of inequalities with the graph of its solution. [The graphs are labeled (a), (b), (c), and (d).] (a) (c) y 2 −6 −2 2 −6 y 2 −6 −2 2 (b) (d) y 2 −6 −2 2 −6 y 2 −6 −2 2 x x x x 85. 87. x2 y2 ≤ 16 x y ≥ 4 x2 y2 ≥ 16 x y ≥ 4 Skills Review 86. 88. x2 y2 ≤ 16 x y ≤ 4 x2 y2 ≥ 16 x y ≤ 4 In Exercises 89–94, find the equation of the line passing through the two points. 89. 91. 93. 2, 6, 4, 4 3 7 4, 2, 2, 5 3.4, 5.2, 2.6, 0.8 90. 92. 94. 8, 0, 3, 1 1 11 2, 0, 2 , 12 4.1, 3.8, 2.9, 8.2 95. Data Analysis: Cell Phone Bills The average monthly (in dollars) in the United States from is the year, are shown as data points (Source: Cellular Telecommunications & Internet y t cell phone bills 1998 to 2003, where t, y. Association) 1998, 39.43, 2001, 47.37, 1999, 41.24, 2002, 48.40, 2000, 45.27 2003, 49.91 (a) Use the regression feature of a graphing utility to find a linear model, a quadratic model, and an exponential t 8 model for the data. Let correspond to 1998. (b) Use a graphing utility to plot the data and the models in the same viewing window. (c) Which model is the best fit for the data? (d) Use the model from part (c) to predict the average monthly cell phone bill in 2008. 333202_0706.qxd 12/5/05 9:46 AM Page 552 552 Chapter 7 Systems of Equations and Inequalities 7.6 Linear Programming What you should learn • Solve linear programming problems. • Use linear programming to model and solve real-life problems. Why you should learn it Linear programming is often useful in making real-life economic decisions. For example, Exercise 44 on page 560 shows how you can determine the optimal cost of a blend of gasoline and compare it with the national average. Linear Programming: A Graphical Approach Many applications in business and economics involve a process called optimization, in which you are asked to find the minimum or maximum of a quantity. In this section, you will study an optimization strategy called linear programming. A two-dimensional linear programming problem consists of a linear objective function and a system of linear inequalities called constraints. The objective function gives the quantity that is to be maximized (or minimized), and the constraints determine the set of feasible solutions. For example, suppose you are asked to maximize the value of z ax by Objective function subject to a set of constraints that determines the shaded region in Figure 7.31. y Feasible solutions x FIGURE 7.31 Tim Boyle/Getty Images Because every point in the shaded region satisfies each constraint, it is not clear how you should find the point that yields a maximum value of Fortunately, it can be shown that if there is an optimal solution, it must occur at one of the vertices. This means that you can find the maximum value of z by testing z at each of the vertices. z. Optimal Solution of a Linear Programming Problem If a linear programming problem has a solution, it must occur at a vertex of the set of feasible solutions. If there is more than one solution, at least one of them must occur at such a vertex. In either case, the value of the objective function is unique. Some guidelines for solving a linear programming problem in two variables are listed at the top of the next page. 333202_0706.qxd 12/5/05 9:46 AM Page 553 Section 7.6 Linear Programming 553 Solving a Linear Programming Problem 1. Sketch the region corresponding to the system of constraints. (The points inside or on the boundary of the region are feasible solutions.) 2. Find the vertices of the region. 3. Test the objective function at each of the vertices and select the values of the variables that optimize the objective function. For a bounded region, both a minimum and a maximum value will exist. (For an unbounded region, if an optimal solution exists, it will occur at a vertex.) Example 1 Solving a Linear Programming Problem Find the maximum value of z 3x 2y subject to the following constraints. x ≥ 0 y ≥ 0 x 2y ≤ 4 x y ≤ 1 Objective function Constraints Solution The constraints form the region shown in Figure 7.32. At the four vertices of this region, the objective function has the following values. z 30 20 0 z 31 20 3 z 32 21 8 z 30 22 4 0, 0: 1, 0: 2, 1: 0, 2: At At At At Maximum value of z So, the maximum value of z is 8, and this occurs when x 2 and y 1. Now try Exercise 5. In Example 1, try testing some of the interior points in the region. You will see that the corresponding values of are less than 8. Here are some examples. At 1, 1: z z 31 21 5 At 1 2, 3 2 : z 31 2 23 2 9 2 To see why the maximum value of the objective function in Example 1 must occur at a vertex, consider writing the objective function in slope-intercept form y 3 2 x z 2 Family of lines z2 y 3 2. is the -intercept of the objective function. This
equation represents a where family of lines, each of slope Of these infinitely many lines, you want the one that has the largest -value while still intersecting the region determined by the constraints. In other words, of all the lines whose slope is you want the one that has the largest -intercept and intersects the given region, as shown in Figure 7.33. From the graph you can see that such a line will pass through one (or more) of the vertices of the region. 3 2, y z y 4 3 2 1 (0, 2) x = 0 x + 2y = 4 (2, 1) x − y = 1 (1, 0) (0, 0) y = 0 2 3 FIGURE 7.32 FIGURE 7.33 x x 333202_0706.qxd 12/5/05 9:46 AM Page 554 554 Chapter 7 Systems of Equations and Inequalities (1, 5) (0, 4) (0, 2) y 5 4 3 2 1 (6, 3) (3, 0) (5, 0) x 1 2 3 4 5 6 FIGURE 7.34 . Historical Note George Dantzig (1914 – ) was the first to propose the simplex method, or linear programming, in 1947. This technique defined the steps needed to find the optimal solution to a complex multivariable problem. The next example shows that the same basic procedure can be used to solve a problem in which the objective function is to be minimized. Example 2 Minimizing an Objective Function Find the minimum value of z 5x 7y Objective function x ≥ 0 where 2x 3x x 2x and 3y ≥ y ≤ y ≤ 5y ≤ y ≥ 0, 6 15 4 27 subject to the following constraints. Constraints Solution The region bounded by the constraints is shown in Figure 7.34. By testing the objective function at each vertex, you obtain the following. At At At At At At 0, 2: 0, 4: 1, 5: 6, 3: 5, 0: 3, 0: Minimum value of z z 50 72 14 z 50 74 28 z 51 75 40 z 56 73 51 z 55 70 25 z 53 70 15 So, the minimum value of z is 14, and this occurs when x 0 and y 2. Now try Exercise 13. Example 3 Maximizing an Objective Function Find the maximum value of z 5x 7y Objective function where x ≥ 0 y ≥ 0, and 3y ≥ 6 y ≤ 15 y ≤ 4 5y ≤ 27 2x 3x x 2x subject to the following constraints. Constraints Solution This linear programming problem is identical to that given in Example 2 above, except that the objective function is maximized instead of minimized. Using the values of at the vertices shown above, you can conclude that the maximum z value of z is z 56 73 51 and occurs when x 6 and y 3. Now try Exercise 15. 333202_0706.qxd 12/5/05 9:46 AM Page 555 y (0, 4) (2, 4) 4 3 2 1 z =12 for any point along this line segment. (5, 1) (0, 0) (5, 0) x 1 2 3 4 5 FIGURE 7.35 Section 7.6 Linear Programming 555 It is possible for the maximum (or minimum) value in a linear programming problem to occur at two different vertices. For instance, at the vertices of the region shown in Figure 7.35, the objective function z 2x 2y has the following values. At At At At At 0, 0: 0, 4: 2, 4: 5, 1: 5, 0: z 20 20 10 z 20 24 18 z 22 24 12 z 25 21 12 z 25 20 10 Objective function Maximum value of z Maximum value of z and 2, 4 5, 1; In this case, you can conclude that the objective function has a maximum value not only at the vertices it also has a maximum value (of 12) at any point on the line segment connecting these two vertices. Note that the objective function in slope-intercept form has the same slope as the line through the vertices 5, 1. Some linear programming problems have no optimal solutions. This can occur if the region determined by the constraints is unbounded. Example 4 illustrates such a problem. y x 1 2 z 2, 4 and Example 4 An Unbounded Region Find the maximum value of z 4x 2y Objective function where y ≥ 0, x ≥ 0 and x 2y ≥ 4 3x y ≥ 7 x 2y ≤ 7 subject to the following constraints. Constraints y 5 4 3 2 1 (1, 4) (2, 1) (4, 0) x 1 2 3 4 5 FIGURE 7.36 Solution The region determined by the constraints is shown in Figure 7.36. For this unbounded region, there is no maximum value of To see this, note that the point Substituting this point into the objective function, you get z 4x 20 4x. lies in the region for all values of z. x ≥ 4. x, 0 x to be large, you can obtain values of By choosing that are as large as you want. So, there is no maximum value of However, there is a minimum value of z. z. z At At At 1, 4: 2, 1: 4, 0: z 41 24 12 z 42 21 10 z 44 20 16 Minimum value of z So, the minimum value of z is 10, and this occurs when x 2 and y 1. Now try Exercise 17. 333202_0706.qxd 12/5/05 9:46 AM Page 556 556 Chapter 7 Systems of Equations and Inequalities Applications Example 5 shows how linear programming can be used to find the maximum profit in a business application. Example 5 Optimal Profit A candy manufacturer wants to maximize the profit for two types of boxed chocolates. A box of chocolate covered creams yields a profit of $1.50 per box, and a box of chocolate covered nuts yields a profit of $2.00 per box. Market tests and available resources have indicated the following constraints. 1. The combined production level should not exceed 1200 boxes per month. 2. The demand for a box of chocolate covered nuts is no more than half the demand for a box of chocolate covered creams. 3. The production level for chocolate covered creams should be less than or equal to 600 boxes plus three times the production level for chocolate covered nuts. Solution x Let be the be the number of boxes of chocolate covered creams and let number of boxes of chocolate covered nuts. So, the objective function (for the combined profit) is given by y P 1.5x 2y. Objective function The three constraints translate into the following linear inequalities. 1. 2. 3. x y ≤ 1200 y ≤ 1 2x x ≤ 600 3y x y ≤ 1200 x 2y ≤ 0 x 3y ≤ 600 Maximum Profit y (800, 400) (1050, 150) (0, 0) (600, 0) 400 800 1200 Boxes of chocolate covered creams 400 300 200 100 FIGURE 7.37 x x ≥ 0 can be negative, you also have the two additional Because neither constraints of Figure 7.37 shows the region determined by the constraints. To find the maximum profit, test the values of at the vertices of the region. y nor y ≥ 0. and P At 0, 0: P 1.50 0 At 800, 400: P 1.5800 2400 2000 At 1050, 150: P 1.51050 2150 1875 900 At 600, 0: P 1.5600 20 20 Maximum profit x So, the maximum profit is $2000, and it occurs when the monthly production consists of 800 boxes of chocolate covered creams and 400 boxes of chocolate covered nuts. Now try Exercise 39. In Example 5, if the manufacturer improved the production of chocolate covered creams so that they yielded a profit of $2.50 per unit, the maximum profit could then be found using the objective function By testing the values of at the vertices of the region, you would find that the maximum profit was $2925 and that it occurred when P 2.5x 2y. and y 150. x 1050 P 333202_0706.qxd 12/5/05 9:46 AM Page 557 Section 7.6 Linear Programming 557 Example 6 Optimal Cost The liquid portion of a diet is to provide at least 300 calories, 36 units of vitamin A, and 90 units of vitamin C. A cup of dietary drink X costs $0.12 and provides 60 calories, 12 units of vitamin A, and 10 units of vitamin C. A cup of dietary drink Y costs $0.15 and provides 60 calories, 6 units of vitamin A, and 30 units of vitamin C. How many cups of each drink should be consumed each day to obtain an optimal cost and still meet the daily requirements? Solution As in Example 9 in Section 7.5, let be the number of cups of dietary drink X and let be the number of cups of dietary drink Y. y x For calories: For vitamin A: For vitamin C: 60x 60y ≥ 300 12x 6y ≥ 36 10x 30y ≥ 90 x ≥ 0 y ≥ 0 Constraints y 8 6 4 2 (0, 6) (1, 4) (3, 2) (9, 0) x 2 4 6 8 10 FIGURE 7.38 The cost C is given by C 0.12x 0.15y. Objective function The graph of the region corresponding to the constraints is shown in Figure 7.38. Because you want to incur as little cost as possible, you want to determine the minimum cost. To determine the minimum cost, test at each vertex of the region. C At At At At 0, 6: 1, 4: 3, 2: 9, 0: C 0.120 0.156 0.90 C 0.121 0.154 0.72 C 0.123 0.152 0.66 C 0.129 0.150 1.08 Minimum value of C So, the minimum cost is $0.66 per day, and this occurs when 3 cups of drink X and 2 cups of drink Y are consumed each day. Now try Exercise 43. W RITING ABOUT MATHEMATICS Creating a Linear Programming Problem Sketch the region determined by the following constraints. x 2y ≤ Constraints Find, if possible, an objective function of the form maximum at each indicated vertex of the region. z ax by that has a a. 0, 4 b. 2, 3 c. 5, 0 d. 0, 0 Explain how you found each objective function. 333202_0706.qxd 12/5/05 9:46 AM Page 558 558 Chapter 7 Systems of Equations and Inequalities 7.6 Exercises VOCABULARY CHECK: Fill in the blanks. 1. In the process called ________, you are asked to find the maximum or minimum value of a quantity. 2. One type of optimization strategy is called ________ ________. 3. The ________ function of a linear programming problem gives the quantity that is to be maximized or minimized. 4. The ________ of a linear programming problem determine the set of ________ ________. 5. If a linear programming problem has a solution, it must occur at a ________ of the set of feasible solutions. PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. In Exercises 1–12, find the minimum and maximum values of the objective function and where they occur, subject to the indicated constraints. (For each exercise, the graph of the region determined by the constraints is provided.) 1. Objective function: z 4x 3y Constraints0, 5) y 6 5 4 3 2 1 2. Objective function: z 2x 8y Constraints: x ≥ 0 y ≥ 0 2x y ≤ 4 y (0, 4) 4 3 2 (0, 0) (5, 0) 1 2 3 4 5 6 x (0, 0) −1 (2, 0) x 1 2 3 3. Objective function: z 3x 8y Constraints: (See Exercise 1.) 5. Objective function: z 3x 2y Constraints: x ≥ 0 y ≥ 0 x 3y ≤ 15 4x y ≤ 16 4. Objective function: z 7x 3y Constraints: (See Exercise 2.) 6. Objective function: z 4x 5y Constraints: x ≥ 0 2x 3y ≥ 6 3x y ≤ 9 x 4y ≤ 16 y 5 4 3 2 1 (0, 5) (3, 4) (0, 0) (4, 00, 4) (0, 2) (4, 3) (3, 0) x 1 2 3 4 5 FIGURE FOR 5 FIGURE FOR 6 7. Objective function: z 5x 0.5y Constraints: (See Exercise 5.) 9. Objective function: z 10x 7y Constraints: 0 ≤ x ≤ 60 0 ≤ y ≤ 45 5x 6y ≤ 420 8. Objective funct
ion: z 2x y Constraints: (See Exercise 6.) 10. Objective function: z 25x 35y Constraints: x ≥ y ≥ 0 8x 9y ≤ 7200 8x 9y ≥ 3600 0 y 60 40 20 (0, 45) (30, 45) (60, 20) (0, 0) (60, 0) 20 40 60 x y 800 400 (0, 800) (0, 400) (900, 0) x 400 (450, 0) 11. Objective function: z 25x 30y Constraints: (See Exercise 9.) 12. Objective function: z 15x 20y Constraints: (See Exercise 10.) 333202_0706.qxd 12/5/05 9:46 AM Page 559 In Exercises 13–20, sketch the region determined by the constraints. Then find the minimum and maximum values of the objective function and where they occur, subject to the indicated constraints. 13. Objective function: z 6x 10y Constraints: x x ≥ 0 y ≥ 0 x 2x 5y ≤ 10 15. Objective function: z 9x 24y Constraints: (See Exercise 13.) 17. Objective function: z 4x 5y Constraints: x ≥ 0 y ≥ 0 x 5y ≥ 8 3x 5y ≥ 30 19. Objective function: z 2x 7y Constraints: (See Exercise 17.) x 14. Objective function: z 7x 8y Constraints 16. Objective function: z 7x 2y Constraints: (See Exercise 14.) x x 1 18. Objective function: z 4x 5y Constraints: x ≥ 0 y ≥ 0 2x 2y ≤ 10 x 2y ≤ 6 20. Objective function: z 2x y Constraints: (See Exercise 18.) In Exercises 21–24, use a graphing utility to graph the region determined by the constraints. Then find the minimum and maximum values of the objective function and where they occur, subject to the constraints. 21. Objective function: 22. Objective function: z 4x y Constraints: x ≥ 0 y ≥ 0 x 2y ≤ 40 2x 3y ≥ 72 23. Objective function: z x 4y Constraints: (See Exercise 21.) z x Constraints: x ≥ 0 y ≥ 0 2x 3y ≤ 60 2x y ≤ 28 4x y ≤ 48 24. Objective function: z y Constraints: (See Exercise 22.) Section 7.6 Linear Programming 559 In Exercises 25–28, find the maximum value of the objective function and where it occurs, subject to the constraints x ≥ 0, and 4x 3y ≤ 30. 3x y ≤ 15, y ≥ 0, 25. 26. 27. 28. z 2x y z 5x y z x y z 3x y In Exercises 29–32, find the maximum value of the objective the function and where constraints and 2x 2y ≤ 21. to x y ≤ 18, x 4y ≤ 20, it occurs, subject x ≥ 0, y ≥ 0, 29. 30. 31. 32. z x 5y z 2x 4y z 4x 5y z 4x y In Exercises 33–38, the linear programming problem has an unusual characteristic. Sketch a graph of the solution region for the problem and describe the unusual characteristic. Find the maximum value of the objective function and where it occurs. 33. Objective function: z 2.5x y Constraints: x ≥ 0 y ≥ 0 3x 5y ≤ 15 5x 2y ≤ 10 35. Objective function: z x 2y Constraints: x ≥ 0 y ≥ 0 x ≤ 10 x y ≤ 7 37. Objective function: z 3x 4y Constraints 2x y ≤ 4 34. Objective function: z x y Constraints 2y ≤ 4 36. Objective function: z x y Constraints 3x y ≥ 3 38. Objective function: z x 2y Constraints: x ≥ 0 y ≥ 0 x 2y ≤ 4 2x 2y ≤ 4 333202_0706.qxd 12/5/05 9:46 AM Page 560 560 Chapter 7 Systems of Equations and Inequalities 39. Optimal Profit A manufacturer produces two models of bicycles. The times (in hours) required for assembling, painting, and packaging each model are shown in the table. Process Hours, model A Hours, model B Assembling Painting Packaging 2 4 1 2.5 1 0.75 The total times available for assembling, painting, and packaging are 4000 hours, 4800 hours, and 1500 hours, respectively. The profits per unit are $45 for model A and $50 for model B. What is the optimal production level for each model? What is the optimal profit? 40. Optimal Profit A manufacturer produces two models of bicycles. The times (in hours) required for assembling, painting, and packaging each model are shown in the table. Process Assembling Painting Packaging Hours, model A Hours, model B 2.5 2 0.75 3 1 1.25 The total times available for assembling, painting, and packaging are 4000 hours, 2500 hours, and 1500 hours, respectively. The profits per unit are $50 for model A and $52 for model B. What is the optimal production level for each model? What is the optimal profit? 41. Optimal Profit A merchant plans to sell two models of MP3 players at costs of $250 and $300. The $250 model yields a profit of $25 per unit and the $300 model yields a profit of $40 per unit. The merchant estimates that the total monthly demand will not exceed 250 units. The merchant does not want to invest more than $65,000 in inventory for these products. What is the optimal inventory level for each model? What is the optimal profit? 42. Optimal Profit A fruit grower has 150 acres of land available to raise two crops, A and B. It takes 1 day to trim an acre of crop A and 2 days to trim an acre of crop B, and there are 240 days per year available for trimming. It takes 0.3 day to pick an acre of crop A and 0.1 day to pick an acre of crop B, and there are 30 days available for picking. The profit is $140 per acre for crop A and $235 per acre for crop B. What is the optimal acreage for each fruit? What is the optimal profit? 43. Optimal Cost A farming cooperative mixes two brands of cattle feed. Brand X costs $25 per bag and contains two units of nutritional element A, two units of element B, and two units of element C. Brand Y costs $20 per bag and contains one unit of nutritional element A, nine units of element B, and three units of element C. The minimum requirements of nutrients A, B, and C are 12 units, 36 units, and 24 units, respectively. What is the optimal number of bags of each brand that should be mixed? What is the optimal cost? Model It 44. Optimal Cost According to AAA (Automobile Association of America), on January 24, 2005, the national average price per gallon for regular unleaded (87-octane) gasoline was $1.84, and the price for premium unleaded (93-octane) gasoline was $2.03. (a) Write an objective function that models the cost of the blend of mid-grade unleaded gasoline (89octane). (b) Determine the constraints for the objective function in part (a). (c) Sketch a graph of the region determined by the constraints from part (b). (d) Determine the blend of regular and premium unleaded gasoline that results in an optimal cost of mid-grade unleaded gasoline. (e) What is the optimal cost? (f) Is the cost lower than the national average of $1.96 per gallon for mid-grade unleaded gasoline? 45. Optimal Revenue An accounting firm has 900 hours of staff time and 155 hours of reviewing time available each week. The firm charges $2500 for an audit and $350 for a tax return. Each audit requires 75 hours of staff time and 10 hours of review time. Each tax return requires 12.5 hours of staff time and 2.5 hours of review time. What numbers of audits and tax returns will yield an optimal revenue? What is the optimal revenue? 46. Optimal Revenue The accounting firm in Exercise 45 lowers its charge for an audit to $2000. What numbers of audits and tax returns will yield an optimal revenue? What is the optimal revenue? 333202_0706.qxd 12/5/05 9:46 AM Page 561 47. Investment Portfolio An investor has up to $250,000 to invest in two types of investments. Type A pays 8% annually and type B pays 10% annually. To have a well-balanced portfolio, the investor imposes the following conditions. At least one-fourth of the total portfolio is to be allocated to type A investments and at least one-fourth of the portfolio is to be allocated to type B investments. What is the optimal amount that should be invested in each type of investment? What is the optimal return? 48. Investment Portfolio An investor has up to $450,000 to invest in two types of investments. Type A pays 6% annually and type B pays 10% annually. To have a well-balanced portfolio, the investor imposes the following conditions. At least one-half of the total portfolio is to be allocated to type A investments and at least one-fourth of the portfolio is to be allocated to type B investments. What is the optimal amount that should be invested in each type of investment? What is the optimal return? Synthesis True or False? the statement is true or false. Justify your answer. In Exercises 49 and 50, determine whether 49. If an objective function has a maximum value at the you can conclude that it also has 8, 3, 4, 7 vertices and a maximum value at the points 4.5, 6.5 and 7.8, 3.2. 50. When solving a linear programming problem, if the objective function has a maximum value at more than one vertex, you can assume that there are an infinite number of points that will produce the maximum value. In Exercises 51 and 52, determine values of t such that the objective function has maximum values at the indicated vertices. 51. Objective function: z 3x t y 52. Objective function: z 3x t y Constraints: (b) x ≥ 0 y ≥ 0 x 3y ≤ 15 4x y ≤ 16 0, 5 (a) 3, 4 Constraints: x ≥ 0 y ≥ 0 x 2y ≤ 4 x y ≤ 1 2, 1 (a) 0, 2 (b) Section 7.6 Linear Programming 561 Think About It In Exercises 53–56, find an objective function that has a maximum or minimum value at the indicated vertex of the constraint region shown below. (There are many correct answers.) y 6 5 3 2 1 A(0, 4) B(4, 3) C(5, 0) −1 1 432 6 x 53. The maximum occurs at vertex A. 54. The maximum occurs at vertex B. 55. The maximum occurs at vertex C. 56. The minimum occurs at vertex C. Skills Review In Exercises 57–60, simplify the complex fraction. 57. 59 58. 60 2x 2 4x 2 1 2 In Exercises 61–66, solve the equation algebraically. Round the result to three decimal places. 61. 62. 63. 64. 65. 66. e 2x 2e x 15 0 e 2x 10e x 24 0 862 e x4 192 75 150 e x 4 7 ln 3x 12 lnx 92 2 In Exercises 67 and 68, solve the system of linear equations and check any solution algebraically. 67. 2x 6y z 17 5y z 8 x 2y 3z 23 68. 7x 3y 5z 28 4x 4z 16 7x 2y z 0 333202_070R.qxd 12/5/05 9:48 AM Page 562 562 Chapter 7 Systems of Equations and Inequalities 7 Chapter Summary What did you learn? Section 7.1 Use the method of substitution to solve systems of linear equations Review Exercises 1– 4 in two variables (p. 496). Use the method of substitution to solve systems of nonlinear equations in two variables (p. 499). Use a graphical approach to solve systems of equations in two variables (p. 500). Use systems of equations to model and solve real-life problems (p. 501). Section 7
.2 Use the method of elimination to solve systems of linear equations in two variables (p.507). Interpret graphically the numbers of solutions of systems of linear equations in two variables (p. 510). Use systems of linear equations in two variables to model and solve real-life problems (p. 513). Section 7.3 Use back-substitution to solve linear systems in row-echelon form (p. 519). Use Gaussian elimination to solve systems of linear equations (p. 520). Solve nonsquare systems of linear equations (p. 524). Use systems of linear equations in three or more variables to model and solve real-life problems (p. 525). Section 7.4 Recognize partial fraction decompositions of rational expressions (p. 533). Find partial fraction decompositions of rational expressions (p. 534). Section 7.5 Sketch the graphs of inequalities in two variables (p. 541). Solve systems of inequalities (p. 543). Use systems of inequalities in two variables to model and solve real-life problems (p. 546). Section 7.6 Solve linear programming problems (p. 552). Use linear programming to model and solve real-life problems (p. 556). 5–8 9–14 15–18 19–26 27–30 31, 32 33, 34 35–38 39, 40 41– 48 49–52 53–60 61–64 65–72 73–76 77–82 83–86 333202_070R.qxd 12/5/05 9:48 AM Page 563 7 Review Exercises In Exercises 1–8, solve the system by the method of 7.1 substitution. In Exercises 19–26, solve the system by the method 7.2 of elimination. Review Exercises 563 1. 3. 5. 7. 1.25x y 4.5y x y 2 x y 0 0.5x x2 y2 9 x y 1 y 2x2 y x 4 2x2 0.75 2.5 2. 4. 6. 8. x 3 0 3 5 4 5 2x 3y y x 2 5y 1 x 5y x2 y2 169 3x 2y 39 x y 3 x y2 1 In Exercises 9–12, solve the system graphically. 9. 11. 10 6 2x y x 5y y y 2x2 4x 1 x2 4x 3 10. 12. 2x 3y 5y 3 28 8x y2 2y x 0 x y 0 In Exercises 13 and 14, use a graphing utility to solve the system of equations. Find the solution accurate to two decimal places. 13. 14. 2e x y y y 2ex 0 lnx 1 3 4 1 2 x y 15. Break-Even Analysis You set up a scrapbook business and make an initial investment of $50,000. The unit cost of a scrapbook kit is $12 and the selling price is $25. How many kits must you sell to break even? 16. Choice of Two Jobs You are offered two sales jobs at a pharmaceutical company. One company offers an annual salary of $35,000 plus a year-end bonus of 1.5% of your total sales. The other company offers an annual salary of $32,000 plus a year-end bonus of 2% of your total sales. What amount of sales will make the second offer better? Explain. 17. Geometry The perimeter of a rectangle is 480 meters and its length is 150% of its width. Find the dimensions of the rectangle. 18. Geometry The perimeter of a rectangle is 68 feet and its times its length. Find the dimensions of the 8 width is 9 rectangle. 19. 21. 23. 25. 2x y 2 6x 8y 39 0.2x 0.3y 0.14 0.4x 0.5y 0.20 3x 2y 0 3x 2 y 5 10 1.25x 2y 8y 3.5 14 5x 20. 22. 24. 26. 40x 30y 24 14 20x 50y 12x 42y 17 30x 18y 19 7x 12y 63 2x 3y 2 21 1.5x 2.5y 8.5 6x 10y 24 In Exercises 27–30, match the system of linear equations with its graph. Describe the number of solutions and state whether the system is consistent or inconsistent. [The graphs are labeled (a), (b), (c), and (d).] (a) (c) −4 y 4 −2 −4 y 2 −2 −2 −4 −6 x 2 4 x 4 6 (b) y 4 −4 −4 (d) y 4 2 −4 x 4 x 6 27. 29. x 5y 4 x 3y 6 3x y 7 6x 2y 8 28. 30. 3x y 7 9x 3y 21 2x y 3 x 5y 4 Supply and Demand equilibrium point of demand and supply equations. In Exercises 31 and 32, find the Demand p 37 0.0002x p 120 0.0001x 31. 32. Supply p 22 0.00001x p 45 0.0002x 333202_070R.qxd 12/5/05 9:48 AM Page 564 564 Chapter 7 Systems of Equations and Inequalities In Exercises 33 and 34, use back-substitution to solve 7.3 the system of linear equations. In Exercises 43 and 44, find the equation of the circle x 2 y 2 Dx Ey F 0 33. 34. y z z x 4y 3z x 7y 8z y 9z z 3 1 5 85 35 3 In Exercises 35–38, use Gaussian elimination to solve the system of equations. 35. 36. 37. 38. y 3y 2x 4x x 3x 4x x x 2y 2x 2x 3y x 3y 3x 3x 2y y 2y 2y 4 4 16 6z z 2z z 13 5z 23 2z 14 z 6 7 3z 11 6z 9 11z 16 7z 11 In Exercises 39 and 40, solve the nonsquare system of equations. 39. 40. 5x 12y 7z 16 3x 7y 4z 9 2x 5y 19z 34 3x 8y 31z 54 In Exercises 41 and 42, find the equation of the parabola y ax 2 bx c that passes through the points. To verify your result, use a graphing utility to plot the points and graph the parabola. 41. y 42. 4 (2, 5) −4 x 4 (1, −2) (0, −5) y 24 12 (−5, 6) −12 −6 (2, 20) 6 (1, 0) x that passes through the points. To verify your result, use a graphing utility to plot the points and graph the circle. 43. y 1 −5 (2, 1) x 1 432 (5, − 2) (− 1, − 2) 44. y (1, 4) 2 (4, 3) −6 −2 2 4 x (− 2, − 5) −8 45. Data Analysis: Online Shopping The table shows the (in millions) of people shopping (Source: projected numbers online in the United States from 2003 to 2005. eMarketer) y Year Online shoppers, y 2003 2004 2005 101.7 108.4 121.1 (a) Use the technique demonstrated in Exercises 67–70 in Section 7.3 to set up a system of equations for the data and to find a least squares regression parabola that x 3 x models the data. Let corresponding to 2003. represent the year, with (b) Use a graphing utility to graph the parabola and the data in the same viewing window. How well does the model fit the data? (c) Use the model to estimate the number of online shoppers in 2008. Does your answer seem reasonable? 46. Agriculture A mixture of 6 gallons of chemical A, 8 gallons of chemical B, and 13 gallons of chemical C is required to kill a destructive crop insect. Commercial spray X contains 1, 2, and 2 parts, respectively, of these chemicals. Commercial spray Y contains only chemical C. Commercial spray Z contains chemicals A, B, and C in equal amounts. How much of each type of commercial spray is needed to get the desired mixture? 47. Investment Analysis An inheritance of $40,000 was divided among three investments yielding $3500 in interest per year. The interest rates for the three investments were 7%, 9%, and 11%. Find the amount placed in each investment if the second and third were $3000 and $5000 less than the first, respectively. 333202_070R.qxd 12/5/05 9:48 AM Page 565 48. Vertical Motion An object moving vertically is at the given heights at the specified times. Find the position equation for the object. s 1 (a) At (b) At seconds, 2 at2 v0t s0 s 134 second, s 86 s 6 s 184 s 116 s 16 seconds, seconds, seconds, second, feet feet feet feet feet feet At At At At 7.4 In Exercises 49–52, write the form of the partial fraction decomposition for the rational expression. Do not solve for the constants. 49. 51. 3 x2 20x 3x 4 x3 5x2 50. 52. x 8 x2 3x 28 x 2 xx2 22 In Exercises 53–60, write the partial fraction decomposition of the rational expression. 53. 55. 57. 59. 4 x x2 6x 8 x2 x2 2x 15 x2 2x x3 x2 x 1 3x2 4x x2 12 54. 56. 58. 60. x x2 3x 2 9 x2 9 4x 3x 12 4x2 x 1x2 1 In Exercises 61–64, sketch the graph of the 7.5 inequality. 61. y ≤ 5 1 2 x 63. y 4x2 > 1 62. 3y x ≥ 7 64. y ≤ 3 x2 2 In Exercises 65–72, sketch the graph and label the vertices of the solution set of the system of inequalities. 65. 67. 3x 2y ≤ 160 y ≤ 180 x ≥ 0 y ≥ 0 x 3x 2y ≥ 24 x 2y ≥ 12 2 ≤ x ≤ 15 y ≤ 15 69. y < x 1 y > x2 1 66. 68. 70. 2x y ≤ 16 x ≥ 0 y ≥ 0 2x 3y ≤ 24 2x y ≥ 16 x 3y ≥ 18 0 ≤ x ≤ 25 0 ≤ y ≤ 25 y ≤ 6 2x x2 y ≥ x 6 Review Exercises 565 71. 72. 2x 3y ≥ 0 2x y ≤ 8 y ≥ 0 x2 y2 ≤ 9 x 32 y2 ≤ 9 73. Inventory Costs A warehouse operator has 24,000 square feet of floor space in which to store two products. Each unit of product I requires 20 square feet of floor space and costs $12 per day to store. Each unit of product II requires 30 square feet of floor space and costs $8 per day to store. The total storage cost per day cannot exceed $12,400. Find and graph a system of inequalities describing all possible inventory levels. 74. Nutrition A dietitian is asked to design a special dietary supplement using two different foods. Each ounce of food X contains 12 units of calcium, 10 units of iron, and 20 units of vitamin B. Each ounce of food Y contains 15 units of calcium, 20 units of iron, and 12 units of vitamin B. The minimum daily requirements of the diet are 300 units of calcium, 280 units of iron, and 300 units of vitamin B. (a) Write a system of inequalities describing the different amounts of food X and food Y that can be used. (b) Sketch a graph of the region in part (a). (c) Find two solutions to the system and interpret their meanings in the context of the problem. Supply and Demand In Exercises 75 and 76, (a) graph the systems representing the consumer surplus and producer surplus for the supply and demand equations and (b) find the consumer surplus and producer surplus. Demand p 160 0.0001x p 130 0.0002x 75. 76. Supply p 70 0.0002x p 30 0.0003x In Exercises 77– 82, sketch the region determined by 7.6 the constraints. Then find the minimum and maximum values of the objective function and where they occur, subject to the indicated restraints. 77. Objective function: z 3x 4y Constraints: x ≥ 0 y ≥ 0 2x 5y ≤ 50 4x y ≤ 28 78. Objective function: z 10x 7y Constraints: x ≥ 0 y ≥ 0 2x y ≥ 100 x y ≥ 75 333202_070R.qxd 12/5/05 9:48 AM Page 566 566 Chapter 7 Systems of Equations and Inequalities 79. Objective function: z 1.75x 2.25y Constraints: x ≥ 0 y ≥ 0 2x y ≥ 25 3x 2y ≥ 45 81. Objective function: z 5x 11y Constraints: x ≥ 0 y ≥ 0 x 3y ≤ 12 3x 2y ≤ 15 80. Objective function: z 50x 70y Constraints: x ≥ 0 y ≥ 0 x 2y ≤ 1500 5x 2y ≤ 3500 82. Objective function: z 2x y Constraints 5x 2y ≥ 20 83. Optimal Revenue A student is working part time as a hairdresser to pay college expenses. The student may work no more than 24 hours per week. Haircuts cost $25 and require an average of 20 minutes, and permanents cost $70 and require an average of 1 hour and 10 minutes. What combination of haircuts and/or permanents will yield an optimal revenue? What is the optimal revenue? 84. Optimal Profit A shoe manufacturer produces a walking shoe and a running shoe yielding profits of $18 and $24, respectively. Each shoe must go through three
processes, for which the required times per unit are shown in the table. Process Process Process II III I Hours for walking shoe Hours for running shoe Hours available per day 4 2 24 1 2 9 1 1 8 What is the optimal production level for each type of shoe? What is the optimal profit? 85. Optimal Cost A pet supply company mixes two brands of dry dog food. Brand X costs $15 per bag and contains eight units of nutritional element A, one unit of nutritional element B, and two units of nutritional element C. Brand Y costs $30 per bag and contains two units of nutritional element A, one unit of nutritional element B, and seven units of nutritional element C. Each bag of mixed dog food must contain at least 16 units, 5 units, and 20 units of nutritional elements A, B, and C, respectively. Find the numbers of bags of brands X and Y that should be mixed to produce a mixture meeting the minimum nutritional requirements and having an optimal cost. What is the optimal cost? 86. Optimal Cost Regular unleaded gasoline and premium unleaded gasoline have octane ratings of 87 and 93, respectively. For the week of January 3, 2005, regular unleaded gasoline in Houston, Texas averaged $1.63 per gallon. For the same week, premium unleaded gasoline averaged $1.83 per gallon. Determine the blend of regular and premium unleaded gasoline that results in an optimal cost of midgrade unleaded (89-octane) gasoline. What is the optimal (Source: Energy Information Administration) cost? Synthesis True or False? the statement is true or false. Justify your answer. In Exercises 87 and 88, determine whether 87. The system 26 represents the region covered by an isosceles trapezoid. 88. It is possible for an objective function of a linear programming problem to have exactly 10 maximum value points. In Exercises 89–92, find a system of linear equations having the ordered pair as a solution. (There are many correct answers.) 89. 90. 91. 92. 6, 8 5, 4 3, 3 4 1, 9 4 In Exercises 93–96, find a system of linear equations having the ordered triple as a solution. (There are many answers.) 93. 94. 95. 96. 4, 1, 3 3, 5, 6 2, 2 5, 3 4, 2, 8 3 97. Writing Explain what is meant by an inconsistent system of linear equations. 98. How can you tell graphically that a system of linear equa- tions in two variables has no solution? Give an example. 99. Writing Write a brief paragraph describing any advantages of substitution over the graphical method of solving a system of equations. 333202_070R.qxd 12/5/05 9:48 AM Page 567 7 Chapter Test Chapter Test 567 Take this test as you would take a test in class. When you are finished, check your work against the answers given in the back of the book. In Exercises 1– 3, solve the system by the method of substitution. 1. x y 4x 5y 7 8 2. y x 1 y x 13 3. 2x y2 0 x y 4 In Exercises 4– 6, solve the system graphically. 4. 2x 3y 0 2x 3y 12 5. y 9 x2 y x 3 6. y ln x 7x 2y 11 12 6 In Exercises 7–10, solve the linear system by the method of elimination. 7. 9. 2x 3y 5x 4y x 17 15 2y 3z z 3y z 2x 8. 10. 2.5x y 6 3x 4y 2 3x 2y z 17 x y z 4 x y z 3 11 3 8 In Exercises 11–14, write the partial fraction decomposition of the rational expression. 11. 2x 5 x2 x 2 12. 3x2 2x 4 x22 x 13. x2 5 x3 x 14. x2 4 x3 2x In Exercises 15–17, sketch the graph and label the vertices of the solution of the system of inequalities. 15. 2x y ≤ 4 2x y ≥ 0 x ≥ 0 16. y < x2 x 4 y > 4x 17. x2 y2 ≤ x ≥ y ≥ 16 1 3 18. Find the maximum and minimum values of the objective function z 20x 12y and where they occur, subject to the following constraints. x ≥ 0 y ≥ 0 x 4y ≤ 32 3x 2y ≤ 36 Constraints 19. A total of $50,000 is invested in two funds paying 8% and 8.5% simple interest. The yearly interest is $4150. How much is invested at each rate? 20. Find the equation of the parabola y ax 2 bx c passing through the points 0, 6, 2, 2, and 3, 9 2 . 21. A manufacturer produces two types of television stands. The amounts (in hours) of time for assembling, staining, and packaging the two models are shown in the table at the left. The total amounts of time available for assembling, staining, and packaging are 4000, 8950, and 2650 hours, respectively. The profits per unit are $30 (model I) and $40 (model II). What is the optimal inventory level for each model? What is the optimal profit? Model Model I 0.5 2.0 0.5 II 0.75 1.5 0.5 Assembling Staining Packaging TABLE FOR 21 333202_070R.qxd 12/5/05 9:48 AM Page 568 Proofs in Mathematics An indirect proof can be useful in proving statements of the form “ implies ” q. p → q q is Recall that the conditional statement is false only when q is false. To prove a conditional statement indirectly, assume that false. If this assumption leads to an impossibility, then you have proved that the conditional statement is true. An indirect proof is also called a proof by contradiction. is true and is true and p p p You can use an indirect proof to prove the following conditional statement, “If a is a positive integer and a2 is divisible by 2, then a is divisible by 2,” as follows. First, assume that aq, is true and is odd and can be written as 2. If so, a2 “ is a positive integer and is divisible by 2,” a “ is divisible by 2,” is false. This means that is not divisible by n is an integer. where a 2n 1, ap, a a 2n 1 a2 4n2 4n 1 a2 22n2 2n 1 Definition of an odd integer Square each side. Distributive Property So, by the definition of an odd integer, and you can conclude that a is divisible by 2. a2 is odd. This contradicts the assumption, Example Using an Indirect Proof Use an indirect proof to prove that 2 is an irrational number. Solution Begin by assuming that as the quotient of two integers and 2 a is not an irrational number. Then 2 can be written bb 0 that have no common factors. 2 a b 2 a2 b2 2b2 a2 This implies that 2 is a factor of as is an integer. 2c, c a2. where 2b2 2c2 2b2 4c2 b2 2c2 Assume that 2 is a rational number. Square each side. Multiply each side by b2. So, 2 is also a factor of a, and can be written a Substitute 2c for a. Simplify. Divide each side by 2. and also a factor of So, 2 is a factor of both and This contradicts the assumption that and have no common factors. This implies that 2 is a factor of a So, you can conclude that b is an irrational number. 2 b. a b. b2 568 333202_070R.qxd 12/5/05 9:48 AM Page 569 P.S. Problem Solving This collection of thought-provoking and challenging exercises further explores and expands upon concepts learned in this chapter. 1. A theorem from geometry states that if a triangle is inscribed in a circle such that one side of the triangle is a diameter of the circle, then the triangle is a right triangle. Show that this theorem is true for the circle x2 y2 100 and the triangle formed by the lines and y 1 y 0, 2 x 5, k2 infinite number of solutions. and k1 2. Find y 2x 20. such that the system of equations has an 3x 5y 8 2x k1y k2 3. Consider the following system of linear equations in and x y. ax by e cx dy f Under what conditions will the system have exactly one solution? 4. Graph the lines determined by each system of linear equations. Then use Gaussian elimination to solve each system. At each step of the elimination process, graph the corresponding lines. What do you observe? (a) (b) x 4y 3 5x 6y 13 2x 3y 7 4x 6y 14 5. A system of two equations in two unknowns is solved and has a finite number of solutions. Determine the maximum number of solutions of the system satisfying each condition. (a) Both equations are linear. (b) One equation is linear and the other is quadratic. (c) Both equations are quadratic. 6. In the 2004 presidential election, approximately 118.304 million voters divided their votes among three presidential candidates. George W. Bush received 3,320,000 votes more than John Kerry. Ralph Nader received 0.3% of the votes. Write and solve a system of equations to find the total number of votes cast for each candidate. Let represent the total votes cast for Bush, the total votes cast for Kerry, and N the total votes cast for Nader. (Source: CNN.com) K B 7. The Vietnam Veterans Memorial (or “The Wall”) in Washington, D.C. was designed by Maya Ying Lin when she was a student at Yale University. This monument has two vertical, triangular sections of black granite with a common side (see figure). The bottom of each section is level with the ground. The tops of the two sections can be approximately modeled by the equations 2x 50y 505 and 2x 50y 505 x when the -axis is superimposed at the base of the wall. Each unit in the coordinate system represents 1 foot. How high is the memorial at the point where the two sections meet? How long is each section? −2x + 50y = 505 2x + 50y = 505 Not drawn to scale C2H6 8. Weights of atoms and molecules are measured in atomic mass units (u). A molecule of (ethane) is made up of two carbon atoms and six hydrogen atoms and weighs (propane) is made up of 30.07 u. A molecule of C3H8 three carbon atoms and eight hydrogen atoms and weighs 44.097 u. Find the weights of a carbon atom and a hydrogen atom. 9. To connect a DVD player to a television set, a cable with special connectors is required at both ends. You buy a six-foot cable for $15.50 and a three-foot cable for $10.25. Assuming that the cost of a cable is the sum of the cost of the two connectors and the cost of the cable itself, what is the cost of a four-foot cable? Explain your reasoning. 10. A hotel 35 miles from an airport runs a shuttle service to and from the airport. The 9:00 A.M. bus leaves for the airport traveling at 30 miles per hour. The 9:15 A.M. bus leaves for the airport traveling at 40 miles per hour. Write a system of linear equations that represents distance as a function of time for each bus. Graph and solve the system. How far from the airport will the 9:15 A.M. bus catch up to the 9:00 A.M. bus? 569 333202_070R.qxd 12/5/05 9:49 AM Page 570 11. Solve each system of equations by letting X 1x, Y 1y, and Z 1z. 12 x 3 x 12 y 4 y 7 0 (a) (b 13 z 4 10 8 12. What values should be given t
o a, 1, 2, 3 c and so that the lin- b, as its only solution? ear system shown has x 2y 3z a x y z b 2x 3y 2z c Equation 1 Equation 2 Equation 3 13. The following system has one solution: x 1, y 1, and z 2. 4x 2y 5z 16 x y 0 x 3y 2z 6 Solve the system given by (a) Equation 1 and Equation 2, (b) Equation 1 and Equation 3, and (c) Equation 2 and Equation 3. (d) How many solutions does each of these systems have? 14. Solve the system of linear equations algebraically. x1 3x1 2x1 2x1 x2 2x2 x2 2x2 2x2 2x3 4x3 x3 4x3 4x3 2x4 4x4 x4 5x4 4x4 6x5 12x5 3x5 15x5 13x5 6 14 3 10 13 15. Each day, an average adult moose can process about 32 kilograms of terrestrial vegetation (twigs and leaves) and aquatic vegetation. From this food, it needs to obtain about 1.9 grams of sodium and 11,000 calories of energy. Aquatic vegetation has about 0.15 gram of sodium per kilogram and about 193 calories of energy per kilogram, whereas terrestrial vegetation has minimal sodium and about four times more energy than aquatic vegetation. Write and graph a system of inequalities that describes the amounts of terrestrial and aquatic vegetation, respectively, for the daily diet of an average adult moose. (Source: Biology by Numbers) and a t 570 16. For a healthy person who is 4 feet 10 inches tall, the recommended minimum weight is about 91 pounds and increases by about 3.7 pounds for each additional inch of height. The recommended maximum weight is about 119 pounds and increases by about 4.8 pounds for each additional inch of height. (Source: Dietary Guidelines Advisory Committee) (a) Let x be the number of inches by which a person’s height exceeds 4 feet 10 inches and let be the person’s weight in pounds. Write a system of inequalities that describes the possible values of for a healthy person. and y x y (b) Use a graphing utility to graph the system of inequalities from part (a). (c) What is the recommended weight range for someone 6 feet tall? 17. The cholesterol in human blood is necessary, but too much cholesterol can lead to health problems. A blood cholesterol test gives three readings: LDL (“bad”) cholesterol, HDL (“good”) cholesterol, and total cholesterol (LDL HDL). It is recommended that your LDL cholesterol level be less than 130 milligrams per deciliter, your HDL cholesterol level be at least 35 milligrams per deciliter, and your total cholesterol level be no more than 200 milligrams per deciliter. (Source: WebMD, Inc.) (a) Write a system of linear inequalities for the represent HDL recommended cholesterol levels. Let y cholesterol and let represent LDL cholesterol. x (b) Graph the system of inequalities from part (a). Label any vertices of the solution region. (c) Are the following cholesterol levels within recommendations? Explain your reasoning. LDL: 120 milligrams per deciliter HDL: 90 milligrams per deciliter Total: 210 milligrams per deciliter (d) Give an example of cholesterol levels in which the LDL cholesterol level is too high but the HDL and total cholesterol levels are acceptable. (e) Another recommendation is that the ratio of total cholesterol to HDL cholesterol be less than 4. Find a point in your solution region from part (b) that meets this recommendation, and explain why it meets the recommendation. 333202_0800.qxd 12/5/05 10:52 AM Page 571 Matrices and Determinants 8.1 Matrices and Systems of Equations 8.2 8.3 8.4 8.5 Operations with Matrices The Inverse of a Square Matrix The Determinant of a Square Matrix Applications of Matrices and Determinants 88 Matrices can be used to analyze financial information such as the profit a fruit farmer makes on two fruit crops AT I O N S Matrices have many real-life applications. The applications listed below represent a small sample of the applications in this chapter. • Electrical Network, Exercise 82, page 585 • Profit, Exercise 67, page 600 • Long-Distance Plans, Exercise 66, page 634 • Data Analysis: Snowboarders, • Investment Portfolio, Exercise 90, page 585 Exercises 67–70, page 609 • Agriculture, Exercise 61, page 599 • Data Analysis: Supreme Court, Exercise 58, page 630 571 333202_0801.qxd 12/5/05 10:59 AM Page 572 572 Chapter 8 Matrices and Determinants 8.1 Matrices and Systems of Equations What you should learn • Write matrices and identify their orders. • Perform elementary row operations on matrices. • Use matrices and Gaussian elimination to solve systems of linear equations. • Use matrices and Gauss- Jordan elimination to solve systems of linear equations. Why you should learn it You can use matrices to solve systems of linear equations in two or more variables. For instance, in Exercise 90 on page 585, you will use a matrix to find a model for the number of people who participated in snowboarding in the United States from 1997 to 2001. Matrices In this section, you will study a streamlined technique for solving systems of linear equations. This technique involves the use of a rectangular array of real numbers called a matrix. The plural of matrix is matrices. Definition of Matrix n If rectangular array m and are positive integers, an m n n (read “ by ”) matrix is a m Row 1 Row 2 Row 3 . . . Row m a21 a11 a31... am1 Column 1 Column 2 Column 3 . a12 a22 a13 a23 . Column n a1n a2n . . . . . . . . . . . . . a3n... amn a32... am2 a33... am3 in which each entry, n rows and columns. Matrices are usually denoted by capital letters. of the matrix is a number. An m n ai j, matrix has m a ij. For instance, The entry in the ith row and jth column is denoted by the double subscript refers to the entry in the second row, third column. notation m n, m n. n A matrix having m rows and columns is said to be of order If a11, a22, a33, . . . n. the matrix is square of order For a square matrix, the entries are the main diagonal entries. a23 Example 1 Order of Matrices Determine the order of each matrix. a. 2 c. 0 0 0 0 b. d Solution a. This matrix has one row and one column. The order of the matrix is b. This matrix has one row and four columns. The order of the matrix is c. This matrix has two rows and two columns. The order of the matrix is d. This matrix has three rows and two columns. The order of the matrix is 1 1. 1 4. 2 2. 3 2. The HM mathSpace® CD-ROM and Eduspace® for this text contain additional resources related to the concepts discussed in this chapter. Now try Exercise 1. A matrix that has only one row is called a row matrix, and a matrix that has only one column is called a column matrix. 333202_0801.qxd 12/5/05 10:59 AM Page 573 Section 8.1 Matrices and Systems of Equations 573 The vertical dots in an augmented matrix separate the coefficients of the linear system from the constant terms. A matrix derived from a system of linear equations (each written in standard form with the constant term on the right) is the augmented matrix of the system. Moreover, the matrix derived from the coefficients of the system (but not including the constant terms) is the coefficient matrix of the system. System: 5 3 6 4y 3y x 2x x 1 1 1 2 1 2 3z z 4z Augmented Matrix: Coefficient Matrix: ... ... ... 5 3 6 Note the use of 0 for the missing coefficient of the -variable in the third equation, and also note the fourth column of constant terms in the augmented matrix. y When forming either the coefficient matrix or the augmented matrix of a system, you should begin by vertically aligning the variables in the equations and using zeros for the coefficients of the missing variables. Example 2 Writing an Augmented Matrix Write the augmented matrix for the system of linear equations. x 3y w y 4z 2w x 5z 6w 2x 4y 3z 9 2 0 4 What is the order of the augmented matrix? Solution Begin by rewriting the linear system and aligning the variables. x 3y w 9 y 4z 2w 2 x 5z 6w 0 2x 4y 3z 4 R1 R2 R3 R4 1 Next, use the coefficients and constant terms as the matrix entries. Include zeros for the coefficients of the missing variables. 0 4 5 3 ... ... ... ... The augmented matrix has four rows and five columns, so it is a The notation is represented by matrix. is used to designate each row in the matrix. For example, Row 1 R1 Rn Now try Exercise 9. 333202_0801.qxd 12/5/05 10:59 AM Page 574 574 Chapter 8 Matrices and Determinants Elementary Row Operations In Section 7.3, you studied three operations that can be used on a system of linear equations to produce an equivalent system. 1. Interchange two equations. 2. Multiply an equation by a nonzero constant. 3. Add a multiple of an equation to another equation. In matrix terminology, these three operations correspond to elementary row operations. An elementary row operation on an augmented matrix of a given system of linear equations produces a new augmented matrix corresponding to a new (but equivalent) system of linear equations. Two matrices are row-equivalent if one can be obtained from the other by a sequence of elementary row operations. Elementary Row Operations 1. Interchange two rows. 2. Multiply a row by a nonzero constant. 3. Add a multiple of a row to another row. Although elementary row operations are simple to perform, they involve a lot of arithmetic. Because it is easy to make a mistake, you should get in the habit of noting the elementary row operations performed in each step so that you can go back and check your work. Example 3 Elementary Row Operations a. Interchange the first and second rows of the original matrix. Original Matrix New Row-Equivalent Matrix R2 R1 . Multiply the first row of the original matrix by 1 2. Original Matrix 2 0 2 4 3 2 6 3 1 New Row-Equivalent Matrix 1 1 2 2R1 0 3 2 2 →1 3 3 1 1 5 c. Add times the first row of the original matrix to the third row. Original Matrix New Row-Equivalent Matrix 1 0 0 2 3 3 4 2 13 3 1 8 2R1 R3 → Note that the elementary row operation is written beside the row that is changed. Now try Exercise 25. 2 1 5 2 Te c h n o l o g y Most graphing utilities can perform elementary row operations on matrices. Consult the user’s guide for your graphing utility for specific keystrokes. After performing a ro
w operation, the new row-equivalent matrix that is displayed on your graphing utility is stored in the answer variable. You should use the answer variable and not the original matrix for subsequent row operations. 333202_0801.qxd 12/5/05 10:59 AM Page 575 Section 8.1 Matrices and Systems of Equations 575 In Example 3 in Section 7.3, you used Gaussian elimination with backsubstitution to solve a system of linear equations. The next example demonstrates the matrix version of Gaussian elimination. The two methods are essentially the same. The basic difference is that with matrices you do not need to keep writing the variables. Example 4 Comparing Linear Systems and Matrix Operations 9 5 17 R1 R2 Linear System x 2y x 3y 2x 5y 3z 5z 9 4 17 Add the first equation to the second equation. x 2y 3z y 3z 2x 5y 5z 2 Add to the third equation. times the first equation x 2y 3z y 3z y z 9 5 1 Add the second equation to the third equation. x 2y 3z y 3z 2z 9 5 4 Multiply the third equation by 1 2. Associated Augmented Matrix 1 1 2 2 3 5 3 0 5 . .. . .. . .. 9 4 17 →1 0 2 9 5 17 R1 3 3 5 Add the first row to the . second row 2 1 5 2 Add to the third row 2 1 1 R2 . .. . .. . .. times the first row . 2R1 R3 . .. 9 . .. 5 . .. 1 3 3 1 1 0 0 2R1 R3 → Add the second row to the . R2 R3 third row . 2 .. 3 . .. 1 3 . .. 0 2 1 9 5 4 0 0 R2 R3 → Multiply the third row by 1 2 R3 1 2 . 2 1 0 3 3 1 . .. . .. . .. 9 5 2 Remember that you should check a solution by substituting x, y, the values of into each equation of the original system. For example, you can check the solution to Example 4 as follows. and z Equation 1: 1 21 32 9 ✓ Equation 2: 1 31 4 ✓ Equation 3: 21 51 52 17 ✓ x 2y 3z 9 y 3z 5 z 2 1 0 0 1 2R3 → At this point, you can use back-substitution to find x and y. y 32 5 Substitute 2 for z. y 1 Solve for y. x 21 32 9 x 1 y 1, x 1, The solution is Substitute 1 for y and 2 for z. Solve for x. and z 2. Now try Exercise 27. 333202_0801.qxd 12/5/05 10:59 AM Page 576 576 Chapter 8 Matrices and Determinants The last matrix in Example 4 is said to be in row-echelon form. The term echelon refers to the stair-step pattern formed by the nonzero elements of the matrix. To be in this form, a matrix must have the following properties. Row-Echelon Form and Reduced Row-Echelon Form A matrix in row-echelon form has the following properties. 1. Any rows consisting entirely of zeros occur at the bottom of the matrix. 2. For each row that does not consist entirely of zeros, the first nonzero entry is 1 (called a leading 1). 3. For two successive (nonzero) rows, the leading 1 in the higher row is farther to the left than the leading 1 in the lower row. A matrix in row-echelon form is in reduced row-echelon form if every column that has a leading 1 has zeros in every position above and below its leading 1. Example 5 Row-Echelon Form Determine whether each matrix is in row-echelon form. If it is, determine whether the matrix is in reduced row-echelon form.1 a. c. e. d. f Solution The matrices in (a), (c), (d), and (f) are in row-echelon form. The matrices in (d) and (f) are in reduced row-echelon form because every column that has a leading 1 has zeros in every position above and below its leading 1. The matrix in (b) is not in row-echelon form because a row of all zeros does not occur at the bottom of the matrix. The matrix in (e) is not in row-echelon form because the first nonzero entry in Row 2 is not a leading 1. Now try Exercise 29. Every matrix is row-equivalent to a matrix in row-echelon form. For instance, in Example 5, you can change the matrix in part (e) to row-echelon form by multiplying its second row by 1 2. 333202_0801.qxd 12/5/05 10:59 AM Page 577 Section 8.1 Matrices and Systems of Equations 577 Gaussian Elimination with Back-Substitution Gaussian elimination with back-substitution works well for solving systems of linear equations by hand or with a computer. For this algorithm, the order in which the elementary row operations are performed is important. You should operate from left to right by columns, using elementary row operations to obtain zeros in all entries directly below the leading 1’s. Example 6 Gaussian Elimination with Back-Substitution Solve the system x 2x x y z 2y z 4y z 4y 7z 2w 3w w 3 2 2 19 . Solution 2R1 R1 R3 R4 6R2 R4 1 3R3 13R4 1 R2 0 2 1 1 2 1 0 R1 1 1 1 → 13 0 2 1 1 . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. 3 2 2 19 2 3 2 19 2 3 6 21 2 3 6 39 2 3 2 3 Write augmented matrix. Interchange R1 and R2 so first column has leading 1 in upper left corner. Perform operations on R3 and R4 so first column has zeros below its leading 1. Perform operations on R4 so second column has zeros below its leading 1. Perform operations on R3 and R4 so third and fourth columns have leading 1’s. The matrix is now in row-echelon form, and the corresponding system is x 2y z y z z 2w w w 2 3 2 3 . Using back-substitution, the solution is x 1, y 2, z 1, and w 3. Now try Exercise 51. 333202_0801.qxd 12/5/05 10:59 AM Page 578 578 Chapter 8 Matrices and Determinants The procedure for using Gaussian elimination with back-substitution is summarized below. Gaussian Elimination with Back-Substitution 1. Write the augmented matrix of the system of linear equations. 2. Use elementary row operations to rewrite the augmented matrix in row-echelon form. 3. Write the system of linear equations corresponding to the matrix in row-echelon form, and use back-substitution to find the solution. When solving a system of linear equations, remember that it is possible for the system to have no solution. If, in the elimination process, you obtain a row with zeros except for the last entry, it is unnecessary to continue the elimination process. You can simply conclude that the system has no solution, or is inconsistent. Example 7 A System with No Solution Solve the system x x 2x 3x y 2z 4 z 6 3y 5z 4 2y z 1 . Solution R1 2R1 3R1 R2 R3 R4 R2 R3 1 2 3 1 1 → . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. 4 6 4 1 4 2 4 11 4 2 2 11 Write augmented matrix. Perform row operations. Perform row operations. Note that the third row of this matrix consists of zeros except for the last entry. This means that the original system of linear equations is inconsistent. You can see why this is true by converting back to a system of linear equations. x y 2z y z 0 5y 7z 4 2 2 11 Because the third equation is not possible, the system has no solution. Now try Exercise 57. 333202_0801.qxd 12/5/05 10:59 AM Page 579 Te c h n o l o g y matrix, see For a demonstration of a graphical approach to Gauss-Jordan elimi2 3 nation on a the Visualizing Row Operations Program available for several models of graphing calculators at our website college.hmco.com. The advantage of using GaussJordan elimination to solve a system of linear equations is that the solution of the system is easily found without using back-substitution, as illustrated in Example 8. Section 8.1 Matrices and Systems of Equations 579 Gauss-Jordan Elimination With Gaussian elimination, elementary row operations are applied to a matrix to obtain a (row-equivalent) row-echelon form of the matrix. A second method of elimination, called Gauss-Jordan elimination, after Carl Friedrich Gauss and Wilhelm Jordan (1842–1899), continues the reduction process until a reduced row-echelon form is obtained. This procedure is demonstrated in Example 8. Example 8 Gauss-Jordan Elimination Use Gauss-Jordan elimination to solve the system x 2y x 3y 2x 5y 3z 5z 9 4 17 . Solution In Example 4, Gaussian elimination was used to obtain the row-echelon form of the linear system above. . .. 2 . .. 1 . .. Now, apply elementary row operations until you obtain zeros above each of the leading 1’s, as follows. 2R2 R1 9R3 3R3 R1 R2 0 0 →1 → . .. . .. . .. . .. . .. . .. 19 5 2 1 1 2 Perform operations on R1 so second column has a zero above its leading 1. Perform operations on R1 and R2 so third column has zeros above its leading 1. The matrix is now in reduced row-echelon form. Converting back to a system of linear equations, you have x y z 1 1. 2 Now you can simply read the solution, 1, 1, 2. written as the ordered triple Now try Exercise 59. x 1, y 1, and z 2, which can be The elimination procedures described in this section sometimes result in fractional coefficients. For instance, in the elimination procedure for the system 2x 5y 3x 2y 3x 3y 5z 3z 17 11 6 you may be inclined to multiply the first row by to produce a leading 1, which will result in working with fractional coefficients. You can sometimes avoid fractions by judiciously choosing the order in which you apply elementary row operations. 1 2 333202_0801.qxd 12/5/05 10:59 AM Page 580 580 Chapter 8 Matrices and Determinants Recall from Chapter 7 that when there are fewer equations than variables in a system of equations, then the system has either no solution or infinitely many solutions. Example 9 A System with an Infinite Number of Solutions Solve the system. 2x 4y 3x 5y 2z 0 1 Solution 2 3 →1 3 0 →1 →1 0 → 2R1 3R1 R2 R2 R1 2R2 . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. The corresponding system of equations is x 5z 2 . y 3z 1 x y Solving for and x 5z 2 z, in terms of you have y 3z 1. and To write a solution to the system that does not use any of the three variables of the system, let z a. represent any real number and let a y x In Example 9, and are solved for in terms of the third variable z. To write a solution to the system that does not use any of the three variables of the system, a let represent any real number x and let and The solution can then be written in terms of which is not one of the variables of the system. Then solve for z a. a, y. in the equations for and x y. Now substitute z a for x 5z 2 5a 2 y 3z 1 3a 1 So, the solution set can be written as an ordered triple with the form 5a 2, 3a 1, a where a is any real number. Remember that a solution set of this form represents a an infinite number of solutions. Tr
y substituting values for to obtain a few solutions. Then check each solution in the original equation. Now try Exercise 65. It is worth noting that the row-echelon form of a matrix is not unique. That is, two different sequences of elementary row operations may yield different row-echelon forms. This is demonstrated in Example 10. 333202_0801.qxd 12/5/05 10:59 AM Page 581 Section 8.1 Matrices and Systems of Equations 581 Example 10 Comparing Row-Echelon Forms Compare the following row-echelon form with the one found in Example 4. Is it the same? Does it yield the same solution? x 2y x 3y 2x 5y 3z 5z R2 1 2 1 2 1 2 1 R1 R1 R1 2R1 R2 R3 R2 R3 1 2R3 9 4 17 . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. 17 4 9 17 4 9 17 4 5 9 4 5 4 4 5 2 Solution This row-echelon form is different from that obtained in Example 4. The corresponding system of linear equations for this row-echelon matrix is x 3y y 4 3z 5 . z 2 Using back-substitution on this system, you obtain the solution x 1, y 1, and z 2 which is the same solution that was obtained in Example 4. Now try Exercise 77. You have seen that the row-echelon form of a given matrix is not unique; however, the reduced row-echelon form of a given matrix is unique. Try applying Gauss-Jordan elimination to the row-echelon matrix in Example 10 to see that you obtain the same reduced row-echelon form as in Example 8. 333202_0801.qxd 12/5/05 10:59 AM Page 582 582 Chapter 8 Matrices and Determinants 8.1 Exercises The HM mathSpace® CD-ROM and Eduspace® for this text contain step-by-step solutions to all odd-numbered exercises. They also provide Tutorial Exercises for additional help. VOCABULARY CHECK: Fill in the blanks. 1. A rectangular array of real numbers than can be used to solve a system of linear equations is called a ________. 2. A matrix is ________ if the number of rows equals the number of columns. 3. For a square matrix, the entries a11, a22, a33, . . . , ann are the ________ ________ entries. 4. A matrix with only one row is called a ________ matrix and a matrix with only one column is called a ________ matrix. 5. The matrix derived from a system of linear equations is called the ________ matrix of the system. 6. The matrix derived from the coefficients of a system of linear equations is called the ________ matrix of the system. 7. Two matrices are called ________ if one of the matrices can be obtained from the other by a sequence of elementary row operations. 8. A matrix in row-echelon form is in ________ ________ ________ if every column that has a leading 1 has zeros in every position above and below its leading 1. 9. The process of using row operations to write a matrix in reduced row-echelon form is called ________ ________. PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. In Exercises 1– 6, determine the order of the matrix. 1. 7 0 2 36 3 33 9 3. 5. 45 20 2. 4. 6 15 3 6 7 0 3 7 In Exercises 7–12, write the augmented matrix for the system of linear equations. 7. 9. 11. 3y 3y 4x x x 10y 5x 3y 2x y 7x 5 12 2z 2 4z 0 6 5y z 13 8z 10 19x 8. 10. 12. 7x 4y 22 5x 9y 15 x 8y 9x 5z 15z 8z 2y 3z 25y 11z 7x 3x y 8 38 20 20 5 In Exercises 13–18, write the system of linear equations y, represented by the augmented matrix. (Use variables z, w, x, and 1 2 7 8 2 0 6 13. 14. 15. 7 4 2 3 5 3 if applicable.) 0 2 5 2 0 0 1 3 12 7 2 16. 17. 18. 4 11 12 18 10 1 0 2 0 0 5 3 6 11 18 25 29 0 10 4 10 25 7 23 21 In Exercises 19–22, fill in the blank(s) using elementary row operations to form a row-equivalent matrix. 19. 21 10 4 1 8 1 3 5 3 1 4 10 12 1 3 6 20. 22 333202_0801.qxd 12/5/05 10:59 AM Page 583 In Exercises 23–26, identify the elementary row operation(s) being performed to obtain the new row-equivalent matrix. Original Matrix 2 3 5 1 8 1 Original Matrix 4 1 3 4 7 3 Original Matrix 1 5 7 3 5 1 Original Matrix New Row-Equivalent Matrix 39 13 8 3 New Row-Equivalent Matrix 3 5 1 0 4 5 1 New Row-Equivalent Matrix 7 5 27 New Row-Equivalent Matrix 3 1 7 6 5 27 11 4 5 6 3 2 7 6 23. 24. 25. 26. 27. Perform the sequence of row operations on the matrix. What did the operations accomplisha) Add (b) Add 2 3 1 (d) Multiply 2 (c) Add times times times R2 times by to to R1 R1 R2 to 1 5. to R2. R3. R3. (e) Add R1. 28. Perform the sequence of row operations on the matrix. R2 What did the operations accomplish? 7 0 3 4 1 2 4 1 to R3 (a) Add (c) Add 3 times R4. R1 (b) Interchange R1 times R2 7 (e) Multiply (d) Add by and R4. R3. to R4. to R1 1 2. (f) Add the appropriate multiples of R2 to R1, R3, and R4. In Exercises 29–32, determine whether the matrix is in row-echelon form. If it is, determine if it is also in reduced row-echelon form. 29 Section 8.1 Matrices and Systems of Equations 583 30. 31. 32 10 0 In Exercises 33–36, write the matrix in row-echelon form. (Remember that the row-echelon form of a matrix is not unique.) 33. 35 18 5 10 14 1 8 0 34. 36 10 10 1 5 3 0 1 2 3 14 8 7 23 24 In Exercises 37–42, use the matrix capabilities of a graphing utility to write the matrix in reduced row-echelon form 37. 39. 40. 41. 42. 38. 1 5 2 3 15 6 2 9 10 11 1 5 2 10 1 1 2 10 5 9 3 14 2 8 0 30 12 4 4 32 z, y, x, 43. and In Exercises 43–46, write the system of linear equations represented by the augmented matrix. Then use back-substitution to solve. (Use variables if applicable.) 1 0 1 46 44. 45. 5 1 0 0 0 0 333202_0801.qxd 12/5/05 11:00 AM Page 584 584 Chapter 8 Matrices and Determinants In Exercises 47–50, an augmented matrix that represents a system of linear equations (in variables if applicable) has been reduced using Gauss-Jordan elimination. Write the solution represented by the augmented matrix. and x, y, z, 47. 48. 49. 50 10 4 10 4 5 3 0 In Exercises 51–70, use matrices to solve the system of equations (if possible). Use Gaussian elimination with back-substitution or Gauss-Jordan elimination. 4 34 5 7 5 10 3z 24 z 14 6 2 28 14 2x 6y 16 2x 3y 7 x y 2x 4y 5x 5y 2x 3y x 3y 2x 6y 2x y 2x 3x 2x 2y 7x 5y 2y 3y y x x x x z z 4x 8x 15 10 14 z 2y 2z y 4z y 3z 3 3y 7z 5 9y 15z 9 51. 53. 55. 57. 59. 61. 63. 65. 67. 68. 69. 52. 54. 56. 58. 60. 62. 64. 66. 1.5 z z z 2x 3x 3x 2x y y 2y y 2y 27 13 22 9 3 3z 2z z 2 5 4 14 21 19 28 0 5 x 2y 7 2x y 8 3x 2y x 3y 2x 6y x 2y x 2y 2x 4y x x x 2y 4y y x 5z 3 2z 1 z 0 x 2y z 2w 8 3x 7y 6z 9w 26 4x 12y 7z 20w 22 3x 9y 5z 28w 30 x y 22 4 32 3x 4y 4x 8y 3z 2z z x 2x y y x 70. In Exercises 71–76, use the matrix capabilities of a graphing utility to reduce the augmented matrix corresponding to the system of equations, and solve the system. 71. 72. 73. 74. 75. 76. 3x 4y x 5y 5x 2y x 5y 2z x 5y z 3x 15y 3z x y 4z 2 2x 5y 20z 10 x 2y 8z 4 6 6 3 9 z 2w w 2z 6w z w 3x 3y 12z 6 2x 10y 2z 2x y x 2y 2z 4w x x 2y 3x 6y 5z 12w x 3y 3z 2w 6x y z w z w 0 z 2w 0 z 0 z 3w 0 w 0 z 2w 0 x y y y 3y 5y 2x 3x 6 1 3 3 11 30 5 9 In Exercises 77–80, determine whether the two systems of linear equations yield the same solution. If so, find the solution using matrices. 77. (a) 78. (a) 79. (a) 80. (a) y z z y 5z z x 2y z x 3y 4z x 4y 5z x 3y z y 7z z y 6z z (b) (b) (b) (b) 6 16 3 11 4 2 27 54 8 19 18 4 y 6 8 3 11 4 2 y 3z z 3z z x y 2z x 4y x 6y z 15 x y 3z y 5z 42 z 8 15 14 4 y 2z z x 2y 0 x y 6 3x 2y 8 81. Use the system x 3y z 3 x 5y 5z 1 2x 6y 3z 8 to write two different matrices in row-echelon form that yield the same solution. 333202_0801.qxd 12/5/05 11:00 AM Page 585 82. Electrical Network The currents in an electrical network are given by the solution of the system I1 3I1 I2 4I2 I2 I3 3I3 0 18 6 Section 8.1 Matrices and Systems of Equations 585 89. Mathematical Modeling A videotape of the path of a ball thrown by a baseball player was analyzed with a grid covering the TV screen. The tape was paused three times, and the position of the ball was measured each time. The are coordinates obtained are shown in the table. ( and measured in feet.) x y I1, I2, I3 where system of equations using matrices. and are measured in amperes. Solve the 83. Partial Fractions Use a system of equations to write the partial fraction decomposition of the rational expression. Solve the system using matrices. 4x 2 x 12x 1 A x 1 B x 1 C x 12 84. Partial Fractions Use a system of equations to write the partial fraction decomposition of the rational expression. Solve the system using matrices. 8x2 x 12x 1 A x 1 B x 1 C x 12 85. Finance A small shoe corporation borrowed $1,500,000 to expand its line of shoes. Some of the money was borrowed at 7%, some at 8%, and some at 10%. Use a system of equations to determine how much was borrowed at each rate if the annual interest was $130,500 and the amount borrowed at 10% was 4 times the amount borrowed at 7%. Solve the system using matrices. 86. Finance A small software corporation borrowed $500,000 to expand its software line. Some of the money was borrowed at 9%, some at 10%, and some at 12%. Use a system of equations to determine how much was borrowed at each rate if the annual interest was $52,000 and the amount borrowed at 10% was times the amount borrowed at 9%. Solve the system using matrices. 21 2 In Exercises 87 and 88, use a system of equations to find the specified equation that passes through the points. Solve the system using matrices. Use a graphing utility to verify your results. 87. Parabola: 88. Parabola: y ax 2 bx c y 24 (3, 20) (2, 13) (1, 8) −8 −4 4 8 12 x y ax 2 bx c y 12 8 (1, 9) (2, 8) (3, 5) −8 −4 8 12 x Horizontal distance, x Height, y 0 15 30 5.0 9.6 12.4 (a) Use a system of equations to find the equation of the that passes through the parabola three points. Solve the system using matrices. y ax 2 bx c (b) Use a graphing utility to graph the parabola. (c) Graphically approximate the maximum height of the ball and the point at which the ball struck the ground. (d) Analytically find the maximum height of the ball and the point at which the ball struck the ground. (e) Compare your results from parts (c) and (d). Model It 90. Data Analysis: Snowboarders The tab
le shows the numbers of people (in millions) in the United States who participated in snowboarding for selected years from 1997 to 2001. (Source: National Sporting Goods Association) y Year Number, y 1997 1999 2001 2.8 3.3 5.3 (a) Use a system of equations to find the equation of y at2 bt c that passes through the parabola t 7 the points. Let corresponding to 1997. Solve the system using matrices. represent the year, with t (b) Use a graphing utility to graph the parabola. (c) Use the equation in part (a) to estimate the number of people who participated in snowboarding in 2003. How does this value compare with the actual 2003 value of 6.3 million? y (d) Use the equation in part (a) to estimate in the year 2008. Is the estimate reasonable? Explain. 333202_0801.qxd 12/5/05 11:00 AM Page 586 586 Chapter 8 Matrices and Determinants Network Analysis In Exercises 91 and 92, answer the questions about the specified network. (In a network it is assumed that the total flow into each junction is equal to the total flow out of each junction.) 91. Water flowing through a network of pipes (in thousands of cubic meters per hour) is shown in the figure. 600 600 x3 x 1 x6 x4 x2 x7 x5 500 500 (a) Solve this system using matrices for the water flow xi, i 1, 2, . . . , 7. represented by (b) Find the network flow pattern when 0. x 7 0 x6 and (c) Find the network flow pattern when 0. x6 1000 and x5 92. The flow of traffic (in vehicles per hour) through a network of streets is shown in the figure. 300 200 x2 x 1 x5 x3 x 4 150 350 represented by (a) Solve this system using matrices for the traffic flow xi, i 1, 2, . . . , 5. 200 150 (b) Find the traffic flow when (c) Find the traffic flow when 50. 0. and and x3 x3 x2 x2 Synthesis True or False? statement is true or false. Justify your answer. In Exercises 93–95, determine whether the 93. 5 1 94. The matrix 0 3 2 6 7 0 is a 4 2 matrix is in reduced row-echelon form. 95. The method of Gaussian elimination reduces a matrix until a reduced row-echelon form is obtained. 96. Think About It The augmented matrix represents z and ) that a system of linear equations (in variables has been reduced using Gauss-Jordan elimination. Write a system of equations with nonzero coefficients that is represented by the reduced matrix. (There are many correct answers.) y, x 97. Think About It (a) Describe the row-echelon form of an augmented matrix that corresponds to a system of linear equations that is inconsistent. (b) Describe the row-echelon form of an augmented matrix that corresponds to a system of linear equations that has an infinite number of solutions. 98. Describe the three elementary row operations that can be performed on an augmented matrix. 99. What is the relationship between the three elementary row operations performed on an augmented matrix and the operations that lead to equivalent systems of equations? 100. Writing In your own words, describe the difference between a matrix in row-echelon form and a matrix in reduced row-echelon form. Skills Review In Exercises 101–106, sketch the graph of the function. Do not use a graphing utility. fx 2x2 4x 3x x2 fx x2 2x 1 102. 101. x2 1 103. 104. 105. 106. f x 2 x1 gx 3x2 hx lnx 1 f x 3 ln x 333202_0802.qxd 12/5/05 10:57 AM Page 587 8.2 Operations with Matrices Section 8.2 Operations with Matrices 587 What you should learn • Decide whether two matrices are equal. • Add and subtract matrices and multiply matrices by scalars. • Multiply two matrices. • Use matrix operations to model and solve real-life problems. Why you should learn it Matrix operations can be used to model and solve real-life problems. For instance, in Exercise 70 on page 601, matrix operations are used to analyze annual health care costs. Equality of Matrices In Section 8.1, you used matrices to solve systems of linear equations. There is a rich mathematical theory of matrices, and its applications are numerous. This section and the next two introduce some fundamentals of matrix theory. It is standard mathematical convention to represent matrices in any of the following three ways. , . , or bij aij brackets, such as Representation of Matrices 1. A matrix can be denoted by an uppercase letter such as A, B, or C. 2. A matrix can be denoted by a representative element enclosed in cij 3. A matrix can be denoted by a rectangular array of numbers such as a13 a23 a33 ... am3 a11 a12 a22 a32 ... am2 a1n a2n a3n ... amn a21 a31 ... am1 . A aij . . . . . . . . . . . . Two matrices and A aij for and 1 ≤ i ≤ m B bij and 1 ≤ j ≤ n. are equal if they have the same order In other words, two matrices m n are equal if their corresponding entries are equal. bij aij Example 1 Equality of Matrices Solve for a11 a21 a11, a12, a21, and a12 a22 2 3 in the following matrix equation. a22 1 0 © Royalty-Free/Corbis Solution Because two matrices are equal only if their corresponding entries are equal, you can conclude that a11 2, a12 1, a21 3, and a22 0. Now try Exercise 1. Be sure you see that for two matrices to be equal, they must have the same order and their corresponding entries must be equal. For instance.5 but 333202_0802.qxd 12/5/05 10:57 AM Page 588 588 Chapter 8 Matrices and Determinants Matrix Addition and Scalar Multiplication In this section, three basic matrix operations will be covered. The first two are matrix addition and scalar multiplication. With matrix addition, you can add two matrices (of the same order) by adding their corresponding entries Historical Note Arthur Cayley (1821–1895), a British mathematician, invented matrices around 1858. Cayley was a Cambridge University graduate and a lawyer by profession. His groundbreaking work on matrices was begun as he studied the theory of transformations. Cayley also was instrumental in the development of determinants. Cayley and two American mathematicians, Benjamin Peirce (1809–1880) and his son Charles S. Peirce (1839–1914), are credited with developing “matrix algebra.” Definition of Matrix Addition B bij If m n A aij and matrix given by bij A B aij . are matrices of order m n, their sum is the The sum of two matrices of different orders is undefined. Example 2 Addition of Matrices . 0 1 1 1 c. 3 2 d. The sum of and 1 3 4 is undefined because A is of order 3 3 and B is of order 3 2. Now try Exercise 7(a). In operations with matrices, numbers are usually referred to as scalars. In this text, scalars will always be real numbers. You can multiply a matrix by a scalar by multiplying each entry in by c. A A c Definition of Scalar Multiplication If c A aij m n is the matrix given by . cA caij matrix and m n is an c is a scalar, the scalar multiple of by A 333202_0802.qxd 12/5/05 10:57 AM Page 589 Section 8.2 Operations with Matrices 589 Exploration Consider matrices A, B, and C below. Perform the indicated operations and compare the results. Find b. Find B A. and then add C to c. Find the resulting matrix. Find B C, then add A to the resulting matrix. 2B, two resulting matrices. Find A B, resulting matrix by 2. then multiply the and 2A then add the 1 , 7 2 6 A B A B, The symbol A Moreover, if 1B. A and 1A. sum of A, represents the negation of which is the scalar product A B and represents the That is, are of the same order, then A B A B A 1B. Subtraction of matrices The order of operations for matrix expressions is similar to that for real numbers. In particular, you perform scalar multiplication before matrix addition and subtraction, as shown in Example 3(c). Example 3 Scalar Multiplication and Matrix Subtraction For the following matrices, find (a) 3A , (b) B, and (c and B 2 1 1 0 4 3 3A B. 0 3 2 Solution a. b. c. 9 6 3 2 33 32 3A 3 2 32 6 B 1 2 2 3A 10 7 4 1 2 2 0 1 32 30 31 6 0 3 12 3 6 34 31 32 12 3 6 6 4 0 12 6 4 Scalar multiplication Multiply each entry by 3. Simplify. Definition of negation Multiply each entry by 1. 0 4 3 0 3 2 Matrix subtraction Subtract corresponding entries. Now try Exercises 7(b), (c), and (d). It is often convenient to rewrite the scalar multiple by factoring out of every entry in the matrix. For instance, in the following example, the scalar has been factored out of the matrix. 3 1 2 1 1 21 cA 1 2 1 2 5 c 333202_0802.qxd 12/5/05 10:57 AM Page 590 590 Chapter 8 Matrices and Determinants The properties of matrix addition and scalar multiplication are similar to those of addition and multiplication of real numbers. Properties of Matrix Addition and Scalar Multiplication Let c matrices and let and be scalars. m n d B,A, C and be cdA cdA) 1A A cA B cA cB c dA cA dA 1. 2. 3. 4. 5. 6. Commutative Property of Matrix Addition Associative Property of Matrix Addition Associative Property of Scalar Multiplication Scalar Identity Property Distributive Property Distributive Property Note that the Associative Property of Matrix Addition allows you to write expressions such as without ambiguity because the same sum occurs no matter how the matrices are grouped. This same reasoning applies to sums of four or more matrices. A B C Example 4 Addition of More than Two Matrices By adding corresponding entries, you obtain the following sum of four matrices Now try Exercise 13. Example 5 Using the Distributive Property Perform the indicated matrix operations. 4 3 32 4 2 7 0 1 Solution 32 4 4 3 0 1 2 7 32 4 6 12 6 21 Now try Exercise 15. 2 7 6 21 34 0 3 1 12 0 9 3 6 24 In Example 5, you could add the two matrices first and then multiply the matrix by 3, as follows. Notice that you obtain the same result. 6 2 24 7 32 7 6 21 32 4 4 3 2 8 0 1 Te c h n o l o g y Most graphing utilities have the capability of performing matrix operations. Consult the user’s guide for your graphing utility for specific keystrokes. Try using a graphing utility to find the sum of the matrices A 2 1 3 0 and B 1 2 . 4 5 333202_0802.qxd 12/5/05 10:57 AM Page 591 Section 8.2 Operations with Matrices 591 One important property of addition of real numbers is that the number 0 is for any real number For matrices, a m n zero O A O A. c 0 c the additive identity. Tha
t is, A similar property holds. That is, if is an matrix consisting entirely of zeros, then O matrices. For example, the following matrices are the additive identities for the set of all 2 3 is the additive identity for the set of all In other words, c. is the matrix and m n m n 2 2 and O 0 0 matrices. 0 0 0 0 and O 0 0 0 0 Remember that matrices are denoted by capital letters. So, when you solve for X, you are solving for a matrix that makes the matrix equation true. 2 3 zero matrix 2 2 zero matrix The algebra of real numbers and the algebra of matrices have many similarities. For example, compare the following solutions. Real Numbers (Solve for x.) Matrices (Solve for X.) The algebra of real numbers and the algebra of matrices also have important differences, which will be discussed later. Example 6 Solving a Matrix Equation Solve for X A 1 0 in the equation and 2 3 3X A B, B 3 2 where . 4 1 Solution Begin by solving the equation for X to obtain 3X B A X 1 3 B A. Now, using the matrices 3 2 A and , you have 2 3 Substitute the matrices. Subtract matrix A from matrix B. Multiply the matrix by 1 3. Now try Exercise 25. 333202_0802.qxd 12/5/05 10:57 AM Page 592 592 Chapter 8 Matrices and Determinants Matrix Multiplication The third basic matrix operation is matrix multiplication. At first glance, the definition may seem unusual. You will see later, however, that this definition of the product of two matrices has many practical applications. Definition of Matrix Multiplication B bij If AB matrix and m n matrix is an A aij m p is an AB cij is an n p matrix, the product where ci j ai1b1j ai2b2 j ai3b3j . . . ainbnj. The definition of matrix multiplication indicates a row-by-column multiplication, where the entry in the th row and is i by the corresponding obtained by multiplying the entries in the th row of B entries in the th column of and then adding the results. The general pattern for matrix multiplication is as follows. j th column of the product AB A j i b11 b21 b31 . .. bn1 b12 b22 b32 . .. bn2 . . . . . . . . . . . . b1j b2j b3j . .. bnj . . . . . . . . . . . . b1p b2p b3p . .. bnp c11 c21 . .. ci1 . .. cm1 c12 c22 . .. ci2 . .. cm2 . . . . . . . . . . . . c1j c2j . .. cij . .. cmj . . . . . . . . . . . . c1p c2p . .. cip . .. cmp a11 a21 a31 . .. ai1 . .. am1 a12 a22 a32 . .. ai2 . .. am2 a13 a23 a33 . .. ai3 . .. am3 . . . . . . . . . . . . . . . a1n a2n a3n . .. ain . .. amn ai1b1j ai2b2j ai3b3j . . . ainbnj cij Example 7 Finding the Product of Two Matrices AB First, note that the product equal to the number of rows of Moreover, the product A find the entries of the product, multiply each row of as follows. A is defined because the number of columns of 3 2. B. AB by each column of has order is To B AB 1 13 34 9 43 24 53 04 4 15 1 6 10 12 31 42 21 52 01 Now try Exercise 29. 333202_0802.qxd 12/5/05 10:57 AM Page 593 Exploration Use the following matrices to find AB, BA, What do your results tell you about matrix multiplication, commutativity, and associativity? ABC. ABC, and Section 8.2 Operations with Matrices 593 Be sure you understand that for the product of two matrices to be defined, the number of columns of the first matrix must equal the number of rows of the second matrix. That is, the middle two indices must be the same. The outside two indices give the order of the product, as shown below. A m n B n p AB m p Equal Order of AB Example 8 Finding the Product of Two Matrices Find the product AB where A 1 2 0 1 3 2 and B 2 1 1 . 4 0 1 Solution Note that the order of has order 2 2. A is 2 3 and the order of B is 3 2. So, the product AB AB 1 2 0 1 3 2 2 12 01 31 22 11 21 1 1 4 0 1 14 00 31 24 10 21 5 3 7 6 Now try Exercise 31. Example 9 Patterns in Matrix Multiplication a 10 5 9 3 1 c. The product AB for the following matrices is not defined and Now try Exercise 33. 333202_0802.qxd 12/5/05 10:57 AM Page 594 594 Chapter 8 Matrices and Determinants Example 10 Patterns in Matrix Multiplication a Now try Exercise 45. In Example 10, note that the two products are different. Even if BA are defined, matrix multiplication is not, in general, commutative. That is, for most matrices, This is one way in which the algebra of real numbers and the algebra of matrices differ. AB BA. and AB Properties of Matrix Multiplication Let and be matrices and let be a scalar. c C B,A, ABC ABC AB C AB AC A B)C AC BC cAB cAB AcB 1. 2. 3. 4. Associative Property of Multiplication Distributive Property Distributive Property Associative Property of Scalar Multiplication Definition of Identity Matrix The is called the identity matrix of order n and is denoted by n n matrix that consists of 1’s on its main diagonal and 0’s elsewhere 1 0 0 . .. 0 In . 0 1 0 . .. 0 0 0 1 . .. . .. 1 Identity matrix Note that an identity matrix must be square. When the order is understood to be you can denote simply by I. n, In If A is an n n For example, InA A. matrix, the identity matrix has the property that AIn A and and AI A IA A 333202_0802.qxd 12/5/05 10:57 AM Page 595 Section 8.2 Operations with Matrices 595 Applications Matrix multiplication can be used to represent a system of linear equations. Note how the system a11x1 a21x1 a31x1 a12x2 a22x2 a32x2 a13x3 a23x3 a33x3 b1 b2 b3 can be written as the matrix equation X of the system, and and B are column matrices. AX B, where A is the coefficient matrix . .. B represents A The notation the augmented matrix formed B is adjoined to when matrix . .. X matrix The notation represents the reduced rowechelon form of the augmented matrix that yields the solution to the system. I A. a11 a21 a31 a12 a22 a32 A a13 a23 a33 x1 x2 x3 b1 b2 b3 X B Example 11 Solving a System of Linear Equations Consider the following system of linear equations. x1 x3 2x3 2x3 2x2 x2 3x2 4 4 2 2x1 a. Write this system as a matrix equation, b. Use Gauss-Jordan elimination on the augmented matrix AX B. A B to solve for the matrix X. Solution a. In matrix form, AX B, the system can be written as follows x1 x2 x3 4 4 2 b. The augmented matrix is formed by adjoining matrix B to matrix A ... ... ... Using Gauss-Jordan elimination, you can rewrite this equation as ... ... ... So, the solution of the system of linear equations is x3 and the solution of the matrix equation is 1, 1, x1 x2 2, and X x1 x2 x3 1 2 1 . Now try Exercise 55. 333202_0802.qxd 12/5/05 10:57 AM Page 596 596 Chapter 8 Matrices and Determinants Example 12 Softball Team Expenses Two softball teams submit equipment lists to their sponsors. Women’s Team Men’s Team Bats Balls Gloves 12 45 15 15 38 17 Each bat costs $80, each ball costs $6, and each glove costs $60. Use matrices to find the total cost of equipment for each team. Solution The equipment lists E and the costs per item C can be written in matrix form as E 12 45 15 15 38 17 and C 80 6 60. The total cost of equipment for each team is given by the product CE 80 6 6012 45 15 15 38 17 8012 645 6015 8015 638 6017 2130 2448. So, the total cost of equipment for the women’s team is $2130 and the total cost of equipment for the men’s team is $2448. Notice that you cannot find the total cost using the product is not defined. That is, the number of E columns of EC (2 columns) does not equal the number of rows of because (1 row). EC C Now try Exercise 63. W RITING ABOUT MATHEMATICS Problem Posing Write a matrix multiplication application problem that uses the matrix A 20 17 42 30 . 33 50 Exchange problems with another student in your class. Form the matrices that represent the problem, and solve the problem. Interpret your solution in the context of the problem. Check with the creator of the problem to see if you are correct. Discuss other ways to represent and/or approach the problem. 333202_0802.qxd 12/5/05 10:57 AM Page 597 Section 8.2 Operations with Matrices 597 8.2 Exercises VOCABULARY CHECK: In Exercises 1–4, fill in the blanks. 1. Two matrices are ________ if all of their corresponding entries are equal. 2. When performing matrix operations, real numbers are often referred to as ________. 3. A matrix consisting entirely of zeros is called a ________ matrix and is denoted by ________. 4. The n n matrix consisting of 1’s on its main diagonal and 0’s elsewhere is called the ________ matrix of order n. In Exercises 5 and 6, match the matrix property with the correct form. m n, and and are scalars. d c B,A, and are matrices of order C 5. (a) (b) (c) (d) (e) 6. (a) (b) (c) (d) 1A A A B C A B C c dA cA dA cdA cdA A B B A A O A cAB AcB AB C AB AC ABC ABC (i) Distributive Property (ii) Commutative Property of Matrix Addition (iii) Scalar Identity Property (iv) Associative Property of Matrix Addition (v) Associative Property of Scalar Multiplication (i) Distributive Property (ii) Additive Identity of Matrix Addition (iii) Associative Property of Multiplication (iv) Associative Property of Scalar Multiplication PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. y. x 1. 2 y In Exercises 1–4, find and 2 4 22 7 5 x 12 8 2. x 7 5 y 16 x 2 3 0 1 7 3. 4. 13 8 16 2x 6 3 0 4 13 2 1 7 4 13 2 8 2y 2 5 15 4 4 6 0 3 2x y 2 2x 1 15 3y 5 8 18 2 4 3x 0 3 8 11 A B, (b) A B, 6. 5. and (d) In Exercises 5–12, if possible, find (a) 3A 2B. (c , 3A 10 . 7. 3 1 4 2 9. 10. 11. 12 10 In Exercises 13–18, evaluate the expression. 13. 5 3 14 10 14 11 2 8 6 7 1 333202_0802.qxd 12/5/05 10:57 AM Page 598 598 Chapter 8 Matrices and Determinants 15. 16. 17. 18. 1 2 44 0 5 2 30 18 4 9 2 1 3 3 0 14 6 24 3 7 1 7 1 4 13 4 6 8 65 1 9 6 3 0 3 2 11 1 3 9 5 1 1 In Exercises 19–22, use the matrix capabilities of a graphing utility to evaluate the expression. Round your results to three decimal places, if necessary. 0 2 19. 2 5 3 4 1 7 55 14 22 3.211 12 6 1.004 0.055 1 2 20. 21. 22. 63 2 11 22 19 13 20 6 1.630 6.829 4.914 3.889 20 9 5 14 8 7 5.256 9.768 15 6 0 31 3.090 8.335 4.251 19 10 10 16 24 In Exercises 23–26, solve fo X in the equation, given A [2 1 3 1 0 4] 23. 25. X 3A 2B 2X 3A B and B [ 0 2 4 3 0 1]. 24. 2
6. 2X 2A B 2A 4B 2X In Exercises 27–34, if possible, find the result. AB and state the order of 27. 28. 29. 30. 31. 32 17 , 13 10 12 , 33. 34 11 16 0 4 4 0 B 6 2 1 6 In Exercises 35– 40, use the matrix capabilities of a graphing utility to find if possible. AB, 35. 36. 37. 38. 39. 40. 2 10 A 5 A 11 14 6 6 5 5 12 10 2 3 1 5 , , 4 12 9 A 3 12 5 8 15 1 6 9 1 A 2 21 13 10 12 16 8 4 B 1 B 12 5 15 B 3 , B 2 24 16 8 0 15 14 7 32 8 6 5 0.5 1.6 2 4 9 1 15 10 4 6 14 21 10 A 9 100 B 52 40 A 15 4 8 , 18 75 10 50 38 250 85 35 18 12 22 , 45 27 82 60 B 7 8 22 16 1 24 In Exercises 41– 46, if possible, find (a) (Note: A2 AA. AB, (b) BA, and (c) A2 41. 42. 43. 44. 45 46 In Exercises 47–50, evaluate the expression. Use the matrix capabilities of a graphing utility to verify your answer. 1 47 333202_0802.qxd 12/5/05 10:57 AM Page 599 36 48. 49. 0 4 50 In Exercises 51–58, (a) write the system of linear equations and (b) use Gauss-Jordan as a matrix equation, elimination on the augmented matrix to solve for the matrix [A B] AX B, 51. 53. 55. 56. 57. 58. 3x3 x3 5x3 3x3 4 3x2 6x1 x2 36 x1 2x1 X. 4 x1 x2 2x1 x2 0 2x1 x1 x1 x1 5x2 x2 2x2 x2 3x2 6x2 2x2 3x2 5x2 x2 2x2 x2 x1 x1 x1 3x1 x1 x3 2x3 x3 5x3 4x3 5x3 52. 54. 2x1 3x2 5 x1 4x2 10 4x1 13 9x2 x1 3x2 12 9 6 17 9 6 5 20 8 16 17 11 40 59. Manufacturing A corporation has three factories, each of which manufactures acoustic guitars and electric guitars. The number of units of guitars produced at factory in one aij day is represented by . A 70 35 in the matrix 50 100 25 70 j Find the production levels if production is increased by 20%. 60. Manufacturing A corporation has four factories, each of which manufactures sport utility vehicles and pickup trucks. The number of units of vehicle produced at factory j in one day is represented by . A 100 40 in the matrix 90 20 30 60 70 60 aij i Find the production levels if production is increased by 10%. Section 8.2 Operations with Matrices 599 61. Agriculture A fruit grower raises two crops, apples and peaches. Each of these crops is sent to three different outlets for sale. These outlets are The Farmer’s Market, The Fruit Stand, and The Fruit Farm. The numbers of bushels of apples sent to the three outlets are 125, 100, and 75, respectively. The numbers of bushels of peaches sent to the three outlets are 100, 175, and 125, respectively. The profit per bushel for apples is $3.50 and the profit per bushel for peaches is $6.00. (a) Write a matrix i of each crop what each entry (b) Write a matrix A that represents the number of bushels that are shipped to each outlet State aij B that represents the profit per bushel of of the matrix of the matrix represents. j. bij each fruit. State what each entry represents. (c) Find the product matrix represents. BA and state what each entry of the 62. Revenue A manufacturer of electronics produces three models of portable CD players, which are shipped to two warehouses. The number of units of model that are j in the matrix shipped to warehouse is represented by i aij A 5,000 6,000 8,000 . 4,000 10,000 5,000 The prices per unit are represented by the matrix B $39.50 $44.50 $56.50. Compute BA and interpret the result. 63. Inventory A company sells five models of computers through three retail outlets. The inventories are represented by S. Model Outlet The wholesale and retail prices are represented by T. Price Wholesale Retail T $840 $1200 $1450 $2650 $3050 $1100 $1350 $1650 $3000 $3200 A B C D E Model Compute ST and interpret the result. 333202_0802.qxd 12/5/05 10:57 AM Page 600 600 Chapter 8 Matrices and Determinants 64. Voting Preferences The matrix From R P 0.6 0.2 0.2 D 0.1 0.7 0.2 I 0.1 0.1 0.8 R D I To i j is called a stochastic matrix. Each entry represents the proportion of the voting population that changes from party represents the proportion that remains loyal to the party from one election to the next. Compute and interpret to party and P2. pij pii j, i 65. Voting Preferences Use a graphing utility to find P6, P7, P5, you detect a pattern as and P8 P 4, for the matrix given in Exercise 64. Can is raised to higher powers? P3, P 66. Labor/Wage Requirements A company that manufactures boats has the following labor-hour and wage requirements. Labor per boat Department Selling price Profit B 2.65 2.85 3.05 0.65 0.70 0.85 Skim milk 2% milk Whole milk (a) Compute AB and interpret the result. (b) Find the dairy mart’s total profit from milk sales for the weekend. 68. Profit At a convenience store, the numbers of gallons of 87-octane, 89-octane, and 93-octane gasoline sold over the A. weekend are represented by Octane 87 89 A 580 560 860 840 420 1020 93 320 160 540 Friday Saturday Sunday The selling prices per gallon and the profits per gallon for the three grades of gasoline sold by the convenience store are represents by B. Cutting Assembly Packaging S 1.0 hr 1.6 hr 2.5 hr 0.5 hr 1.0 hr 2.0 hr 0.2 hr 0.2 hr 1.4 hr Small Medium Large Boat size Selling price Profit B 1.95 2.05 2.15 0.32 0.36 0.40 87 89 93 Octane Wages per hour Plant A B T $12 $9 $8 $10 $8 $7 Cutting Assembly Packaging Department Compute ST and interpret the result. 67. Profit At a local dairy mart, the numbers of gallons of skim milk, 2% milk, and whole milk sold over the weekend are represented by A. Skim milk 2% Whole milk milk A 40 60 76 64 82 96 52 76 84 Friday Saturday Sunday The selling prices (in dollars per gallon) and the profits (in dollars per gallon) for the three types of milk sold by the dairy mart are represented by B. (a) Compute AB and interpret the result. (b) Find the convenience store’s profit from gasoline sales for the weekend. 69. Exercise The numbers of calories burned by individuals of different body weights performing different types of aerobic exercises for a 20-minute time period are shown in matrix A. Calories burned 120-lb person 150-lb person A 109 127 64 136 159 79 Bicycling Jogging Walking (a) A 120-pound person and a 150-pound person bicycled for 40 minutes, jogged for 10 minutes, and walked for 60 minutes. Organize the time spent exercising in a matrix B. (b) Compute BA and interpret the result. 333202_0802.qxd 12/5/05 10:57 AM Page 601 Model It 70. Health Care The health care plans offered this year by a local manufacturing plant are as follows. For individuals, the comprehensive plan costs $694.32, the HMO standard plan costs $451.80, and the HMO Plus plan costs $489.48. For families, the comprehensive plan costs $1725.36, the HMO standard plan costs $1187.76 and the HMO Plus plan costs $1248.12. The plant expects the costs of the plans to change next year as follows. For individuals, the costs for the comprehensive, HMO standard, and HMO Plus plans will be $683.91, $463.10, and $499.27, respectively. For families, the costs for the comprehensive, HMO standard, and HMO Plus plans will be $1699.48, $1217.45, and $1273.08, respectively. A (a) Organize the information using two matrices and B, represents the health care plan costs for where this year and represents the health care plan costs for next year. State what each entry of each matrix represents. B A (b) Compute A B and interpret the result. (c) The employees receive monthly paychecks from which the health care plan costs are deducted. Use the matrices from part (a) to write matrices that show how much will be deducted from each employees’ paycheck this year and next year. (d) Suppose the costs of each plan instead increase by 4% next year. Write a matrix that shows the new monthly payment. Synthesis True or False? the statement is true or false. Justify your answer. In Exercises 71 and 72, determine whether 71. Two matrices can be added only if they have the same order. 6 2 72 In Exercises 73– 80, let matrices 3 2, Think About It and respectively. be of orders D Determine whether the matrices are of proper order to perform the operation(s). If so, give the order of the answer. C,B,A, 2 3, 2 3, 2 2, and 73. 75. 77. 79. A 2C AB BC D DA 3B 74. 76. 78. 80. B 3C BC CB D BC DA Section 8.2 Operations with Matrices 601 81. Think About It and a, b, then c If and are real numbers such that a b. c 0 ac bc, C and is not are nonzero matrices such that necessarily equal to Illustrate this using the following matrices. A 0 0 However, if AC BC, C 2 2 B 1 1 B,A, A then , , 1 1 3 3 0 0 B. b and b 0. a or AB O, are real numbers such that are it is not necessarily true that Illustrate this using the following However, if and A B 82. Think About It then If ab 0, a 0 matrices such that A O B O. or matrices 83. Exploration Let and be unequal diagonal matrices of the same order. (A diagonal matrix is a square matrix in which each entry not on the main diagonal is zero.) Determine the products for several pairs of such matrices. Make a conjecture about a quick rule for such products. AB 84. Exploration Let and let i . 0 and (a) Find i 3, A2, and i 4. A3, and A4. Identify any similarities with i 2, (b) Find and identify B2. Skills Review In Exercises 85–90, solve the equation. 85. 86. 87. 88. 89. 90. 3x2 20x 32 0 8x2 10x 3 0 4x3 10x2 3x 0 3x 3 22x2 45x 0 3x3 12x2 5x 20 0 2x3 5x2 12x 30 0 In Exercises 91–94, solve the system of linear equations both graphically and algebraically. 92. 91. x 4y 9 5x 8y 39 8x 3y 17 6x 7y 27 x 2y 5 8 3x y 94. 6x 13y 11 9x 5y 41 93. 333202_0803.qxd 12/5/05 11:01 AM Page 602 602 Chapter 8 Matrices and Determinants 8.3 The Inverse of a Square Matrix What you should learn • Verify that two matrices are inverses of each other. • Use Gauss-Jordan elimination to find the inverses of matrices. • Use a formula to find the inverses of 2 2 • Use inverse matrices to solve systems of linear equations. matrices. Why you should learn it You can use inverse matrices to model and solve real-life problems. For instance, in Exercise 72 on page 610, an inverse matrix is used to find a linear model for the number of licensed drivers in the United States. Jon Love/Getty Images The Inverse of a Matrix This section further develops the algebra of matrices. To begin, consid
er the real x, To solve this equation for multiply each side of the number equation equation by ax b. (provided that a 0 ). a1 ax b a1ax a1b 1x a1b x a1b a1 The number definition of the multiplicative inverse of a matrix is similar. is called the multiplicative inverse of a because a1a 1. The A matrix and let be the Definition of the Inverse of a Square Matrix Let be an exists a matrix AA1 In A1 n n A1 A1A is called the inverse of The symbol such that n n then A1 A. In identity matrix. If there is read “ A inverse.” Example 1 The Inverse of a Matrix Show that is the inverse of where A, B A 1 1 2 1 and B 1 1 . 2 1 Solution To show that B AB 1 1 BA 1 1 2 1 is the inverse of 1 2 1 1 1 1 A, show that AB I BA as follows. 0 1 0 1 As you can see, an inverse. Note that not all square matrices have an inverse. AB I BA. This is an example of a square matrix that has Now try Exercise 1. Recall that it is not always true that A So, in Example 1, you need only to check that AB I2. even if both products are AB In , it can be defined. However, if BA In . shown that are both square matrices and and B AB BA, 333202_0803.qxd 12/5/05 11:01 AM Page 603 Section 8.3 The Inverse of a Square Matrix 603 Finding Inverse Matrices A A A and is of order BA is called invertible (or nonsingular); otherwise, If a matrix has an inverse, A is called singular. A nonsquare matrix cannot have an inverse. To see this, note ), the products that if AB are of different orders and so cannot be equal to each other. Not all square matrices have inverses (see the matrix at the bottom of page 605). If, however, a matrix does have an inverse, that inverse is unique. Example 2 shows how to use a system of equations to find the inverse of a matrix. is of order m n (where n m m n and B Example 2 Finding the Inverse of a Matrix Find the inverse of A 1 1 4 3 . Solution To find the inverse of A, try to solve the matrix equation AX I for X. X I A 1 1 4x21 3x21 x11 x11 4 3 x11 x12 x22 x21 4x22 3x22 x12 x12 1 0 1 0 0 1 0 1 Equating corresponding entries, you obtain two systems of linear equations. x11 x11 x12 x12 4x21 3x21 4x22 3x22 1 0 0 1 Linear system with two variables, x11 and x21. Linear system with two variables, x12 and x22. Solve the first system using elementary row operations to determine that 4 x11 and x21 Therefore, the inverse of From the second system you can determine that 1. and x12 is A 3 1. x22 X A1 3 1 . 4 1 You can use matrix multiplication to check this result. Check AA1 1 1 A1A ✓ ✓ Now try Exercise 13. 333202_0803.qxd 12/5/05 11:01 AM Page 604 604 Chapter 8 Matrices and Determinants In Example 2, note that the two systems of linear equations have the same coefficient matrix 1 1 4 3 A. . .. . .. and 1 1 4 3 . .. . .. Rather than solve the two systems represented by 1 0 0 1 separately, you can solve them simultaneously by adjoining the identity matrix to the coefficient matrix to obtain I A 1 1 4 3 . .. . .. 1 0 . 0 1 This “doubly augmented” matrix can be represented as By applying Gauss-Jordan elimination to this matrix, you can solve both systems with a single elimination process. A I. Te c h n o l o g y Most graphing utilities can find the inverse of a square matrix. To do so, you may have to use the inverse key . Consult the user’s guide for your graphing utility for specific keystrokes . .. . .. . .. . .. . .. . .. 1 0 1 1 3 1 R1 4R2 R2 R1 0 1 0 1 4 1 A I, So, from the “doubly augmented” matrix I A1. you obtain the matrix A 1 1 4 3 . .. . .. A1 3 1 4 1 . .. . .. This procedure (or algorithm) works for any square matrix that has an inverse. Finding an Inverse Matrix Let be a square matrix of order A n. 1. Write the and the n 2n matrix that consists of the given matrix on the left n n identity matrix on the right to obtain I A A I. 2. If possible, row reduce A I. A entire matrix not possible, is not invertible. I to using elementary row operations on the A The result will be the matrix I A1. If this is 3. Check your work by multiplying to see that AA1 I A1A. 333202_0803.qxd 12/8/05 10:44 AM Page 605 Section 8.3 The Inverse of a Square Matrix 605 Example 3 Finding the Inverse of a Matrix Find the inverse of Solution Begin by adjoining the identity matrix to 1 0 2 .. . . .. . .. . .. A to form the matrix 0 1 0 . 0 0 1 I A1, as follows. Use elementary row operations to obtain the form R1 6R1 R2 R2 R3 R1 4R2 R3 R3 R3 R1 R2 → 1 1 → . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. A1 Be sure to check your solution because it is easy to make algebraic errors when using elementary row operations. So, the matrix A is invertible and its inverse is A1 Confirm this result by multiplying A and A1 to obtain I, as follows. Check AA1 Now try Exercise 21. The process shown in Example 3 applies to any A matrix When using this algorithm, if the matrix does not reduce to the identity matrix, then does not have an inverse. For instance, the following matrix has no inverse. n n A above has no inverse, adjoin the identity matrix to to To confirm that matrix form and perform elementary row operations on the matrix. After doing so, you will see that it is impossible to obtain the identity matrix on the left. Therefore, is not invertible. A I A 333202_0803.qxd 12/5/05 11:01 AM Page 606 606 Chapter 8 Matrices and Determinants Exploration Use a graphing utility with matrix capabilities to find the inverse of the matrix 3 . A 1 2 6 What message appears on the screen? Why does the graphing utility display this message? The Inverse of a 2 2 Matrix Using Gauss-Jordan elimination to find the inverse of a matrix works well (even 2 2 as a computer technique) for matrices of order matrices, however, many people prefer to use a formula for the inverse rather 2 2 than Gauss-Jordan elimination. This simple formula, which works only for A matrices, is explained as follows. If or greater. For matrix given by 3 3 2 2 is a A a c b d is invertible if and only if A then inverse is given by ad bc 0. Moreover, if ad bc 0, the A1 1 ad bc d c ad bc The denominator will study determinants in the next section. b a . Formula for inverse of matrix A is called the determinant of the 2 2 matrix You A. Example 4 Finding the Inverse of a 2 2 Matrix If possible, find the inverse of each matrix. a. Solution a. For the matrix A, apply the formula for the inverse of a 2 2 matrix to obtain ad bc 32 12 4. Because this quantity is not zero, the inverse is formed by interchanging the entries on the main diagonal, changing the signs of the other two entries, and multiplying by the scalar as follows. 1 4, A1 1 2 42 Substitute for a, b, c, d, and the determinant. Multiply by the scalar 1 4. b. For the matrix B, you have ad bc 32 16 0 which means that B is not invertible. Now try Exercise 39. 333202_0803.qxd 12/5/05 11:01 AM Page 607 Section 8.3 The Inverse of a Square Matrix 607 Systems of Linear Equations You know that a system of linear equations can have exactly one solution, of a square infinitely many solutions, or no solution. If the coefficient matrix system (a system that has the same number of equations as variables) is invertible, the system has a unique solution, which is defined as follows. A A System of Equations with a Unique Solution A If AX B has a unique solution given by is an invertible matrix, the system of linear equations represented by Te c h n o l o g y X A1B. To solve a system of equations with a graphing utility, enter the B matrices and editor. Then, using the inverse key, solve for in the matrix X. A A x 1 B ENTER The screen will display the solution, matrix X. Example 5 Solving a System Using an Inverse You are going to invest $10,000 in AAA-rated bonds, AA-rated bonds, and B-rated bonds and want an annual return of $730. The average yields are 6% on AAA bonds, 7.5% on AA bonds, and 9.5% on B bonds. You will invest twice as much in AAA bonds as in B bonds. Your investment can be represented as x 0.06x x y 0.075y z 0.095z 2z 10,000 730 0 x, where respectively. Use an inverse matrix to solve the system. represent the amounts invested in AAA, AA, and B bonds, and y, z Solution Begin by writing the system in the matrix form AX B. 1 0.06 1 1 0.075 0 1 0.095 2 x y z 10,000 730 0 Then, use Gauss-Jordan elimination to find A1. A1 15 21.5 7.5 Finally, multiply by B X A1B 200 300 100 A1 2 3.5 1.5 on the left to obtain the solution. 15 21.5 7.5 200 300 100 2 3.5 1.5 10,000 4000 4000 2000 730 0 y 4000, x 4000, The solution to the system is invest $4000 in AAA bonds, $4000 in AA bonds, and $2000 in B bonds. z 2000. and So, you will Now try Exercise 67. 333202_0803.qxd 12/5/05 11:01 AM Page 608 608 Chapter 8 Matrices and Determinants 8.3 Exercises VOCABULARY CHECK: Fill in the blanks. 1. In a ________ matrix, the number of rows equals the number of columns. 2. If there exists an n n matrix A1 such that AA1 In A1 A, then A1 is called the ________ of A. 3. If a matrix has an inverse, it is called invertible or ________; if it does not have an inverse, A it is called ________. 4. If A is an invertible matrix, the system of linear equations represented by AX B has a unique solution given by X ________. PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. 1. 3. 2 , In Exercises 1–10, show that B is the inverse of A 17 11 3 2 3 5 11 . 5. 6 11 4 7 4 2 9 1 5 3 14 10 , 34 , 12 33 . 8. 9. 10. 4 8 17. 16. 14. 13. 18. 11. 15. 12. 2 7 1 1 0 3 2 3 33 19 1 0 3 4 In Exercises 11–26, find the inverse of the matrix (if it exists). 11 15 20. 23. 25. 26. 24. 22. 19. 21. 3 2 6 0 3 3 3 2 In Exercises 27–38, use the matrix capabilities of a graphing utility to find the inverse of the matrix (if it exists). 27. 29. 31 10 15 2 0 3 1 4 3 2 1 2 28. 30. 5 3 10 3 32 11 6 2 5 2 333202_0803.qxd 12/5/05 11:01 AM Page 609 Section 8.3 The Inverse of a Square Matrix 609 33. 35. 37. 0.3 0.5 0..2 0.2 0.4 0.3 0.2 0. 34. 36. 38. 0.7 1 0..3 0.2 0.9 14 6 7 10 2 3 5 11 1 2 2 4 39. 2 2 In Exercises 39–44, use the formula on page 606 to find the inverse of the matrix (if it exists
). 2 5 7 8 3 2 6 4 12 3 2 5 7 1 12 5 44. 40. 43. 42. 41 In Exercises 45– 48, use the inverse matrix found in Exercise 13 to solve the system of linear equations. 45. 47. x 2y 5 2x 3y 10 x 2y 4 2x 3y 2 46. 48. x 2y 0 2x 3y 3 x 2y 2x 3y 1 2 In Exercises 49 and 50, use the inverse matrix found in Exercise 21 to solve the system of linear equations. 49. x y z 0 3x 5y 4z 5 3x 6y 5z 2 50. x y z 3x 5y 4z 3x 6y 5z 1 2 0 51. 3x1 2x1 x1 In Exercises 51 and 52, use the inverse matrix found in Exercise 38 to solve the system of linear equations. 2x4 3x4 5x4 11x4 2x4 3x4 5x4 11x4 x3 2x3 2x3 4x3 x3 2x3 2x3 4x3 2x2 5x2 5x2 4x2 2x2 5x2 5x2 4x2 x1 x1 1 2 0 3 0 1 1 2 3x1 2x1 x1 52. In Exercises 53– 60, use an inverse matrix to solve (if possible) the system of linear equations. 53. 3x 4y 5x 3y 2 4 54. 18x 12y 13 30x 24y 23 55. 57. 59. 1.6 0.8y 4y 5 8 y 2 4 y 12 0.4x 2x 1 4 x 3 2 x 3 4x y z 2x 2y 3z 5x 2y 6z 3 5 10 1 56. 58. 60. 4 0.6y 1.4y y 20 2 y 51 0.2x x 5 6 x 3 x 4x 2y 3z 2x 2y 5z 8x 5y 2z 7 2.4 8.8 2 16 4 In Exercises 61–66, use the matrix capabilities of a graphing utility to solve (if possible) the system of linear equations. 61. 63. 64. 65. 66. 62. 2x 3y 5z 4 3x 5y 9z 7 5x 9y 17z 13 29 37 24 151 86 187 2 3 4 2x 2y 3z x 7y 8z 4x y 3z x 5y z 5x 3y 2z 3x 2y z 8x 7y 10z 7x 2x 12x 3y 5z 15x 9y 2z 3y y 2w w z 2w w w 2z 2w 5z w 3w 5y 4y 2y 2x 4x x x 2x x y 41 13 12 8 11 7 3 1 Investment Portfolio In Exercises 67–70, consider a person who invests in AAA-rated bonds, A-rated bonds, and B-rated bonds. The average yields are 6.5% on AAA bonds, 7% on A bonds, and 9% on B bonds. The person invests z twice as much in B bonds as in A bonds. Let represent the amounts invested in AAA, A, and B bonds, respectively. and x, y, 0.065x x y 0.07y 2y z 0.09z z (total investment) (annual return) 0 Use the inverse of the coefficient matrix of this system to find the amount invested in each type of bond. Total Investment Annual Return 67. $10,000 68. $10,000 69. $12,000 70. $500,000 $705 $760 $835 $38,000 333202_0803.qxd 12/5/05 11:01 AM Page 610 610 Chapter 8 Matrices and Determinants 71. Circuit Analysis Consider the circuit shown in the figure. in amperes, are the solution of I2, I3, and I1, The currents the system of linear equations 4I3 4I3 I3 2I1 I2 I2 E1 E2 0 I1 E2 E1 and where are voltages. Use the inverse of the coefficient matrix of this system to find the unknown currents for the voltages. 2Ω d + _ I1 E1 I3 I2 4Ω 1Ω b + _ E2 a c (a) (b) E1 E1 14 volts, E2 24 volts, E2 28 volts 23 volts Model It 72. Data Analysis: Licensed Drivers The table shows the numbers (in millions) of licensed drivers in the United States for selected years 1997 to 2001. (Source: U.S. Federal Highway Administration) y Year Drivers, y 1997 1999 2001 182.7 187.2 191.3 (a) Use the technique demonstrated in Exercises 57–62 in Section 7.2 to create a system of linear equations for the data. Let represent the year, with t 7 corresponding to 1997. t (b) Use the matrix capabilities of a graphing utility to find an inverse matrix to solve the system from part (a) and find the least squares regression line y at b. (c) Use the result of part (b) to estimate the number of licensed drivers in 2003. (d) The actual number of licensed drivers in 2003 was 196.2 million. How does this value compare with your estimate from part (c)? Model It (co n t i n u e d ) (e) Use the result of part (b) to estimate when the number of licensed drivers will reach 208 million. Synthesis True or False? the statement is true or false. Justify your answer. In Exercises 73 and 74, determine whether 73. Multiplication of an invertible matrix and its inverse is commutative. 74. If you multiply two square matrices and obtain the identity matrix, you can assume that the matrices are inverses of one another. 75. If A is a 2 2 matrix if and only if inverse is ad bc 0. , A a b d c ad bc 0, then If A is invertible verify that the A1 1 ad bc d c . b a 76. Exploration Consider matrices of the form A a11 0 0 0 0 a22 0 0 0 0 a33 ann (a) Write a A. matrix and a Find the inverse of each. 2 2 3 3 matrix in the form of (b) Use the result of part (a) to make a conjecture about the inverses of matrices in the form of A. Skills Review In Exercises 77 and 78, solve the inequality and sketch the solution on the real number line. 77. x 7 ≥ 2 78. 2x 1 < 3 In Exercises 79– 82, solve the equation. Approximate the result to three decimal places. 79. 81. 3x2 315 log2 x 2 4.5 80. 82. 2000ex5 400 ln x lnx 1 0 83. Make a Decision To work an extended application analyzing the number of U.S. households with color televisions from 1985 to 2005, visit this text’s website at college.hmco.com. Source: Nielsen Media Research) (Data 333202_0804.qxd 12/5/05 11:03 AM Page 611 Section 8.4 The Determinant of a Square Matrix 611 8.4 The Determinant of a Square Matrix What you should learn • Find the determinants of 2 2 matrices. • Find minors and cofactors of square matrices. • Find the determinants of square matrices. Why you should learn it Determinants are often used in other branches of mathematics. For instance, Exercises 79–84 on page 618 show some types of determinants that are useful when changes in variables are made in calculus. The Determinant of a 2 2 Matrix Every square matrix can be associated with a real number called its determinant. Determinants have many uses, and several will be discussed in this and the next section. Historically, the use of determinants arose from special number patterns that occur when systems of linear equations are solved. For instance, the system a1x b1y c1 a2x b2y c2 has a solution x c1b2 a1b2 c2b1 a2b1 a1b2 and y a1c2 a1b2 a 2c1 a 2b1 provided that Note that the denominators of the two fractions are the same. This denominator is called the determinant of the coefficient matrix of the system. a2b1 0. Coefficient Matrix A a1 a 2 b1 b2 Determinant detA a1b2 a 2b1 The determinant of the matrix can also be denoted by vertical bars on both sides of the matrix, as indicated in the following definition. A Definition of the Determinant of a 2 The determinant of the matrix 2 Matrix A a1 a2 b1 b2 is given by detA A a1 a 2 b1 b2 a1b2 a 2b1. detA A In this text, A. are used interchangeably to represent the determinant of Although vertical bars are also used to denote the absolute value of a real number, the context will show which use is intended. and A convenient method for remembering the formula for the determinant of a 2 2 matrix is shown in the following diagram. detA a1 a 2 b1 b2 a1b2 a 2b1 Note that the determinant is the difference of the products of the two diagonals of the matrix. 333202_0804.qxd 12/5/05 11:03 AM Page 612 612 Chapter 8 Matrices and Determinants Example 1 The Determinant of a 2 2 Matrix Find the determinant of each matrix. a. b. c detA 2 2 1 Solution 3 2 Exploration Use a graphing utility with matrix capabilities to find the determinant of the following matrix. A 1 1 3 2 0 2 What message appears on the screen? Why does the graphing utility display this message? a. b. c. 22 13 4 3 7 detB 2 4 detC 0 2 1 2 3 2 4 22 41 4 4 0 04 23 0 3 3 2 Now try Exercise 5. Notice in Example 1 that the determinant of a matrix can be positive, zero, or negative. The determinant of a matrix of order 1 1 is defined simply as the entry of the matrix. For instance, if A 2, then detA 2. Te c h n o l o g y Most graphing utilities can evaluate the determinant of a matrix. For instance, you can evaluate the determinant of A 2 1 3 2 by entering the matrix as The result should be 7, as in Example 1(a). Try evaluating the determinants of other matrices. Consult the user’s guide for your graphing utility for specific keystrokes. and then choosing the determinant feature. A 333202_0804.qxd 12/5/05 11:03 AM Page 613 Section 8.4 The Determinant of a Square Matrix 613 Minors and Cofactors To define the determinant of a square matrix of order convenient to introduce the concepts of minors and cofactors. 3 3 or higher, it is Sign Pattern for Cofactors .. . .. . 3 3 matrix 4 4 .. . n n matrix .. . matrix . . . . . . . . . . . . . . . .. . A is a square matrix, the minor Minors and Cofactors of a Square Matrix Mi j ai j of the entry If is the determinant i j of the matrix obtained by deleting the th row and th column of The cofactor of the entry A. is Ci j ai j Ci j 1ijMi j. In the sign pattern for cofactors at the left, notice that odd positions (where is even) have is odd) have negative signs and even positions (where i j i j positive signs. Example 2 Finding the Minors and Cofactors of a Matrix Find all the minors and cofactors of Solution To find the minor determinant of the resulting matrix. M11, delete the first row and first column of A and evaluate the M11 2 1 11 02 1 Similarly, to find M12, delete the first row and second column M12 2 1 31 42 5 Continuing this pattern, you obtain the minors. M11 M21 M31 1 2 5 M12 M22 M32 5 4 3 M13 M23 M33 4 8 6 3 3 matrix shown at the upper left. Now, to find the cofactors, combine these minors with the checkerboard pattern of signs for a 1 2 5 4 8 6 5 4 C13 C11 C22 C12 C23 C21 3 C33 C32 C31 Now try Exercise 27. 333202_0804.qxd 12/5/05 11:03 AM Page 614 614 Chapter 8 Matrices and Determinants The Determinant of a Square Matrix The definition below is called inductive because it uses determinants of matrices of order to define determinants of matrices of order n 1 n. A is a square matrix (of order Determinant of a Square Matrix 2 2 If or greater), the determinant of the sum of the entries in any row (or column) of multiplied by their respective cofactors. For instance, expanding along the first row yields A A is A a11C11 a12C12 . . . a1nC1n. Applying this definition to find a determinant is called expanding by cofactors. 2 2 matrix Try checking that for a A a1 a2 b1 b2 this definition of the determinant yields defined. A a1b2 a2b1, as previously Example 3 The Determinant of a Matrix of Order 3 3 Find the determinant of Solution Note that this is the same matrix that was in Example 2.
There you found the cofactors of the entries in the first row to be 4. 1, 5, and C13 C11 C12 So, by the definition of a determinant, you have a12C12 A a11C11 a13C13 First-row expansion 01 25 14 14. Now try Exercise 37. In Example 3, the determinant was found by expanding by the cofactors in the first row. You could have used any row or column. For instance, you could have expanded along the second row to obtain a 23C23 a 22C22 32 14 28 14. A a 21C21 Second-row expansion 333202_0804.qxd 12/5/05 11:03 AM Page 615 Section 8.4 The Determinant of a Square Matrix 615 When expanding by cofactors, you do not need to find cofactors of zero entries, because zero times its cofactor is zero. aijCij 0Cij 0 So, the row (or column) containing the most zeros is usually the best choice for expansion by cofactors. This is demonstrated in the next example. Example 4 The Determinant of a Matrix of Order 4 4 Find the determinant of Solution After inspecting this matrix, you can see that three of the entries in the third column are zeros. So, you can eliminate some of the work in the expansion by using the third column. A 3C13 C23, C33, To do this, delete the first row and third column of 0C33 have zero coefficients, you need only find the cofacand evaluate the 0C23 C43 and Because C13. tor determinant of the resulting matrix. 0C43 A 0 3 1131 1 0131 1 2 4 0 3 4 C13 C13 2141 2 3 Expanding by cofactors in the second row yields Delete 1st row and 3rd column. Simplify. 2 2 3151 3 1 4 0 218 317 5. So, you obtain A 3C13 35 15. Now try Exercise 47. Try using a graphing utility to confirm the result of Example 4. 333202_0804.qxd 12/5/05 11:03 AM Page 616 616 Chapter 8 Matrices and Determinants 8.4 Exercises VOCABULARY CHECK: Fill in the blanks. A 1. Both Mij 2. The ________ detA and aij column of the square matrix of the entry A. represent the ________ of the matrix A. is the determinant of the matrix obtained by deleting the th row and th j i 3. The ________ Cij of the entry aij of the square matrix A is given by 1ij Mij. 4. The method of finding the determinant of a matrix of order 2 2 or greater is called ________ by ________. PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. In Exercises 1–16, find the determinant of the matrix. 1. 3. 5. 7. 9. 11. 13. 15. 4. 6. 8. 10. 12. 14. 16 In Exercises 17–22, use the matrix capabilities of a graphing utility to find the determinant of the matrix. 17. 19. 21. 0.2 0.4 0.3 0.9 1 0.1 2.2 3 2 0.2 0.2 0.4 0.7 0.3 4.2 4 6 1 0.2 0.2 0.3 0 1.3 6.1 2 6 4 18. 20. 22. 0.3 0.5 0.1 0.1 2 7.5 0.3 0 0 0.2 0.2 0.4 0.3 0.2 0.4 4.3 0.7 1.2 1 2 2 0.1 6.2 0.6 3 5 0 In Exercises 23–30, find all (a) minors and (b) cofactors of the matrix. 3 2 3 2 11 3 6 7 5 2 1 4 4 5 23. 25. 26. 24. 0 2 27. 29 28. 30 In Exercises 31–36, find the determinant of the matrix by the method of expansion by cofactors. Expand using the indicated row or column. 31. 33. 3 2 5 3 (a) Row 1 4 2 (b) Column 2 3 4 3 0 12 6 5 0 1 1 6 1 32. 34. 3 6 4 4 3 7 (a) Row 2 2 1 8 (b) Column 3 5 0 10 10 30 0 5 10 1 (a) Row 2 (b) Column 2 (a) Row 3 (b) Column 1 35. 6 4 1 8 0 13 36. 10 a) Row 2 (b) Column 2 (a) Row 3 (b) Column 1 In Exercises 37–52, find the determinant of the matrix. Expand by cofactors on the row or column that appears to make the computations easiest. 37 38 333202_0804.qxd 12/5/05 11:03 AM Page 617 Section 8.4 The Determinant of a Square Matrix 617 39. 41. 43. 45. 47. 49. 51. 52. 53. 55. 57. 59. 60 12 11 2 6 0 1 3 4 6 0 2 40. 42. 44. 46. 48. 50. 5 2 0 0 0 In Exercises 53– 60, use the matrix capabilities of a graphing utility to evaluate the determinant 12 14 14 4 12 54. 56. 58 In Exercises 61– 68, (d) find (a) A, (b) B, (c) AB, and AB 61. 62. 63. 64. 65. 66. 67. 68 In Exercises 69 –74, evaluate the determinant(s) to verify the equation. cx z x y y x x w x cw x z y z cz cw z cy z w cx 0 z 2 y xz xz y a b b23a cw 71. 69. 73. 70. 72 78. x 4 3 74. 75. 76. 77. a a 1 3 7 In Exercises 75–78, solve for x 333202_0804.qxd 12/8/05 10:39 AM Page 618 618 Chapter 8 Matrices and Determinants In Exercises 79–84, evaluate the determinant in which the entries are functions. Determinants of this type occur when changes in variables are made in calculus. 92. If A of is obtained from A B by adding a multiple of a row of to another row of or by adding a multiple of a column A to another column of B A. then A, A 1 1 2v 4u 3e3x e2x 1x x 2e2x ln x e3x 1 79. 81. 83. 1 3x 2 ex x 1 ex 3y 2 1 1 xex 1 ln x x ln x xex 80. 82. 84. Synthesis True or False? the statement is true or false. Justify your answer. In Exercises 85 and 86, determine whether 85. If a square matrix has an entire row of zeros, the determi- nant will always be zero. 86. If two columns of a square matrix are the same, the determinant of the matrix will be zero. 87. Exploration Find square matrices and A B to demonstrate that A B A B. 88. Exploration Consider square matrices in which the entries are consecutive integers. An example of such a matrix is 4 7 10 5 8 11 . 6 9 12 (a) Use a graphing utility to evaluate the determinants of four matrices of this type. Make a conjecture based on the results. (b) Verify your conjecture. 89. Writing Write a brief paragraph explaining the differ- ence between a square matrix and its determinant. 90. Think About It A 5, If is it possible to find is a matrix of order 2A? Explain. A 3 3 such that Properties of Determinants In Exercises 91–93, a property of determinants is given ( are square matrices). and State how the property has been applied to the given determinants and use a graphing utility to verify the results. A B 91. If B is obtained from by interchanging two rows of or A, interchanging two columns of B A. then A A 7 6 1 1 2 1 (a) (b is obtained from by multiplying a row by a nonzero or by multiplying a column by a nonzero then B cA. constant constant c c, A (a) (b) 93. If a) (b) (a) (c) 3 0 3 17 10 3 6 3 51 10 2 2 3 3 121 3 7 3 6 8 12 b 94. Exploration A diagonal matrix is a square matrix with all zero entries above and below its main diagonal. Evaluate the determinant of each diagonal matrix. Make a conjecture based on your results. Skills Review In Exercises 95–100, find the domain of the function. 95. f x x3 2x 96. gx 3x 97. hx 16 x2 99. gt lnt 1 98. 100. Ax 3 36 x2 f s 625e0.5s In Exercises 101 and 102, sketch the graph of the solution of the system of inequalities. 101. x x 2x y ≤ ≥ y < 8 3 5 102. x y > y ≤ 7x 4y ≤ 4 1 10 In Exercises 103–106, find the inverse of the matrix (if it exists). 103. 105 104. 5 3 106 333202_0805.qxd 12/5/05 11:05 AM Page 619 Section 8.5 Applications of Matrices and Determinants 619 8.5 Applications of Matrices and Determinants What you should learn • Use Cramer’s Rule to solve systems of linear equations. • Use determinants to find the areas of triangles. • Use a determinant to test for collinear points and find an equation of a line passing through two points. • Use matrices to encode and decode messages. Why you should learn it You can use Cramer’s Rule to solve real-life problems. For instance, in Exercise 58 on page 630, Cramer’s Rule is used to find a quadratic model for the number of U.S. Supreme Court cases waiting to be tried. Cramer’s Rule So far, you have studied three methods for solving a system of linear equations: substitution, elimination with equations, and elimination with matrices. In this section, you will study one more method, Cramer’s Rule, named after Gabriel Cramer (1704–1752). This rule uses determinants to write the solution of a system of linear equations. To see how Cramer’s Rule works, take another look at the solution described at the beginning of Section 8.4. There, it was pointed out that the system a1x b1y c1 a2x b2y c2 has a solution x c1b2 a1b2 c2b1 a2b1 a1b2 and y a1c2 a1b2 a2c1 a2b1 a2b1 provided that can be expressed as a determinant, as follows. 0. Each numerator and denominator in this solution x c1b2 a1b2 c2b1 a2b1 y a1c2 a1b2 a2c1 a2b1 c2 c1 a1 a2 b1 b2 b2 b1 a2 a1 a1 a2 c1 c2 b2 b1 Relative to the original system, the denominator for and is simply the deterD. minant of the coefficient matrix of the system. This determinant is denoted by respectively. They are The numerators for formed by using the column of constants as replacements for the coefficients of x are denoted by as follows. and and and Dy, Dx y, x y y x Coefficient Matrix a1 b1 b2 a2 D a1 a2 b1 b2 Dx c1 c2 b1 b2 Dy a1 a2 c1 c2 © Lester Lefkowitz /Corbis For example, given the system 2x 5y 3 4x 3y 8 the coefficient matrix, D, Dx, and Dy are as follows. Coefficient Matrix Dx 5 3 Dy 2 4 3 8 333202_0805.qxd 12/5/05 11:05 AM Page 620 620 Chapter 8 Matrices and Determinants Cramer’s Rule generalizes easily to systems of equations in variables. The value of each variable is given as the quotient of two determinants. The denominator is the determinant of the coefficient matrix, and the numerator is the determinant of the matrix formed by replacing the column corresponding to the variable (being solved for) with the column representing the constants. For instance, the solution for in the following system is shown. n n x3 a11x1 a21x1 a31x1 a12x2 a22x2 a32x2 a13x3 a23x3 a33x3 b1 b2 b3 x3 A3 A a21 a31 a11 a11 a21 a31 a12 a22 a32 a12 a22 a32 b1 b2 b3 a33 a13 a23 n A, linear equations in variables has a coefficient matrix Cramer’s Rule n If a system of with a nonzero determinant A1 A2 A , A , i Ai where the th column of equations. If the determinant of the coefficient matrix is zero, the system has either no solution or infinitely many solutions. the solution of the system is An A is the column of constants in the system of . . . , x2 x1 xn A Example 1 Using Cramer’s Rule for a 2 2 System Use Cramer’s Rule to solve the system of linear equations. 4x 2y 10 3x 5y 11 Because this determinant is not zero, you can apply Cramer’s Rule. Solution To begin, find the determinant of the coefficient matrix. D 4 3 x Dx D y Dy D 2 5 20 6 14 10 4 50 22 14 5 11 14 2 11 10 3 44 30 14 y 1. and 14 x
2 Now try Exercise 1. 28 14 2 14 14 1 So, the solution is Check this in the original system. 333202_0805.qxd 12/5/05 11:05 AM Page 621 Section 8.5 Applications of Matrices and Determinants 621 Example 2 Using Cramer’s Rule for a 3 3 System Use Cramer’s Rule to solve the system of linear equations. x 2x 3x 2y 4y 3z 1 z 0 4z 2 Solution To find the determinant of the coefficient matrix 1 2 3 2 0 4 3 1 4 expand along the second row, as follows. D 213 2 3 4 0141 3 3 4 1151 3 4 2 4 24 0 12 10 Because this determinant is not zero, you can apply Cramer’s Rule 10 1 0 2 10 2 0 4 10 2, 8 5 x Dx D y Dy D z Dz D 8 10 4 5 15 10 3 2 16 10 8 5 The solution is 4 5, 3 . Check this in the original system as follows. Check 4 5 4 5 24 5 8 5 34 5 12 5 23 2 3 8 5 8 5 43 2 6 38 5 24 5 48 5 32 Now try Exercise 7. Substitute into Equation 1. Equation 1 checks. ✓ Substitute into Equation 2. Equation 2 checks. ✓ Substitute into Equation 3. Equation 3 checks. ✓ Remember that Cramer’s Rule does not apply when the determinant of the coefficient matrix is zero. This would create division by zero, which is undefined. 333202_0805.qxd 12/5/05 11:05 AM Page 622 622 Chapter 8 Matrices and Determinants Area of a Triangle Another application of matrices and determinants is finding the area of a triangle whose vertices are given as points in a coordinate plane. Area of a Triangle The area of a triangle with vertices x1, y1 , x2, y2 , and x3, y3 is x1 1 2 y1 y2 y3 1 1 1 Area ± x2 x3 ± where the symbol yield a positive area. indicates that the appropriate sign should be chosen to Example 3 Finding the Area of a Triangle (4, 3) Find the area of a triangle whose vertices are in Figure 8.1. 1, 0, 2, 2, and 4, 3, as shown Solution x1, y1 Let of the triangle, evaluate the determinant. 1, 0, x2, y2 2, 2, and x3, y3 4, 3. Then, to find the area y 3 2 1 (1, 0) FIGURE 8.1 (2, 2) 1 2 3 4 x x1 x2 x3 y1 y2 y3 1 1 1142 4 2 3 1 1 1 1 0132 1 4 11 0 12 3 1122 Using this value, you can conclude that the area of the triangle is Area 1 2 Choose so that the area is positive. 1 2 3 3 2 square units. Now try Exercise 19. Exploration Use determinants to find the area of a triangle with vertices and and using the formula 7, 1, Confirm your answer by plotting the points in a coordinate plane 3, 1, 7, 5. Area 1 2 baseheight. 333202_0805.qxd 12/5/05 11:05 AM Page 623 y 3 2 1 (0, 1) FIGURE 8.2 (4, 3) (2, 21 (−2, −2) FIGURE 8.3 (7, 5) (1, 1) 1 2 3 4 5 6 7 x Section 8.5 Applications of Matrices and Determinants 623 Lines in a Plane What if the three points in Example 3 had been on the same line? What would have happened had the area formula been applied to three such points? The answer is that the determinant would have been zero. Consider, for instance, the 0, 1, 2, 2, three collinear points as shown in Figure 8.2. The area of the “triangle” that has these three points as vertices is 4, 3, and 20 2 4 1 1 2 3 1 1 20122 1 1 3 1 1 1132 4 1 1 1142 4 2 3 0 12 12 1 2 0. The result is generalized as follows. and x3, y3 are collinear (lie on the same line) Test for Collinear Points x1, y1 Three points if and only if , x2, y2 , x1 x2 x3 y1 y2 y3 1 0. 1 1 Example 4 Testing for Collinear Points Determine whether the points Figure 8.3.) 2, 2, 1, 1, and 7, 5 are collinear. (See Solution Letting x1 x2 x3 x1, y1 y1 y2 y3 x3, y3 2, 2, x2, y2 1 1 1 2 2121 2 1 5 1 7 1 1 and 1, 1, 1 1 2131 1 7 5 24 26 12 6. 7, 5, you have 1 1 1141 7 1 5 Because the value of this determinant is not zero, you can conclude that the three points do not lie on the same line. Moreover, the area of the triangle with vertices square units. at these points is 6 3 1 2 Now try Exercise 31. 333202_0805.qxd 12/5/05 11:05 AM Page 624 624 Chapter 8 Matrices and Determinants The test for collinear points can be adapted to another use. That is, if you are given two points on a rectangular coordinate system, you can find an equation of the line passing through the two points, as follows. y 5 4 (−1, 3) 2 1 (2, 4) −1 1 2 3 4 x FIGURE 8.4 Solution x1, y1 Let for the equation of a line produces 2, 4 x2, y2 and y 4 3 x 2 1 x124 3 1 1 1 0. 1 y13 2 1 1 Two-Point Form of the Equation of a Line An equation of the line passing through the distinct points x2, y2 is given by x x1 x2 1 0. 1 1 y y1 y2 x1, y1 and Example 5 Finding an Equation of a Line Find an equation of the line passing through the two points shown in Figure 8.4. 2, 4 and 1, 3, as 1, 3. Applying the determinant formula To evaluate this determinant, you can expand by cofactors along the first row to obtain the following. 1 1 114 2 1 4 3 0 x11 y13 1110 0 x 3y 10 0 So, an equation of the line is x 3y 10 0. Now try Exercise 39. Note that this method of finding the equation of a line works for all lines, including horizontal and vertical lines. For instance, the equation of the vertical line through 2, 2 2, 0 is x 2 2 1 1 and 1 0 y 0 2 4 2x 0 x 2. 333202_0805.qxd 12/5/05 11:05 AM Page 625 Section 8.5 Applications of Matrices and Determinants 625 Cryptography A cryptogram is a message written according to a secret code. (The Greek word kryptos means “hidden.”) Matrix multiplication can be used to encode and decode messages. To begin, you need to assign a number to each letter in the alphabet (with 0 assigned to a blank space), as follows 19 I 10 J 11 K 12 L 13 M 14 N 15 O 16 P 17 Q 18 R 19 S 20 T 21 U 22 V 23 W 24 X 25 Y 26 Z Then the message is converted to numbers and partitioned into uncoded row matrices, each having entries, as demonstrated in Example 6. n Example 6 Forming Uncoded Row Matrices Write the uncoded row matrices of order 1 3 for the message MEET ME MONDAY. Solution Partitioning the message (including blank spaces, but ignoring punctuation) into groups of three produces the following uncoded row matrices. 0 13 5 0 13 15 14 4 1 25 0 5 5 20 13 Note that a blank space is used to fill out the last uncoded row matrix. Now try Exercise 45. To encode a message, use the techniques demonstrated in Section 8.3 to choose an n n A 1 1 1 invertible matrix such as 2 1 1 2 3 4 and multiply the uncoded row matrices by matrices. Here is an example. A (on the right) to obtain coded row Uncoded Matrix Encoding Matrix A Coded Matrix 13 13 26 21 333202_0805.qxd 12/5/05 11:05 AM Page 626 626 Chapter 8 Matrices and Determinants Example 7 Encoding a Message Use the following invertible matrix to encode the message MEET ME MONDAY Solution The coded row matrices are obtained by multiplying each of the uncoded row matrices found in Example 6 by the matrix as follows. A, Uncoded Matrix Encoding Matrix A Coded Matrix 13 20 5 5 0 0 15 14 1 25 1 1 1 1 5 1 13 1 13 21 13 26 33 53 12 18 23 42 5 20 24 56 23 77 So, the sequence of coded row matrices is 13 26 21 33 53 12 18 23 42 5 20 56 24 23 77. Finally, removing the matrix notation produces the following cryptogram. 13 26 21 33 53 12 18 23 42 5 20 56 24 23 77 Now try Exercise 47. For those who do not know the encoding matrix decoding the cryptogram found in Example 7 is difficult. But for an authorized receiver who knows the decoding is simple. The receiver just needs to multiply the encoding matrix coded row matrices by (on the right) to retrieve the uncoded row matrices. Here is an example. A1 A, A, 13 26 Coded 211 1 0 10 6 1 A1 13 8 5 1 5 5 Uncoded 333202_0805.qxd 12/5/05 11:05 AM Page 627 s i b r o C © Historical Note During World War II, Navajo soldiers created a code using their native language to send messages between battalions. Native words were assigned to represent characters in the English alphabet, and they created a number of expressions for important military terms, like iron-fish to mean submarine. Without the Navajo Code Talkers, the Second World War might have had a very different outcome. Section 8.5 Applications of Matrices and Determinants 627 Example 8 Decoding a Message Use the inverse of the matrix to decode the cryptogram 13 26 21 33 53 12 18 23 42 5 20 56 24 23 77. A1 by using the techniques demonstrated in Section 8.3. Solution First find is the decoding matrix. Then partition the message into groups of three to form the coded row matrices. Finally, multiply each coded row matrix by (on the right). A1 A1 A1 Decoded Matrix 1 0 1 0 13 26 Coded Matrix Decoding Matrix 21 1 33 53 12 1 18 23 42 1 56 1 77 1 10 6 1 10 6 1 10 6 1 10 6 1 10 6 1 5 20 1 0 1 0 1 0 23 13 20 5 15 1 24 5 0 5 13 0 13 14 4 25 0 So, the message is as follows. 13 5 5 20 0 13 5 0 13 15 14 4 1 25 Now try Exercise 53. W RITING ABOUT MATHEMATICS Cryptography Use your school’s library, the Internet, or some other reference source to research information about another type of cryptography. Write a short paragraph describing how mathematics is used to code and decode messages. 333202_0805.qxd 12/5/05 11:05 AM Page 628 628 Chapter 8 Matrices and Determinants 8.5 Exercises VOCABULARY CHECK: Fill in the blanks. 1. The method of using determinants to solve a system of linear equations is called ________ ________. 2. Three points are ________ if the points lie on the same line. 3. The area of a triangle with vertices A x1, y1 , , x2, y2 and x3, y3 is given by ________. 4. A message written according to a secret code is called a ________. 5. To encode a message, choose an invertible matrix A (on the right) to obtain ________ row matrices. by A and multiply the ________ row matrices PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. In Exercises 1–10, use Cramer’s Rule to solve (if possible) the system of equations. 17. y 18. y 4 (0, 4) 6 (1, 6) 1. 3. 5. 7. 9. 2 4 2 4 3x 4y 5x 3y 3x 2y 6x 4y 0.4x 0.8y 1.6 0.2x 0.3y 2.2 4x y z 5 10 1 3 6 11 2x 2y 3z 5x 2y 6z x 2y 3z 2x y z 3x 3y 2z 2. 4. 6. 8. 10. 47 27 17 76 4x 7y x 6y 6x 5y 13x 3y 2.4x 1.3y 4.6x 0.5y 4x 2y 3z 5x 4y z 2x 2y 5z 8x 5y 2z x 2y 2z 3x y z 14.63 11.51 2 16 4 14 10 1 In Exercises 11–14, use a graphing utility and Cramer’s Rule to solve (if possible) the system of equations. 11. 13. 3x 5y 9z 2 5x 9y 17z 4 3x 3y 5z 1 2x y 2z 6 x 2y 3z 0 3x 2y z 6 12. 14. 7 8 8
2x 2y 2z x 3y 4z x 2y z 2x 3y 5z 4 3x 5y 9z 7 5x 9y 17z 13 In Exercises 15–24, use a determinant and the given vertices of a triangle to find the area of the triangle. 15. y 16. y 5 4 3 2 1 (1, 5) (0, 0) (3, 14, 5) (0, 0) x −1 −2 4 1 (5, −2) −4 −2 2 4 x (−2, −3) (2, −3) (−2, 1) (3, −1) 2 −2 x 4 19. y 20. y (6, 10) 8 4 (−4, −5) −8 x (6, −1) 4 3 2 1 (4, 3) ( 1 20 21. 22. 23. 24. 2, 4, 2, 3, 1, 5 0, 2, 1, 4, 3, 5 3, 5, 2, 6, 3, 5 2, 4, 1, 5, 3, 2 In Exercises 25 and 26, find a value of such that the triangle with the given vertices has an area of 4 square units. y 25. 26. 5, 1, 0, 2, 2, y 4, 2, 3, 5, 1, y In Exercises 27 and 28, find a value of such that the triangle with the given vertices has an area of 6 square units. y 27. 28. x 6 2, 3, 1, 0, 5, 3 , 3, y 1, 1, 8, y 333202_0805.qxd 12/5/05 11:05 AM Page 629 Section 8.5 Applications of Matrices and Determinants 629 29. Area of a Region A large region of forest has been infested with gypsy moths. The region is roughly triangular, as shown in the figure. From the northernmost vertex of the region, the distances to the other vertices are 25 miles south and 10 miles east (for vertex ), and 20 miles south and 28 miles east (for vertex ). Use a graphing utility to approximate the number of square miles in this region. C B A W N S E A 20 25 B 10 28 In Exercises 39– 44, use a determinant to find an equation of the line passing through the points. 39. 41. 43. 0, 0, 5, 3 4, 3, 2, 1 1 2, 1 2, 3, 5 40. 42. 44. 0, 0, 2, 2 10, 7, 2, 7 2 3, 4, 6, 12 In Exercises 45 and 46, find the uncoded 1 3 row matrices for the message. Then encode the message using the encoding matrix. Message 45. TROUBLE IN RIVER CITY Encoding Matrix In Exercises 47–50, write a cryptogram for the message using the matrix A. C 46. PLEASE SEND MONEY 30. Area of a Region You own a triangular tract of land, as shown in the figure. To estimate the number of square feet in the tract, you start at one vertex, walk 65 feet east and 50 feet north to the second vertex, and then walk 85 feet west and 30 feet north to the third vertex. Use a graphing utility to determine how many square feet there are in the tract of land]. 47. CALL AT NOON 48. ICEBERG DEAD AHEAD 49. HAPPY BIRTHDAY 50. OPERATION OVERLOAD 85 30 W N S E 65 50 52. In Exercises 51–54, use A1 to decode the cryptogram. 51. A 1 3 2 5 11 21 64 112 25 50 29 53 23 46 40 75 55 92 A 5 7 136 58 178 73 90 36 115 49 199 82 120 51 242 101 173 72 70 28 95 38 115 47 2 3 31. In Exercises 31–36, use a determinant to determine whether the points are collinear. 3, 1, 0, 3, 12, 5 2, 1 , 4, 4, 6, 3 0, 2, 1, 2.4, 1, 1.6 3, 5, 6, 1, 10, 2 0, 1, 4, 2, 2, 5 2 2, 3, 3, 3.5, 1, 2 34. 36. 32. 35. 33. 2 In Exercises 37 and 38, find collinear. y such that the points are 37. 2, 5, 4, y, 5, 2 38. 6, 2, 5, y, 3, 5 1 0 2 0 1 3 19 9 53. A 1 1 6 9 9 41 1 64 A 3 54. 0 4 38 21 31 4 2 5 2 1 3 19 28 9 19 80 25 5 4 112 140 83 19 25 13 72 118 71 20 21 38 35 23 36 42 48 32 76 61 95 333202_0805.qxd 12/5/05 11:05 AM Page 630 630 Chapter 8 Matrices and Determinants In Exercises 55 and 56, decode the cryptogram by using the inverse of the matrix A. Synthesis ] 55. 20 17 12 15 62 143 181 9 59 24 29 65 144 172 56. 13 56 104 1 25 65 61 112 106 17 73 131 11 57. The following cryptogram was encoded with a 2 2 matrix. 13 10 13 15 6 20 40 8 21 1 The last word of the message is _RON. What is the message? 5 10 5 25 5 19 1 16 18 18 Model It 58. Data Analysis: Supreme Court The table shows the numbers of U.S. Supreme Court cases waiting to be tried for the years 2000 through 2002. (Source: Office of the Clerk, Supreme Court of the United States) y Year Number of cases, y 2000 2001 2002 8965 9176 9406 (a) Use the technique demonstrated in Exercises 67–70 in Section 7.3 to create a system of linear equations t 0 for the data. Let corresponding to 2000. represent the year, with t (b) Use Cramer’s Rule to solve the system from part (a) and find the least squares regression parabola y at2 bt c. (c) Use a graphing utility to graph the parabola from part (b). (d) Use the graph from part (c) to estimate when the number of U.S. Supreme Court cases waiting to be tried will reach 10,000. True or False? statement is true or false. Justify your answer. In Exercises 59– 61, determine whether the 59. In Cramer’s Rule, the numerator is the determinant of the coefficient matrix. 60. You cannot use Cramer’s Rule when solving a system of linear equations if the determinant of the coefficient matrix is zero. 61. In a system of linear equations, if the determinant of the coefficient matrix is zero, the system has no solution. 62. Writing At this point in the text, you have learned several methods for solving systems of linear equations. Briefly describe which method(s) you find easiest to use and which method(s) you find most difficult to use. Skills Review In Exercises 63–66, use any method to solve the system of equations. 63. 64. 65. 66. 11 16 x 7y 22 5x y 26 3x 8y 2x 12y x 3y 5z 5x y z 4x 2y z 5x 3y 2z 2x 3y z 4x 10y 5z 14 1 11 7 5 37 In Exercises 67 and 68, sketch the region determined by the constraints. Then find the minimum and maximum values of the objective function and where they occur, subject to the constraints. 67. Objective function: z 6x 4y Constraints: 68. Objective function: z 6x 7y Constraints: x ≥ 0 y ≥ 0 x 6y ≤ 30 6x y ≤ 40 x ≥ 0 y ≥ 0 4x 3y ≥ 24 x 3y ≥ 15 333202_080R.qxd 12/5/05 11:08 AM Page 631 8 Chapter Summary What did you learn? Section 8.1 Write matrices and identify their orders (p. 572). Perform elementary row operations on matrices (p. 574). Use matrices and Gaussian elimination to solve systems of linear equations (p. 577). Use matrices and Gauss-Jordan elimination to solve systems of linear equations (p. 579). Section 8.2 Decide whether two matrices are equal (p. 587). Add and subtract matrices and multiply matrices by scalars (p. 588). Multiply two matrices (p. 592). Use matrix operations to model and solve real-life problems (p. 595). Section 8.3 Verify that two matrices are inverses of each other (p. 602). Use Gauss-Jordan elimination to find the inverses of matrices (p. 603). Use a formula to find the inverses of Use inverse matrices to solve systems of linear equations (p. 607. matrices (p. 606). 2 2 Section 8.4 Find the determinants of Find minors and cofactors of square matrices (p. 613). Find the determinants of square matrices (p. 614). matrices (p. 611). 2 2 Section 8.5 Use Cramer’s Rule to solve systems of linear equations (p. 619). Use determinants to find the areas of triangles (p. 622). Use a determinant to test for collinear points and to find an equation of a line passing through two points (p. 623). Use matrices to encode and decode messages (p. 625). Chapter Summary 631 Review Exercises 1–8 9, 10 11–24 25–30 31–34 35–48 49–62 63–66 67–70 71–78 79–82 83–94 95–98 99–102 103–106 107–110 111–114 115–120 121–124 333202_080R.qxd 12/8/05 10:40 AM Page 632 632 Chapter 8 Matrices and Determinants 8 Review Exercises 8.1 In Exercises 1–4, determine the order of the matrix. 1. 4 0 5 3. 3 2. 6 2 5 8 0 In Exercises 5 and 6, write the augmented matrix for the system of linear equations. 5. 3x 10y 15 5x 4y 22 6. 8x 7y 4z 12 3x 5y 2z 20 5x 3y 3z 26 In Exercises 7 and 8, write the system of linear equations y, represented by the augmented matrix. (Use variables z, and w, x, if applicable.) 7 0 2 1 2 4 16 21 10 7 8 4 9 10 3 3 5 3 2 12 1 7. 8. 4 9 5 13 1 4 In Exercises 9 and 10, write the matrix in row-echelon form. Remember that the row-echelon form of a matrix is not unique. 9 10. 4 3 2 8 1 10 16 2 12 x, y, 0 0 11. 2 1 0 9 2 0 and 3 2 1 9 1 1 In Exercises 11–14, write the system of linear equations represented by the augmented matrix. Then use backz. substitution to solve the system. (Use variables ) 1 1 1 1 4 10 12. 14. 13. 0 0 0 0 0 0 In Exercises 15–24, use matrices and Gaussian elimination with back-substitution to solve the system of equations (if possible). 16. 2x 5y 2 3x 7y 1 6 9 11 14 15. 17. 18. 19. 20. 21. 22. 23. 24. 2 22 2x 3y 3z 4x 2y 3z 6x 6y 12z 13 12x 9y z 2 5x 4y x y 0.3x 0.1y 0.13 0.2x 0.3y 0.25 0.2x 0.1y 0.07 0.4x 0.5y 0.01 2x 3y z 10 22 2 2x 3y 3z 3 2x y x 2y 6z 2x x 2x 5y 15z 3x y 3z z y 2y 3z 2z 3y z 2z 4 5 6z 2 2x 2y 2x y 2y 3y 4y 1 4 6 3x x 3z z z 4x 2x w 2w 3w w 3 0 2w 0 3 In Exercises 25–28, use matrices and Gauss-Jordan elimination to solve the system of equations. 1 2 4 26. 25. 2x 3y z 5x 4y 2z 4x 2y 8z 1 5x 3y 8z 6 x y 2z 4x 4y 4z 5 2x y 9z 28. 3x y 7z x 3y 4z 5x 2y z 5x 2y z x y 4z 27. 8 15 17 20 34 8 333202_080R.qxd 12/8/05 10:41 AM Page 633 In Exercises 29 and 30, use the matrix capabilities of a graphing utility to reduce the augmented matrix corresponding to the system of equations, and solve the system. 29. 30. 44 1 15 58 x 6y 4z w 5x y z 3w 4y z 8w 3x y 5z 2w 4x 12y 2z x 6y 4z x 6y z 2x 10y 2z 20 12 8 10 Review Exercises 633 42. 8 2 0 1 4 6 52 3 6 8 12 0 0 1 12 4 1 8 43. In Exercises 43 and 44, use the matrix capabilities of a graphing utility to evaluate the expression 44. 2 11 3 6 1 0 2 2 7 8 8.2 In Exercises 31–34, find and x y. In Exercises 45–48, solve for X in the equation given 31. 32. 33. 34 12 9 5x 1 9 0 2 0 1 2x 8 4 4y 2 6x 5 4 0 4 3 1 4 44 3 2 6 16 x 10 7 1 5 2y 0 In Exercises 35–38, if possible, find (a) (c) and (d) A B, (b) A B, 4A, A 2 3 A 5 A 5 7 11 7 11 35. 36. 37. 38. A 6 5 A 3B 10 8 12 40 30 3 12 40 20 15 B 3 12 B 4 B 0 B 1 4 20 7, 4 8 In Exercises 39–42, perform the matrix operations. If it is not possible, explain why. 39. 7 1 40. 41. 11 7 21 5 6 10 3 14 5 19 1 87 16 2 2 4 0 20 10 A [4 1 3 0 5 2] 45. 47. X 3A 2B 3X 2A B and B [ 1 2 4 2 1 4]. 46. 48. 6X 4A 3B 2A 5B 3X In Exercises 49–52, find 49. 50. 51. A 2 3 A 5 A 5 7 11 7 11 , 52. A 6 5 AB, B 3 12 B 4 if possible. 10 8 12 40 30 20 15 12 40 B 4 20 B 1 4 8 7, In Exercises 53–60, perform the matrix operations. If it is not possible, explain why. 53. 54 57. 4 6 56. 55 333202_080R.qxd 12/5/05 11:08 AM Page 634 634 Chapter 8 Matrices and Determinants 31 4 58. 59. 60. In Exercises 61 and 62, use
the matrix capabilities of a graphing utility to find the product. 4 11 12 2 4 61. 62 10 2 1 5 3 1 2 2 63. Manufacturing A tire corporation has three factories, each of which manufactures two products. The number of units of product in one day is aij represented by A 80 40 i in the matrix produced at factory 140 80 120 100 . j Find the production levels if production is decreased by 5%. 64. Manufacturing A corporation has four factories, each of which manufactures three types of cordless power tools. The number of units of cordless power tools produced at factory in one day is represented by in the matrix j aij A 80 50 90 70 30 60 90 80 100 . 40 20 50 Find the production levels if production is increased by 20%. 65. Manufacturing A manufacturing company produces three kinds of computer games that are shipped to two that are warehouses. The number of units of game j in the matrix shipped to warehouse is represented by i aij A 8200 6500 5400 . 7400 9800 4800 The price per unit is represented by the matrix B $10.25 $14.50 $17.75. Compute BA and interpret the result. 66. Long-Distance Plans The charges (in dollars per minute) of two long-distance telephone companies for in-state, stateto-state, and international calls are represented by C. Company A C 0.07 0.10 0.28 B 0.095 0.08 0.25 In-state State-to-state International Type of call You plan to use 120 minutes on in-state calls, 80 minutes on state-to-state calls, and 20 minutes on international calls each month. (a) Write a matrix T that represents the times spent on the phone for each type of call. (b) Compute TC and interpret the result. A. B is the inverse of 1 4 8.3 67. 68. 69. 70. 1 2 1 , 2 1 0 2 In Exercises 67–70, show that A 4 , 7 A 5 11 A 1 A 1 B 2 7 B 2 11 , 71. In Exercises 71–74, find the inverse of the matrix (if it exists). 6 5 1 3 2 0 73. 74. 72 In Exercises 75–78, use the matrix capabilities of a graphing utility to find the inverse of the matrix (if it exists). 75. 77 76. 2 1 1 78. 8 4 1 1 4 3 18 0 2 2 4 6 1 16 8 2 4 1 2 0 1 1 333202_080R.qxd 12/5/05 11:08 AM Page 635 In Exercises 79–82, use the formula below to find the inverse of the matrix, it it exists. In Exercises 83–90, use an inverse matrix to solve (if possible) the system of linear equations. b a] A1 1 c ad bc[ d 2 2 20 6 5 2 8 3 4 3 7 8 10 7 1 3 2 3 10 4 4 5 8 13 10 47 8 5 13 24 x 4y 2x 7y 5x y 9x 2y 3x 10y 5x 17y 4x 2y 19x 9y 3x 2y z x 4y 2z 2x y 2z 3x y 5z 2x 9y 5z x 5y 4z x 4y z y z x y 2z 5x y z x y 6z 8x 4y z 6 1 7 12 25 10 13 11 0 14 8 44 79. 80. 81. 82. 83. 84. 85. 86. 87. 88. 89. 90. In Exercises 91–94, use the matrix capabilities of a graphing utility to solve (if possible) the system of linear equations. 91. 92. 93. x 2y 1 3x 4y 5 x 3y 6x 2y 3x 3y 4z y z 4x 3y 4z 23 18 2 1 1 Review Exercises 635 94. x 3y 2z 2x 7y 3z x y 3z 8 19 3 In Exercises 95–98, find the determinant of the 95. 8.4 matrix. 8 2 9 7 50 10 14 12 98. 96. 97. 5 4 11 4 30 5 24 15 99. 100. 6 4 1 4 In Exercises 99–102, find all (a) minors and (b) cofactors of the matrix 101. 102. 3 5 1 4 9 2 6 4 In Exercises 103–106, find the determinant of the matrix. Expand by cofactors on the row or column that appears to make the computations easiest. 103. 104. 2 5 6 5 2 4 3 106. 5 0 6 0 105 333202_080R.qxd 12/5/05 11:08 AM Page 636 636 Chapter 8 Matrices and Determinants In Exercises 107–110, use Cramer’s Rule to solve (if 8.5 possible) the system of equations. In Exercises 123 and 124, decode the cryptogram by using the inverse of the matrix 107. 109. 6 23 5x 2y 11x 3y 2x 3y 5z 4x y z x 4y 6z 11 3 15 108. 110. 7 37 3x 8y 9x 5y 5x 2y z 3x 3y z 2x y 7z 15 7 3 In Exercises 111–114, use a determinant and the given vertices of a triangle to find the area of the triangle. A [5 10 8 4 7 6 3 6 5]. 123. 124. 145 105 92 264 188 160 23 16 15 5 11 2 370 265 225 57 48 33 32 15 20 245 171 147 129 84 78 9 8 5 159 118 100 219 152 133 370 265 225 105 84 63 111. y 112. y Synthesis 8 6 4 2 −2 (5, 8) (5, 0) (1, 0) 4 6 8 113. (−2, 3) y 6 2 (0, 5) −4 −2 −2 −4 2 4 (1, −4) x x (0, 6) (4, 0) 6 2 −4 −2 (−4, 0) 2 4 114. y 3 2 1 3 2( ( , 1 (4, 2) 1 2 3 1 4, − 2 ( ( x x In Exercises 115 and 116, use a determinant to determine whether the points are collinear. 1, 7, 3, 9, 3, 15 0, 5, 2, 6, 8, 1 116. 115. In Exercises 117–120, use a determinant to find an equation of the line passing through the points. 117. 119. 4, 0, 4, 4 5 2, 1 2, 3, 7 118. 120. 2, 5, 6, 1 0.8, 0.2, 0.7, 3.2 In Exercises 121 and 122, find the uncoded row matrices for the message. Then encode the message using the encoding matrix. 1 3 Message 121. LOOK OUT BELOW 122. RETURN TO BASE Encoding Matrix True or False? In Exercises 125 and 126, determine whether the statement is true or false. Justify your answer. 125. It is possible to find the determinant of a a12 a22 c2 a13 a23 126. a33 a31 4 5 matrix. c3 a13 a23 c3 a12 a22 c2 a32 a13 a23 a33 a11 a21 c1 a11 a21 a31 a11 a21 c1 a12 a22 a32 127. Under what conditions does a matrix have an inverse? 128. Writing What is meant by the cofactor of an entry of a matrix? How are cofactors used to find the determinant of the matrix? 129. Three people were asked to solve a system of equations using an augmented matrix. Each person reduced the matrix to row-echelon form. The reduced matrices were 1 0 , 3 1 2 1 1 0 and Can all three be right? Explain. 130. Think About It Describe the row-echelon form of an augmented matrix that corresponds to a system of linear equations that has a unique solution. 131. Solve the equation for . 2 3 8 0 5 333202_080R.qxd 12/5/05 11:08 AM Page 637 8 Chapter Test Chapter Test 637 Take this test as you would take a test in class. When you are finished, check your work against the answers given in the back of the book. In Exercises 1 and 2, write the matrix in reduced row-echelon form. 1. Write the augmented matrix corresponding to the system of equations and solve the system. 4x 3y 2z x y 2z 3x y 4z 14 5 8 4. Find (a) A 5 4 A B, (b) 3 , 4 4 A, (c) B 4 4 3A 2B, 1 0 and (d) AB (if possible). In Exercises 5 and 6, find the inverse of the matrix (if it exists). 5. 6 10 4 5 6. Use the result of Exercise 5 to solve the system. 6x 4y 10 10x 5y 20 y 6 4 (−5, 0) −4 −2 −2 (4, 4) (3, 2) 2 4 x In Exercises 8–10, evaluate the determinant of the matrix. 8. 9 13 4 16 9. 5 2 8 13 4 6 5 10 In Exercises 11 and 12, use Cramer’s Rule to solve (if possible) the system of equations. 11. 7x 6y 2x 11y 9 49 12. 6x y 2z 2x 3y z 4x 4y z 4 10 18 FIGURE FOR 13 13. Use a determinant to find the area of the triangle in the figure. 14. Find the uncoded 1 3 row matrices for the message KNOCK ON WOOD. Then encode the message using the matrix below 15. One hundred liters of a 50% solution is obtained by mixing a 60% solution with a 20% solution. How many liters of each solution must be used to obtain the desired mixture? 333202_080R.qxd 12/5/05 11:08 AM Page 638 Proofs in Mathematics Proofs without words are pictures or diagrams that give a visual understanding of why a theorem or statement is true. They can also provide a starting point for determinant is the writing a formal proof. The following proof shows that a area of a parallelogram. 2 2 (0, d) (a, b + d) (a + c, b + d) (a, d) (a + c, d) (a, b) (0, 0) (a, 0) a c b d ad bc The following is a color-coded version of the proof along with a brief expla- nation of why this proof works. (0, d) (a, b + d) (a + c, b + d) (a, d) (a + c, d) (a, b) (0, 0) (a, 0) a c b d ad bc Area of Area of yellow Area of blue Area of orange Area of pink Area of white quadrilateral Area of Area of orange quadrilateral Area of pink Area of green Area of Area of white quadrilateral Area of blue Area of Area of green quadrilateral Area of Area of yellow From “Proof Without Words” by Solomon W. Golomb, Mathematics Magazine, March 1985. Vol. 58, No. 2, pg. 107. Reprinted with permission. 638 333202_080R.qxd 12/5/05 11:08 AM Page 639 P.S. Problem Solving This collection of thought-provoking and challenging exercises further explores and expands upon concepts learned in this chapter. 1. The columns of matrix T vertices of a triangle. Matrix A 0 T 1 1 1 1 0 show the coordinates of the A is a transformation matrix. 2 4 3 2 (a) Find AT and AAT. Then sketch the original triangle and the two transformed triangles. What transformation does A represent? 4. Let A 1 2 (a) Show that . 2 1 A2 2A 5I O, where I is the identity matrix of order 2. A1 1 5 (b) Show that 2I A. (c) Show in general that for any square matrix satisfying A2 2A 5I O (b) Given the triangle determined by describe the transformation process that produces the triangle determined by and then the triangle determined by AAT, AT T. 2. The matrices show the number of people (in thousands) who lived in each region of the United States in 2000 and the number of people (in thousands) projected to live in each region in 2015. The regional populations are separated into three age categories. (Source: U.S. Census Bureau) 0–17 13,049 16,646 25,569 4,935 12,098 0–17 12,589 15,886 25,916 5,226 14,906 Northeast Midwest South Mountain Pacific Northeast Midwest South Mountain Pacific 2000 18–64 33,175 39,486 62,235 11,210 28,036 2015 18–64 34,081 41,038 68,998 12,626 33,296 65 + 7,372 8,263 12,437 2,031 4,893 65 + 8,165 10,101 17,470 3,270 6,565 (a) The total population in 2000 was 281,435,000 and the projected total population in 2015 is 310,133,000. Rewrite the matrices to give the information as percents of the total population. (b) Write a matrix that gives the projected change in the percent of the population in each region and age group from 2000 to 2015. (c) Based on the result of part (b), which region(s) and age group(s) are projected to show relative growth from 2000 to 2015? 3. Determine whether the matrix is idempotent. A square A2 A. matrix is idempotent if (ac) (b) (d is given by the inverse of A1 1 5 2I A. 5. Two competing companies offer cable television to a city with 100,000 households. Gold Cable Company has 25,000 subscribers and Galaxy Cable Company has 30,000 subscribers.
(The other 45,000 households do not subscribe.) The percent changes in cable subscriptions each year are shown in the matrix below. Percent Changes From Gold 0.70 0.20 0.10 From From NonGalaxy subscriber 0.15 0.80 0.05 0.15 0.15 0.70 Percent Changes To Gold To Galaxy To Nonsubscriber (a) Find the number of subscribers each company will have in 1 year using matrix multiplication. Explain how you obtained your answer. (b) Find the number of subscribers each company will have in 2 years using matrix multiplication. Explain how you obtained your answer. (c) Find the number of subscribers each company will have in 3 years using matrix multiplication. Explain how you obtained your answer. (d) What is happening to the number of subscribers to each company? What is happening to the number of nonsubscribers? 6. Find such that the matrix is equal to its own inverse. x A 3 2 x 3 7. Find such that the matrix is singular. x 3 x A 4 2 8. Find an example of a singular 2 2 matrix satisfying A2 A. 639 333202_080R.qxd 12/5/05 11:08 AM Page 640 10. Verify the following equation. 9. Verify the following equation. a a2 a a3 1 1 1 b b2 1 b b3 1 c c2 a bb cc a c3 a bb cc aa b c a ax2 bx 11. Verify the following equation. 12. Use the equation given in Exercise 11 as a model to find a determinant that is equal to ax3 bx2 cx d. 13. The atomic masses of three compounds are shown in the table. Use a linear system and Cramer’s Rule to find the atomic masses of sulfur (S), nitrogen (N), and fluorine (F). Compound Formula Atomic mass Tetrasulphur tetranitride Sulfur hexafluoride Dinitrogen tetrafluoride S4N4 SF6 N2F4 184 146 104 14. A walkway lighting package includes a transformer, a certain length of wire, and a certain number of lights on the wire. The price of each lighting package depends on the length of wire and the number of lights on the wire. Use the following information to find the cost of a transformer, the cost per foot of wire, and the cost of a light. Assume that the cost of each item is the same in each lighting package. • A package that contains a transformer, 25 feet of wire, and 5 lights costs $20. • A package that contains a transformer, 50 feet of wire, and 15 lights costs $35. • A package that contains a transformer, 100 feet of wire, and 20 lights costs $50. 15. The transpose of a matrix, denoted is formed by writing its columns as rows. Find the transpose of each ABT BTAT. matrix and verify that AT 640 16. Use the inverse of matrix A to decode the cryptogram 23 13 24 14 56 38 41 53 34 37 31 41 34 17 116 13 85 28 32 11 16 63 25 8 20 1 22 17 29 3 61 40 6 17. A code breaker intercepted the encoded message below. 35 28 45 42 10 Let A1 w y 38 30 18 75 55 18 2 35 2 30 22 81 21 60 15 . x z (a) You know that 45 35 A1 10 15 and that 38 30 A1 8 14, is the inverse of the encoding matrix Write and solve two systems y, of equations to find A. x,w, where A1 and z. (b) Decode the message. 18. Let . Use a graphing utility to find Make a conjecture about the determinant of the inverse of a matrix. Compare A1 A1. with n n 19. Let A zero. Find be an A. 20. Consider matrices of the form matrix each of whose rows adds up to A 0 0 0 0 0 a12 0 0 0 0 a13 a23 0 0 0 a14 a24 a34 0 0 ... ... ... ... ... ... a1n a2n a3n an1n 0 (a) Write a of A. 2 2 matrix and a 3 3 matrix in the form (b) Use a graphing utility to raise each of the matrices to higher powers. Describe the result. (c) Use the result of part (b) to make a conjecture about matrix. Use a graphing is a A A 4 4 if powers of utility to test your conjecture. (d) Use the results of parts (b) and (c) to make a conjecture is an about powers of matrix. AA if n n 99 333202_0900.qxd 12/5/05 11:26 AM Page 641 Sequences, Series, and Probability 9.1 9.2 9.3 Sequences and Series Arithmetic Sequences and Partial Sums Geometric Sequences and Series 9.4 Mathematical Induction 9.5 9.6 9.7 The Binomial Theorem Counting Principles Probability Poker has become a popular card game in recent years.You can use the probability theory developed in this chapter to calculate the likelihood of getting different poker hands AT I O N S Sequences, series, and probability have many real-life applications. The applications listed below represent a small sample of the applications in this chapter. • Federal Debt, Exercise 111, page 651 • Data Analysis: Tax Returns, • Lottery, Exercise 61, page 682 Exercise 65, page 700 • Falling Object, • Child Support, Exercises 87 and 88, page 661 Exercise 80, page 690 • Multiplier Effect, • Poker Hand Exercises 113–116, page 671 Exercise 57, page 699 • Defective Units, Exercise 47, page 711 • Population Growth, Exercise 139, page 718 641 333202_0901.qxd 12/5/05 11:28 AM Page 642 642 Chapter 9 Sequences, Series, and Probability 9.1 Sequences and Series What you should learn • Use sequence notation to write the terms of sequences. • Use factorial notation. • Use summation notation to write sums. • Find the sums of infinite series. • Use sequences and series to model and solve real-life problems. Why you should learn it Sequences and series can be used to model real-life problems. For instance, in Exercise 109 on page 651, sequences are used to model the number of Best Buy stores from 1998 through 2003. Scott Olson /Getty Images The HM mathSpace® CD-ROM and Eduspace® for this text contain additional resources related to the concepts discussed in this chapter. Sequences In mathematics, the word sequence is used in much the same way as in ordinary English. Saying that a collection is listed in sequence means that it is ordered so that it has a first member, a second member, a third member, and so on. Mathematically, you can think of a sequence as a function whose domain is the set of positive integers. f 2 a2, f1 a1, f 3 a3, f 4 a4, . . . , f n an, . . . Rather than using function notation, however, sequences are usually written using subscript notation, as indicated in the following definition. Definition of Sequence An infinite sequence is a function whose domain is the set of positive integers. The function values a1, a2, a3, a4, . . . , an, . . . are the terms of the sequence. If the domain of the function consists of the first positive integers only, the sequence is a finite sequence. n On occasion it is convenient to begin subscripting a sequence with 0 instead of 1 so that the terms of the sequence become a0, a1, a2, a3, . . . . Example 1 Writing the Terms of a Sequence Write the first four terms of the sequences given by 3n 2 a. an Solution b. an 3 1n. a. The first four terms of the sequence given by an 3n 2 are 31 2 1 32 2 4 33 2 7 34 2 10. a1 a2 a3 a4 1st term 2nd term 3rd term 4th term 3 1n are an a1 b. The first four terms of the sequence given by 3 11 3 1 2 3 12 3 1 4 3 13 3 1 2 3 14 3 1 4. a2 a3 a4 1st term 2nd term 3rd term 4th term Now try Exercise 1. 333202_0901.qxd 12/5/05 11:28 AM Page 643 Section 9.1 Sequences and Series 643 Example 2 A Sequence Whose Terms Alternate in Sign Write the first five terms of the sequence given by an 1n 2n 1 . Solution The first five terms of the sequence are as follows. 1 2 1 1 a1 1st term 11 21 1 12 22 1 13 23 1 14 24 1 15 25 1 a2 a3 a4 a5 10 1 1 9 2nd term 3rd term 4th term 5th term Now try Exercise 17. Simply listing the first few terms is not sufficient to define a unique th term must be given. To see this, consider the following n sequence—the sequences, both of which have the same first three terms 16 , 1 15 , . . . , 1 2n , . . . , . . . , 6 n 1n2 n 6 , . . . Example 3 Finding the nth Term of a Sequence Write an expression for the apparent th term of each sequence. an n 2, 5, 10, 17, . . . a. 1, 3, 5, 7, . . . b. Solution a. n: 1 2 3 4 Terms: 1 3 5 7 Apparent pattern: Each term is 1 less than twice which implies that . . . n . . . an n, an 2n 1. b. n: 1 2 5 3 10 4 17 . . . n . . . an Terms: 2 Apparent pattern: The terms have alternating signs with those in the even positions being negative. Each term is 1 more than the square of which implies that n, Exploration Write out the first five terms of the sequence whose th term is 1n1 2n 1 an n . Are they the same as the first five terms of the sequence in Example 2? If not, how do they differ? Te c h n o l o g y To graph a sequence using a graphing utility, set the mode to sequence and dot and enter the sequence. The graph of the sequence in Example 3(a) is shown below. You can use the trace feature or value feature to identify the terms. 11 0 0 5 1n1n2 1 an Now try Exercise 37. 333202_0901.qxd 12/5/05 11:28 AM Page 644 644 Chapter 9 Sequences, Series, and Probability Some sequences are defined recursively. To define a sequence recursively, you need to be given one or more of the first few terms. All other terms of the sequence are then defined using previous terms. A well-known example is the Fibonacci sequence shown in Example 4. Example 4 The Fibonacci Sequence: A Recursive Sequence The Fibonacci sequence is defined recursively, as follows. ak2 ak1, Write the first six terms of this sequence. k ≥ 2 where 1, 1, a0 a1 ak a4 The subscripts of a sequence make up the domain of the sequence and they serve to identify the location of a term within the sequence. For examis the fourth term of the ple, an is the nth term sequence, and of the sequence. Any variable can be used as a subscript. The most commonly used variable subscripts in sequence and series notation are and n. k, i, j, Solution a0 1 1 a22 a32 a42 a52 a1 a2 a3 a4 a5 a21 a31 a41 a51 a0 a1 a2 a3 a1 a2 a3 a4 Now try Exercise 51. 0th term is given. 1st term is given. Use recursion formula. Use recursion formula. Use recursion formula. Use recursion formula. Factorial Notation Some very important sequences in mathematics involve terms that are defined with special types of products called factorials. Definition of Factorial If n is a positive integer, n factorial is defined as n. As a special case, zero factorial is defined as 0! 1. that for the first several nonnegative integers. Notice n! Here are some values of is 1 by definitio
n. 0! 0! 1 1! 1 2! 1 2 2 3! 1 2 3 6 4! 1 2 3 4 24 5! 1 2 3 4 5 120 The value of does not have to be very large before the value of extremely large. For instance, 10! 3,628,800. n n! becomes 333202_0901.qxd 12/5/05 11:28 AM Page 645 Section 9.1 Sequences and Series 645 Factorials follow the same conventions for order of operations as do expo- nents. For instance, 2n! 2n! 21 2 3 4 . . . n whereas 2n! 1 2 3 4 . . . 2n. Example 5 Writing the Terms of a Sequence Involving Factorials Write the first five terms of the sequence given by an . 2n n! n 0. Begin with Then graph the terms on a set of coordinate axes. Solution a0 a1 a 2 a3 a4 20 0! 21 1! 22 2! 23 3! 24 4 16 24 2 3 0th term 1st term 2nd term 3rd term 4th term an 4 3 2 1 1 2 3 4 n FIGURE 9.1 Figure 9.1 shows the first five terms of the sequence. Now try Exercise 59. When working with fractions involving factorials, you will often find that the fractions can be reduced to simplify the computations. Example 6 Evaluating Factorial Expressions Evaluate each factorial expression. a. 8! 2! 6! Solution b. 2! 6! 3! 5! c. n! n 1! a. b. c. 8! 2! 6! 2! 6! 3! 5! n! n 1 28 2 6 3 n Now try Exercise 69. Note in Example 6(a) that you can simplify the computation as follows. 8! 2! 6! 8 7 6! 2! 6! 8 7 2 1 28 333202_0901.qxd 12/5/05 11:28 AM Page 646 646 Chapter 9 Sequences, Series, and Probability Te c h n o l o g y Summation Notation Most graphing utilities are able to sum the first n terms of a sequence. Check your user’s guide for a sum sequence feature or a series feature. Summation notation is an instruction to add the terms of a sequence. From the definition at the right, the upper limit of summation tells you where to end the sum. Summation notation helps you generate the appropriate terms of the sequence prior to finding the actual sum, which may be unclear. There is a convenient notation for the sum of the terms of a finite sequence. It is called summation notation or sigma notation because it involves the use of the uppercase Greek letter sigma, written as . Definition of Summation Notation The sum of the first n terms of a sequence is represented by n i1 ai a1 a2 a3 a4 . . . an i is called the index of summation, where summation, and 1 is the lower limit of summation. n is the upper limit of Example 7 Summation Notation for Sums Find each sum. a. 5 i1 3i b. 6 k3 1 k2 c. 8 i0 1 i! Solution 5 a. i1 3i 31 32 33 34 35 31 2 3 4 5 315 45 b. 6 k3 1 k 2 1 32 1 42 1 52 1 62 10 17 26 37 90 c. 8 i0 1 i! 1 0! 1 1! 1 2! 1 6 1 3! 1 24 1 4! 1 120 1 1 6! 5! 1 720 1 7! 1 1 8! 1 5040 40,320 1 1 1 2 2.71828 For this summation, note that the sum is very close to the irrational number e 2.718281828. It can be shown that as more terms of the sequence whose n th term is are added, the sum becomes closer and closer to 1n! e. Now try Exercise 73. In Example 7, note that the lower limit of a summation does not have to be 1. Also note that the index of summation does not have to be the letter For instance, in part (b), the letter is the index of summation. i. k 333202_0901.qxd 12/5/05 11:28 AM Page 647 Section 9.1 Sequences and Series 647 Properties of Sums Variations in the upper and lower limits of summation can produce quite different-looking summation notations for the same sum. For example, the following two sums have the same terms. 3 32i 321 22 23 i1 2 i0 32i1 321 22 23 1. 3. n i1 c cn, c is a constant. 2. n i1 cai cn i1 ai, c is a constant. n i1 ai bi n i1 ai n i1 bi 4. n i1 ai bi n ai i1 n bi i1 For proofs of these properties, see Proofs in Mathematics on page 722. Series Many applications involve the sum of the terms of a finite or infinite sequence. Such a sum is called a series. Definition of Series Consider the infinite sequence a1, a2, a3, . . . , ai, . . . . 1. The sum of the first n terms of the sequence is called a finite series or the nth partial sum of the sequence and is denoted by a1 a2 a3 . . . an n i1 ai. 2. The sum of all the terms of the infinite sequence is called an infinite series and is denoted by a1 a2 a3 . . . ai . . . ai. i1 Example 8 Finding the Sum of a Series For the series i1 3 10i , find (a) the third partial sum and (b) the sum. Solution a. The third partial sum is 3 i1 3 10i 3 101 3 102 3 103 0.3 0.03 0.003 0.333. b. The sum of the series is i1 3 10i . . . 3 105 3 103 3 101 3 102 3 104 0.3 0.03 0.003 0.0003 0.00003 . . 0.33333. . . 1 3 . . Now try Exercise 99. 333202_0901.qxd 12/5/05 11:28 AM Page 648 648 Chapter 9 Sequences, Series, and Probability Application Sequences have many applications in business and science. One such application is illustrated in Example 9. Example 9 Population of the United States For the years 1980 to 2003, the resident population of the United States can be approximated by the model an 226.9 2.05n 0.035n2, n 0, 1, . . . , 23 an is the population (in millions) and n 0 where corresponding to 1980. Find the last five terms of this finite sequence, which represent the U.S. population for the years 1999 to 2003. (Source: U.S. Census Bureau) represents the year, with n Solution The last five terms of this finite sequence are as follows. a19 a20 a21 a22 a23 226.9 2.0519 0.035192 278.5 226.9 2.0520 0.035202 281.9 226.9 2.0521 0.035212 285.4 226.9 2.0522 0.035222 288.9 226.9 2.0523 0.035232 292.6 Now try Exercise 111. 1999 population 2000 population 2001 population 2002 population 2003 population Exploration 3 3 3 A cube is created using 27 unit cubes (a unit cube has a length, width, and height of 1 unit) and only the faces of each cube that are visible are painted blue (see Figure 9.2). Complete the table below to determine how cube have 0 blue faces, 1 blue face, 2 blue many unit cubes of the faces, and 3 blue faces. Do the same for a cube, and a pattern do you observe in the table? Write a formula you could use to determine the column values for an cube and add your results to the table below. What type of cube, a cube. Number of blue cube faces 3 3 3 0 1 2 3 FIGURE 9.2 333202_0901.qxd 12/5/05 11:28 AM Page 649 9.1 Exercises The HM mathSpace® CD-ROM and Eduspace® for this text contain step-by-step solutions to all odd-numbered exercises. They also provide Tutorial Exercises for additional help. Section 9.1 Sequences and Series 649 VOCABULARY CHECK: Fill in the blanks. 1. An ________ ________ is a function whose domain is the set of positive integers. 2. The function values 3. A sequence is a ________ sequence if the domain of the function consists of the first positive integers. 4. If you are given one or more of the first few terms of a sequence, and all other terms of the sequence are are called the ________ of a sequence. a1, a2, a3, a4, . . . n defined using previous terms, then the sequence is said to be defined ________. 5. If n is a positive integer, n ________ is defined as n. 6. The notation used to represent the sum of the terms of a finite sequence is ________ ________ or sigma notation. 7. For the sum in ai , the ________ limit of summation. i1 is called the ________ of summation, n is the ________ limit of summation, and 1 is 8. The sum of the terms of a finite or infinite sequence is called a ________. 9. The ________ ________ ________ of a sequence is the sum of the first n terms of the sequence. PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. In Exercises 1–22, write the first five terms of the sequence. n (Assume that begins with 1.) In Exercises 27–32, use a graphing utility to graph the first 10 terms of the sequence. (Assume that begins with 1.) n 1. 3. 5. an an an 7. an 9. an 11. an 13. an 15. an 17. an 19. 21. an an 3n 1 2n 2n n 2 n 6n 3n 2 1 1 1n n 2 1 3n 1 n32 1n n2 2 3 nn 1n 2 2. 4. 6. an an an 8. an 10. an 5n 3n2 n 4 2n2 1 12. an 1 1n 14. an 16. an 18. an 20. 22. an an 2n 3n 10 n23 1n n n 1 0.3 nn2 6 23. an a25 In Exercises 23–26, find the indicated term of the sequence. 1n1nn 1 4n2 n 3 nn 1n 2 1n3n 2 4n 2n2 3 an a16 26. 25. 24. an an a11 a13 27. an 29. an 31. an 3 4 n 160.5n1 2n n 1 28. an 30. an 32. an 2 4 n 80.75n1 n2 n2 2 In Exercises 33–36, match the sequence with the graph of its first 10 terms. [The graphs are labeled (a), (b), (c), and (d).] (a) (b) an an 2 4 6 8 10 10 8 6 4 2 (c) an 10 8 6 4 2 10 8 6 4 2 (d) an 10 10 n n 2 4 6 8 10 2 4 6 8 10 33. an 35. an 8 n 1 40.5n1 34. an 36. an 8n n 1 4n n! 333202_0901.qxd 12/5/05 11:28 AM Page 650 650 Chapter 9 Sequences, Series, and Probability In Exercises 37–50, write an expression for the apparent th term of the sequence. (Assume that begins with 1.) n n 37. 1, 4, 7, 10, 13, . . . 39. 41. 43. 45. 0, 3, 8, 15, 24, 4, 7, 6 3, 4 1, 3 5, 5 9, . . . 9, 1 4, 1 1, 1 1 16, 25, . . . 2 47. 1, 1, 1, 1, 1, . . . 38. 40. 42. 44. 46. 48. 49. 50. 1 1 1 1 1, 1 1 2, 1 3 2, 1 1 4, 1 7 3, 1 1 8, 1 15 4, 1 1 16, 1 31 3, 7, 11, 15, 19, . . . 2, 4, 6, 8, 10, . . . 1 1 16 , . . . 2, 1 3, 2 81, . . . 24, 1 1, 1 120, . . . 23 24 25 6 120 24 1 4 , 1 8, 27, 8 9, 4 6, 1 2, 1 22 2 5, . . . 32, . . . 1, 2, , , , , . . . In Exercises 51–54, write the first five terms of the sequence defined recursively. 51. 52. 53. 54. 28, 15, 3, 32, a1 a1 a1 a1 4 3 1 ak1 ak1 ak1 ak1 ak ak 2ak 1 2ak In Exercises 55–58, write the first five terms of the sequence defined recursively. Use the pattern to write the th term of the sequence as a function of . (Assume that begins with 1.) n n n 55. 56. 57. 58. 6, 25, 81, 14, a1 a1 a1 a1 ak1 ak1 ak1 ak1 2 5 ak ak 1 3ak 2ak In Exercises 59–64, write the first five terms of the sequence. (Assume that begins with 0.) n 59. an 3n n! 61. an 63. an 1 n 1! 12n 2n! n! n n2 60. an 62. an n 1! 12n1 2n 1! 64. an In Exercises 65–72, simplify the factorial expression. 65. 67. 69. 71. 4! 6! 10! 8! n 1! n! 2n 1! 2n 1! 66. 68. 70. 72. 5! 8! 25! 23! n 2! n! 3n 1! 3n! In Exercises 73–84, find the sum. 73. 75. 77. 79. 81. 83. 5 i1 2i 1 4 k1 10 4 i0 i 2 3 k0 1 k2 1 5 k2 k 12k 3 4 i1 2i 74. 76. 78. 80. 82. 84. 6 i1 3i 1 5 5 k1 5 i0 2i 2 5 j3 1 j 2 3 4 i1 i 12 i 13 4 j0 2 j In Exercises 85–88, use a calculator to find
the sum. 86. 10 j1 3 j 1 85. 87. 88. 6 j1 24 3j 4 k0 4 k0 1k k 1 1k k! In Exercises 89–98, use sigma notation to write the sum. 89. 90. 91. 92. 93. 94. 95. 96. 97. 98. . . . 1 39 . . . 22 1 32 5 1 33 5 1 31 27 81 243 729 1 1 2 1 1 1 12 32 22 1 2 4 . . . 1 128 1 8 1 4 6 . . . 1 202 . . . 1 2 . . . 1 6 10 12 1 15 3 2 8 6 1 42 1 3 5 31 64 120 32 7 16 6 8 15 32 24 16 720 64 In Exercises 99–102, find the indicated partial sum of the series. 99. 101. 51 2 i i1 Fourth partial sum n 41 2 n1 Third partial sum 100. 102. 21 3 i i1 Fifth partial sum n 81 4 n1 Fourth partial sum 333202_0901.qxd 12/5/05 11:28 AM Page 651 In Exercises 103–106, find the sum of the infinite series. Model It (co n t i n u e d ) Section 9.1 Sequences and Series 651 103. 104. 105. 106. k1 i1 i1 k1 6 1 10 i k 1 10 7 1 10 k 2 1 10 i 107. Compound Interest A deposit of $5000 is made in an account that earns 8% interest compounded quarterly. The balance in the account after quarters is given by n An 50001 0.08 4 n , n 1, 2, 3, . . . . (a) Write the first eight terms of this sequence. (b) Find the balance in this account after 10 years by finding the 40th term of the sequence. 108. Compound Interest A deposit of $100 is made each month in an account that earns 12% interest compounded monthly. The balance in the account after months is given by n 1001011.01n 1, n 1, 2, 3, . . . . An (a) Write the first six terms of this sequence. (b) Find the balance in this account after 5 years by finding the 60th term of the sequence. (c) Find the balance in this account after 20 years by finding the 240th term of the sequence. Model It 109. Data Analysis: Number of Stores The table shows of Best Buy stores for the years 1998 the numbers to 2003. an (Source: Best Buy Company, Inc.) Year 1998 1999 2000 2001 2002 2003 Number of an stores, 311 357 419 481 548 608 (a) Use the regression feature of a graphing utility to n find a linear sequence that models the data. Let corresponding to represent the year, with 1998. n 8 (b) Use the regression feature of a graphing utility to find a quadratic sequence that models the data. (c) Evaluate the sequences from parts (a) and (b) for n 8, 9, . . . , 13. Compare these values with those shown in the table. Which model is a better fit for the data? Explain. (d) Which model do you think would better predict the number of Best Buy stores in the future? Use the model you chose to predict the number of Best Buy stores in 2008. 110. Medicine The numbers (in thousands) of AIDS cases reported from 1995 to 2003 can be approximated by the model an 0.0457n3 0.352n2 9.05n 121.4, an n 5, 6, . . . , 13 n where is the year, with (Source: U.S. Centers Prevention) n 5 corresponding to 1995. for Disease Control and (a) Find the terms of this finite sequence. Use the statistical plotting feature of a graphing utility to construct a bar graph that represents the sequence. (b) What does the graph in part (a) say about reported cases of AIDS? 111. Federal Debt From 1990 to 2003, the federal debt of the United States rose from just over $3 trillion to almost $7 (in billions of dollars) from trillion. The federal debt 1990 to 2003 is approximated by the model an 2.7698n3 61.372n2 600.00n 3102.9, an n 0, 1, . . . , 13 n where (Source: U.S. Office of Management and Budget) is the year, with corresponding to 1990. n 0 (a) Find the terms of this finite sequence. Use the statistical plotting feature of a graphing utility to construct a bar graph that represents the sequence. (b) What does the pattern in the bar graph in part (a) say about the future of the federal debt? 333202_0901.qxd 12/5/05 11:28 AM Page 652 652 Chapter 9 Sequences, Series, and Probability 112. Revenue The revenues an (in millions of dollars) for Amazon.com for the years 1996 through 2003 are shown in the figure. The revenues can be approximated by the model an 46.609n2 119.84n 1125.8, n 6, 7, . . . , 13 n is the year, with n 6 where corresponding to 1996. Use this model to approximate the total revenue from 1996 through 2003. Compare this sum with the result of (Source: adding the revenues shown in the figure. Amazon.com) an e u n e v e R 6000 5000 4000 3000 2000 1000 ) ( Synthesis 6 7 9 11 8 Year (6 ↔ 1996) 10 n 12 13 True or False? In Exercises 113 and 114, determine whether the statement is true or false. Justify your answer. 118. Find the arithmetic mean of the following prices per gallon for regular unleaded gasoline at five gasoline stations in a city: $1.899, $1.959, $1.919, $1.939, and $1.999. Use the statistical capabilities of a graphing utility to verify your result. 119. Proof Prove that n i1 xi x 0. 120. Proof Prove that n i1 xi x 2 n i1 x 2 i 1 n n i1 xi2 . In Exercises 121–124, find the first five terms of the sequence. 121. an 123. an xn n! 1n x2n 2n! Skills Review 122. an 124. an 1n x2n1 2n 1 1n x2n1 2n 1! In Exercises 125–128, determine whether the function has an inverse function. If it does, find its inverse function. 125. f x 4x 3 127. hx 5x 1 126. 128. gx 3 x f x x 12 In Exercises 129–132, find (a) and (d) A B, (b) 4B 3A, (c) AB, 113. 4 i2 2i 4 i 2 24 i1 i1 i i1 114. 4 2 j 6 2 j2 j1 j3 Fibonacci Sequence Fibonacci sequence. (See Example 4.) In Exercises 115 and 116, use the 115. Write the first 12 terms of the Fibonacci sequence the first 10 terms of the sequence given by an and bn an1 an , n ≥ 1. 129. 130. 131. 116. Using the definition for bn in Exercise 115, show that bn 132. can be defined recursively by 4 3 12 11 BA. A 6 3 A 10 bn 1 1 bn1 . Arithmetic Mean In Exercises 117–120, use the following definition of the arithmetic mean n measurements x2, n of a set of . . . , xn. x1, x3, xi x x 1 n i1 117. Find the arithmetic mean of the six checking account balances $327.15, $785.69, $433.04, $265.38, $604.12, and $590.30. Use the statistical capabilities of a graphing utility to verify your result. In Exercises 133–136, find the determinant of the matrix. 8 15 A 2 12 5 7 133. 134. 135. A 3 1 A 3 136. A 16 0 4 9 2 4 4 7 9 11 8 1 6 5 3 1 10 3 12 2 2 7 3 1 333202_0902.qxd 12/5/05 11:29 AM Page 653 Section 9.2 Arithmetic Sequences and Partial Sums 653 9.2 Arithmetic Sequences and Partial Sums What you should learn • Recognize, write, and find the nth terms of arithmetic sequences. • Find nth partial sums of arithmetic sequences. • Use arithmetic sequences to model and solve real-life problems. Arithmetic Sequences A sequence whose consecutive terms have a common difference is called an arithmetic sequence. Definition of Arithmetic Sequence A sequence is arithmetic if the differences between consecutive terms are the same. So, the sequence Why you should learn it a1, a2, a3, a4, . . . , an, . . . Arithmetic sequences have practical real-life applications. For instance, in Exercise 83 on page 660, an arithmetic sequence is used to model the seating capacity of an auditorium. is arithmetic if there is a number a3 d a1 The number a2 a4 d a 3 a2 such that . . . d. is the common difference of the arithmetic sequence. Example 1 Examples of Arithmetic Sequences a. The sequence whose n th term is 4n 3 is arithmetic. For this sequence, the common difference between consecutive terms is 4. 7, 11, 15, 19, . . . , 4n 3, . . . Begin with n 1. 11 7 4 b. The sequence whose n th term is 7 5n common difference between consecutive terms is 5. is arithmetic. For this sequence, the © mediacolor’s Alamy 2, 3, 8, 13, . . . , 7 5n, . . . Begin with n 1. 3 2 5 c. The sequence whose th term is n n 3 1 4 common difference between consecutive terms is , , , 1 4. Begin with n 1. is arithmetic. For this sequence, the 5 4 1 1 4 Now try Exercise 1. The sequence 1, 4, 9, 16, . . . , whose th term is n n2, is not arithmetic. The difference between the first two terms is 4 1 3 a1 a2 but the difference between the second and third terms is a3 a2 9 4 5. 333202_0902.qxd 12/5/05 11:29 AM Page 654 654 Chapter 9 Sequences, Series, and Probability an = dn + c an c a1 a2 a3 n FIGURE 9.3 n a1 The alternative recursion form of the th term of an arithmetic sequence can be derived from the pattern below. a1 a1 a1 a1 a1 d 2d 3d 4d a5 a4 a2 a3 4th term 5th term 2nd term 3rd term 1st term 1 less a1 n 1 d an nth term 1 less In Example 1, notice that each of the arithmetic sequences has an th term that is of the form where the common difference of the sequence is An arithmetic sequence may be thought of as a linear function whose domain is the set of natural numbers. dn c, d. n The nth Term of an Arithmetic Sequence The th term of an arithmetic sequence has the form n dn c an Linear form is the common difference between consecutive terms of the d where c a1 sequence and a1 shown in Figure 9.3. Substituting alternative recursion form for the nth term of an arithmetic sequence. A graphical representation of this definition is an yields an dn c d. d for in c an a1 n 1 d Alternative form Example 2 Finding the nth Term of an Arithmetic Sequence Find a formula for the difference is 3 and whose first term is 2. n th term of the arithmetic sequence whose common Solution Because the sequence is arithmetic, you know that the formula for the th term d 3, an is of the form the formula must have the form n Moreover, because the common difference is dn c. Because it follows that an 3n c. 2, a1 d c a1 2 3 1. Substitute 3 for d. Substitute 2 for a1 and 3 for d. So, the formula for the th term is n an 3n 1. The sequence therefore has the following form. 2, 5, 8, 11, 14, . . . , 3n 1, . . . Now try Exercise 21. Another way to find a formula for the th term of the sequence in Example n 2 is to begin by writing the terms of the sequence. a1 2 2 a2 2 3 5 a3 5 3 8 a4 8 3 11 a5 11 3 14 a6 14 3 17 a7 17 3 20 . . . . . . . . . From these terms, you can reason that the th term is of the form n an dn c 3n 1. 333202_0902.qxd 12/5/05 11:29 AM Page 655 in Example 3 by a1 You can find using the alternative recursion n form of the th term of an arithmetic sequence, as follows. n 1d 4 1d 4 15 15 an a1 a1 a4 20 a1 20 a1 5 a1 Section 9.2 Arithmetic Sequen
ces and Partial Sums 655 Example 3 Writing the Terms of an Arithmetic Sequence The fourth term of an arithmetic sequence is 20, and the 13th term is 65. Write the first 11 terms of this sequence. Solution You know that d 13th terms of the sequence are related by 65. 20 and a13 a4 So, you must add the common difference nine times to the fourth term to obtain the 13th term. Therefore, the fourth and a13 9d. a4 20 a4 Using the sequence is as follows. and a13 a4 65, and a13 are nine terms apart. you can conclude that d 5, which implies that a1 5 a2 10 a3 15 a4 20 a5 25 a6 30 a7 35 a8 40 a9 45 a10 50 a11 55 . . . . . . Now try Exercise 37. If you know the th term of an arithmetic sequence and you know the n th term by using the n 1 common difference of the sequence, you can find the recursion formula an1 an d. Recursion formula With this formula, you can find any term of an arithmetic sequence, provided that you know the preceding term. For instance, if you know the first term, you can find the second term. Then, knowing the second term, you can find the third term, and so on. Example 4 Using a Recursion Formula Find the ninth term of the arithmetic sequence that begins with 2 and 9. Solution There are two ways For this sequence, the common difference is to find the ninth term. One way is simply to write out the first nine terms (by repeatedly adding 7). d 9 2 7. 2, 9, 16, 23, 30, 37, 44, 51, 58 Another way to find the ninth term is to first find a formula for the Because the first term is 2, it follows that n th term. c a1 d 2 7 5. Therefore, a formula for the th term is n an 7n 5 which implies that the ninth term is 79 5 58. a9 Now try Exercise 45. 333202_0902.qxd 12/5/05 11:29 AM Page 656 656 Chapter 9 Sequences, Series, and Probability The Sum of a Finite Arithmetic Sequence There is a simple formula for the sum of a finite arithmetic sequence. Note that this formula works only for arithmetic sequences. The Sum of a Finite Arithmetic Sequence The sum of a finite arithmetic sequence with terms is n Sn n 2 a1 an . For a proof of the sum of a finite arithmetic sequence, see Proofs in Mathematics on page 723. Example 5 Finding the Sum of a Finite Arithmetic Sequence Find the sum: 1 3 5 7 9 11 13 15 17 19. Solution To begin, notice that the sequence is arithmetic (with a common difference of 2). Moreover, the sequence has 10 terms. So, the sum of the sequence is Sn a1 an Formula for the sum of an arithmetic sequence 1 19 Substitute 10 for n, 1 for a1, and 19 for an. n 2 10 Historical Note A teacher of Carl Friedrich Gauss (1777–1855) asked him to add all the integers from 1 to 100. When Gauss returned with the correct answer after only a few moments, the teacher could only look at him in astounded silence. This is what Gauss did: . . . 100 3 2 1 100 101 99 101 98 101 . . . 1 . . . 101 Sn Sn 2Sn Sn 100 101 2 5050 520 100. Simplify. Now try Exercise 63. Example 6 Finding the Sum of a Finite Arithmetic Sequence Find the sum of the integers (a) from 1 to 100 and (b) from 1 to N. Solution a. The integers from 1 to 100 form an arithmetic sequence that has 100 terms. So, you can use the formula for the sum of an arithmetic sequence, as follows. Sn Formula for sum of an arithmetic sequence a1 an Substitute 100 for 1 100 1 2 3 4 5 6 . . . 99 100 n 2 100 2 50101 5050 Simplify Substitute N for 1 N an a1 1 for 1 for a1, a1, n, n, Now try Exercise 65. 100 for an. and N for an. Formula for sum of an arithmetic sequence b. Sn 333202_0902.qxd 12/5/05 11:29 AM Page 657 Section 9.2 Arithmetic Sequences and Partial Sums 657 The sum of the first n th partial sum. th partial sum can be found by using the formula for the sum of a finite terms of an infinite sequence is the n n The arithmetic sequence. Example 7 Finding a Partial Sum of an Arithmetic Sequence Find the 150th partial sum of the arithmetic sequence 5, 16, 27, 38, 49, . . . . Solution For this arithmetic sequence, 5 and d 16 5 11. So, a1 c a1 n th term is d 5 11 6 11n 6. and the the sum of the first 150 terms is an Therefore, a150 11150 6 1644, and S150 a150 nth partial sum formula a1 n 2 150 2 751649 123,675. 5 1644 Substitute 150 for n, 5 for a1, and 1644 for a150. Simplify. nth partial sum Now try Exercise 69. Applications Example 8 Prize Money In a golf tournament, the 16 golfers with the lowest scores win cash prizes. First place receives a cash prize of $1000, second place receives $950, third place receives $900, and so on. What is the total amount of prize money? Solution The cash prizes awarded form an arithmetic sequence in which the common difference is c a1 d 50. Because d 1000 50 1050 n th term of the sequence is is term sequence the of and the total amount of prize money is So, the you can determine that the formula for the an a16 50n 1050. 5016 1050 250, S16 1000 950 900 . . . 250 n 2 a16 16th a1 S16 nth partial sum formula 16 2 1000 250 Substitute 16 for n, 1000 for a1, and 250 for a16. 81250 $10,000. Simplify. Now try Exercise 89. 333202_0902.qxd 12/5/05 11:30 AM Page 658 658 Chapter 9 Sequences, Series, and Probability Example 9 Total Sales A small business sells $10,000 worth of skin care products during its first year. The owner of the business has set a goal of increasing annual sales by $7500 each year for 9 years. Assuming that this goal is met, find the total sales during the first 10 years this business is in operation. Solution The annual sales form an arithmetic sequence in which d 7500. So, c a1 d 10,000 7500 2500 10,000 and a1 n and the th term of the sequence is 7500n 2500. an This implies that the 10th term of the sequence is a10 750010 2500 77,500. See Figure 9.4. The sum of the first 10 terms of the sequence is Small Business an a n = 7500n + 2500 S10 1 2 3 4 5 6 7 8 9 10 Year n n 2 10 2 a1 a10 nth partial sum formula 10,000 77,500 Substitute 10 for 10,000 for n, a1, and 77,500 for a10. 587,500 437,500. Simplify. Simplify. So, the total sales for the first 10 years will be $437,500. Now try Exercise 91. W RITING ABOUT MATHEMATICS Numerical Relationships Decide whether it is possible to fill in the blanks in each of the sequences such that the resulting sequence is arithmetic. If so, find a recursion formula for the sequence. a. 7, b. 17, c. 2, 6, , , , d. 4, 7.5, e. 8, 12, , , , , , , , , , 11 , , , , 71 , 162 , , , , , , , , 39 , 60.75 80,000 60,000 40,000 20,000 ) FIGURE 9.4 333202_0902.qxd 12/5/05 11:30 AM Page 659 Section 9.2 Arithmetic Sequences and Partial Sums 659 9.2 Exercises VOCABULARY CHECK: Fill in the blanks. 1. A sequence is called an ________ sequence if the differences between two consecutive terms are the same. This difference is called the ________ difference. 2. The th term of an arithmetic sequence has the form ________. n 3. The formula Sn n 2 a1 an can be used to find the sum of the first n terms of an arithmetic sequence, called the ________ of a ________ ________ ________. PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. In Exercises 1–10, determine whether the sequence is arithmetic. If so, find the common difference. 10, 8, 6, 4, 2, . . . 2. 4, 7, 10, 13, 16, . . . 28. 29. 30. a1 a3 a5 16 85 4, a5 94, a6 190, a10 115 4. 80, 40, 20, 10, 5, 2, 2, 3 2, 1, . . . 3, 5 6. . . . In Exercises 31–38, write the first five terms of the arithmetic sequence. 1. 3. 5. 7. 8. 9. 10. 1, 2, 4, 8, 16, . . . 9 2, 5 4, 3 4, 2, 7 4, . . . 5 1 2 4 3, 3, 6, . . . 3, 1, 5.3, 5.7, 6.1, 6.5, 6.9, . . . ln 1, ln 2, ln 3, ln 4, ln 5, . . . 12, 22, 32, 42, 52, . . . In Exercises 11–18, write the first five terms of the sequence. Determine whether the sequence is arithmetic. If so, find the common difference. (Assume that begins with 1.) n 12. 14. an an 100 3n 1 n 14 11. 13. 15. 16. an an an an 5 3n 3 4n 2 1n 2n1 1n3 n 2n n 17. an 18. an In Exercises 19–30, find a formula for sequence. an for the arithmetic 19. 20. 21. 22. 23. 24. 25. 26. 27. 1, d 3 15, d 4 100, d 8 0, d 2 3 x, d 2x y, d 5y 2, 1, 7 2 , . . . a1 a1 a1 a1 a1 a1 4, 3 10, 5, 0, 5, 10, . . . a1 5, a4 15 31. 32. 33. 34. 35. 36. 37. 38. a1 a1 a1 a1 a1 a4 a8 a3 5, d 6 5, d 3 4 2.6, d 0.4 16.5, d 0.25 2, a12 16, a10 26, a12 19, a15 46 46 42 1.7 In Exercises 39–44, write the first five terms of the arithmetic sequence. Find the common difference and write the th term of the sequence as a function of n. n 39. 40. 41. 42. 43. 44. a1 a1 a1 a1 a1 a1 4 5 ak ak 15, ak1 6, ak1 200, ak1 ak ak 72, ak1 5 ak 8, ak1 0.375, ak1 1 8 ak 10 6 0.25 In Exercises 45–48, the first two terms of the arithmetic sequence are given. Find the missing term. 45. 46. 47. 48. a1 a1 a1 a1 5, 11, a2 3, 13, a2 4.2, a2 0.7, a7 13.8, a10 a9 6.6, a2 a8 333202_0902.qxd 12/5/05 11:30 AM Page 660 660 Chapter 9 Sequences, Series, and Probability In Exercises 49–52, match the arithmetic sequence with its graph. [The graphs are labeled (a), (b), (c), and (d).] (a) an (b) an 24 18 12 6 − 6 (c) an 10 8 6 4 2 −2 2 4 6 8 2 4 6 8 10 8 6 4 2 −2 −4 (d) an 30 24 18 12 6 −6 n n 2 4 6 8 10 2 4 6 8 10 n n 49. 51. an an 3 2 3 4 n 8 4 n 50. 52. an an 3n 5 25 3n 71. 73. 30 n11 n 10 n1 n 400 n1 2n 1 72. 74. 100 n51 n 50 n1 n 250 n1 1000 n In Exercises 75–80, use a graphing utility to find the partial sum. 20 1000 5n 2n 5 50 76. 75. n1 100 n1 60 i1 n 4 2 3i 250 8 77. 79. n0 78. 80. 100 n0 8 3n 16 200 j1 4.5 0.025j Job Offer the given starting salary and the given annual raise. In Exercises 81 and 82, consider a job offer with (a) Determine the salary during the sixth year of employment. (b) Determine the total compensation from the company through six full years of employment. Starting Salary Annual Raise In Exercises 53–56, use a graphing utility to graph the first 10 terms of the sequence. (Assume that begins with 1.) n 81. $32,500 82. $36,800 $1500 $1750 53. 55. an an 15 3 2n 0.2n 3 54. 56. an an 5 2n 0.3n 8 In Exercises 57– 64, find the indicated th partial sum of the arithmetic sequence. n 57. 8, 20, 32, 44, . . . , 58. 2, 8, 14, 20, . . . , n 10 n 25 59. 4.2, 3.7, 3.2, 2.7, . . . , 6
0. 0.5, 0.9, 1.3, 1.7, . . . , n 12 n 10 61. 40, 37, 34, 31, . . . , 63. 62. 75, 70, 65, 60, 100, a25 15, a100 a1 a1 64. . . . , 220, 307, n 10 n 25 n 25 n 100 65. Find the sum of the first 100 positive odd integers. 66. Find the sum of the integers from 10 to 50. In Exercises 67–74, find the partial sum. 67. 69. 50 n1 n 100 n10 6n 68. 70. 100 n1 2n 100 n51 7n 83. Seating Capacity Determine the seating capacity of an auditorium with 30 rows of seats if there are 20 seats in the first row, 24 seats in the second row, 28 seats in the third row, and so on. 84. Seating Capacity Determine the seating capacity of an auditorium with 36 rows of seats if there are 15 seats in the first row, 18 seats in the second row, 21 seats in the third row, and so on. 85. Brick Pattern A brick patio has the approximate shape of a trapezoid (see figure). The patio has 18 rows of bricks. The first row has 14 bricks and the 18th row has 31 bricks. How many bricks are in the patio? 31 14 FIGURE FOR 85 FIGURE FOR 86 86. Brick Pattern A triangular brick wall is made by cutting some bricks in half to use in the first column of every other row. The wall has 28 rows. The top row is one-half brick wide and the bottom row is 14 bricks wide. How many bricks are used in the finished wall? 333202_0902.qxd 12/5/05 11:30 AM Page 661 Section 9.2 Arithmetic Sequences and Partial Sums 661 87. Falling Object An object with negligible air resistance is dropped from a plane. During the first second of fall, the object falls 4.9 meters; during the second second, it falls 14.7 meters; during the third second, it falls 24.5 meters; it falls 34.3 meters. If this during the fourth second, arithmetic pattern continues, how many meters will the object fall in 10 seconds? 88. Falling Object An object with negligible air resistance is dropped from the top of the Sears Tower in Chicago at a height of 1454 feet. During the first second of fall, the object falls 16 feet; during the second second, it falls 48 feet; during the third second, it falls 80 feet; during the fourth second, it falls 112 feet. If this arithmetic pattern continues, how many feet will the object fall in 7 seconds? 89. Prize Money A county fair is holding a baked goods competition in which the top eight bakers receive cash prizes. First places receives a cash prize of $200, second place receives $175, third place receives $150, and so on. (b) Find the total amount of interest paid over the term of the loan. 94. Borrowing Money You borrowed $5000 from your parents to purchase a used car. The arrangements of the loan are such that you will make payments of $250 per month plus 1% interest on the unpaid balance. (a) Find the first year’s monthly payments you will make, and the unpaid balance after each month. (b) Find the total amount of interest paid over the term of the loan. Model It 95. Data Analysis: Personal Income The table shows in the United States (Source: U.S. Bureau of the per capita personal income from 1993 to 2003. Economic Analysis) an (a) Write a sequence an awarded in terms of the place good places. that represents the cash prize in which the baked n (b) Find the total amount of prize money awarded at the competition. 90. Prize Money A city bowling league is holding a tournament in which the top 12 bowlers with the highest three-game totals are awarded cash prizes. First place will win $1200, second place $1100, third place $1000, and so on. (a) Write a sequence an awarded in terms of the place finishes. that represents the cash prize in which the bowler n (b) Find the total amount of prize money awarded at the tournament. 91. Total Profit A small snowplowing company makes a profit of $8000 during its first year. The owner of the company sets a goal of increasing profit by $1500 each year for 5 years. Assuming that this goal is met, find the total profit during the first 6 years of this business. What kinds of economic factors could prevent the company from meeting its profit goal? Are there any other factors that could prevent the company from meeting its goal? Explain. 92. Total Sales An entrepreneur sells $15,000 worth of sports memorabilia during one year and sets a goal of increasing annual sales by $5000 each year for 9 years. Assuming that this goal is met, find the total sales during the first 10 years of this business. What kinds of economic factors could prevent the business from meeting its goals? 93. Borrowing Money You borrowed $2000 from a friend to purchase a new laptop computer and have agreed to pay back the loan with monthly payments of $200 plus 1% interest on the unpaid balance. (a) Find the first six monthly payments you will make, and the unpaid balance after each month. Year Per capita personal income, an 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 $21,356 $22,176 $23,078 $24,176 $25,334 $26,880 $27,933 $29,848 $30,534 $30,913 $31,633 (a) Find an arithmetic sequence that models the data. corresponding represent the year, with n 3 n Let to 1993. (b) Use the regression feature of a graphing utility to find a linear model for the data. How does this model compare with the arithmetic sequence you found in part (a)? (c) Use a graphing utility to graph the terms of the finite sequence you found in part (a). (d) Use the sequence from part (a) to estimate the per capita personal income in 2004 and 2005. (e) Use your school’s library, the Internet, or some other reference source to find the actual per capita personal income in 2004 and 2005, and compare these values with the estimates from part (d). 333202_0902.qxd 12/8/05 10:53 AM Page 662 662 Chapter 9 Sequences, Series, and Probability 96. Data Analysis: Revenue The table shows the annual (in millions of dollars) for Nextel (Source: revenue Communications, Inc. from 1997 to 2003. Nextel Communications, Inc.) an Year Revenue, an 1997 1998 1999 2000 2001 2002 2003 739 1847 3326 5714 7689 8721 10,820 (a) Construct a bar graph showing the annual revenue from 1997 to 2003. (b) Use the linear regression feature of a graphing utility to find an arithmetic sequence that approximates the annual revenue from 1997 to 2003. (c) Use summation notation to represent the total revenue from 1997 to 2003. Find the total revenue. (d) Use the sequence from part (b) to estimate the annual revenue in 2008. (d) Compare the slope of the line in part (b) with the common difference of the sequence in part (a). What can you conclude about the slope of a line and the common difference of an arithmetic sequence? 102. Pattern Recognition (a) Compute the following sums of positive odd integers 11 (b) Use the sums in part (a) to make a conjecture about the sums of positive odd integers. Check your conjecture for the sum 1 3 5 7 9 11 13 . (c) Verify your conjecture algebraically. 103. Think About It The sum of the first 20 terms of an arithmetic sequence with a common difference of 3 is 650. Find the first term. terms of an arithand common difference is Determine the sum if each term is increased by 5. 104. Think About It The sum of the first metic sequence with first term Sn. d Explain. a1 n Synthesis Skills Review True or False? the statement is true or false. Justify your answer. In Exercises 97 and 98, determine whether In Exercises 105–108, find the slope and y-intercept (if possible) of the equation of the line. Sketch the line. 97. Given an arithmetic sequence for which only the first two terms are known, it is possible to find the th term. n 98. If the only known information about a finite arithmetic sequence is its first term and its last term, then it is possible to find the sum of the sequence. 105. 106. 107. 108. 2x 4y 3 9x y 8 x 7 0 y 11 0 99. Writing In your own words, explain what makes a sequence arithmetic. In Exercises 109 and 110, use Gauss-Jordan elimination to solve the system of equations. 100. Writing Explain how to use the first two terms of an 109. arithmetic sequence to find the th term. n 101. Exploration (a) Graph the first 10 terms of the arithmetic sequence an 2 3n. (b) Graph the equation of the line y 3x 2. (c) Discuss any differences between the graph of 110. 3x 6x 2x x 5x 8x y 2y 5y 7z 4z z 10 17 20 4y 3y 2y 10z z 3z 4 31 5 2 3n an and the graph of y 3x 2. 111. Make a Decision To work an extended application analyzing the median sales price of existing one-family homes in the United States from 1987 to 2003, visit this text’s website at college.hmco.com. (Data Source: National Association of Realtors) 333202_0903.qxd 12/5/05 11:32 AM Page 663 9.3 Geometric Sequences and Series Section 9.3 Geometric Sequences and Series 663 What you should learn • Recognize, write, and find the nth terms of geometric sequences. • Find nth partial sums of geometric sequences. • Find the sum of an infinite geometric series. • Use geometric sequences to model and solve real-life problems. Why you should learn it Geometric sequences can be used to model and solve reallife problems. For instance, in Exercise 99 on page 670, you will use a geometric sequence to model the population of China. © Bob Krist/Corbis Geometric Sequences In Section 9.2, you learned that a sequence whose consecutive terms have a common difference is an arithmetic sequence. In this section, you will study another important type of sequence called a geometric sequence. Consecutive terms of a geometric sequence have a common ratio. Definition of Geometric Sequence A sequence is geometric if the ratios of consecutive terms are the same. So, r the sequence such that is geometric if there is a number a1, a2, a3, a4, . . . , an . . . a2 a1 r, a3 a2 r, a4 a3 r, r 0 and so the number r is the common ratio of the sequence. Example 1 Examples of Geometric Sequences a. The sequence whose n th term is 2n is geometric. For this sequence, the common ratio of consecutive terms is 2. 2, 4, 8, 16, . . . , 2n, . . . Begin with n 1. 4 2 2 b. The sequence whose 43n common ratio of consecutive terms is 3. 12, 36, 108, 324, . . . , 43n, . . . n th term is is geometric. For
this sequence, the Begin with n 1. 36 12 3 c. The sequence whose n th term is n 1 is geometric. For this sequence, the 3 1 3. common ratio of consecutive terms is 1 81 , . . . , 1 3 , 1 27 1 3 Begin with , . . . n 1. n 1 9 , , 19 13 1 3 Now try Exercise 1. The sequence 1, 4, 9, 16, . . . , whose th term is n n2, is not geometric. The ratio of the second term to the first term is a2 a1 4 1 4 but the ratio of the third term to the second term is a3 a2 9 4 . 333202_0903.qxd 12/5/05 11:32 AM Page 664 664 Chapter 9 Sequences, Series, and Probability In Example 1, notice that each of the geometric sequences has an n th term that is of the form where the common ratio of the sequence is A geometric sequence may be thought of as an exponential function whose domain is the set of natural numbers. ar n, r. The nth Term of a Geometric Sequence The th term of a geometric sequence has the form n an a1r n1 r is the common ratio of consecutive terms of the sequence. So, every where geometric sequence can be written in the following form. a1, a2, a3, a4, a5, . . . . . , an, . . . . . a1, a1r, a1r2, a1r3, a1r 4 , . . . , a1rn1, . . . n th term of a geometric sequence, you can find the If you know the ran. term by multiplying by That is, an1 r. n 1th Example 2 Finding the Terms of a Geometric Sequence Write the first five terms of the geometric sequence whose first term is whose common ratio is and Then graph the terms on a set of coordinate axes. r 2. a1 3 an 50 40 30 20 10 1 2 3 4 5 n Solution Starting with 3, repeatedly multiply by 2 to obtain the following. a1 a2 a3 a4 a5 3 321 6 322 12 323 24 324 48 1st term 2nd term 3rd term 4th term 5th term FIGURE 9.5 Figure 9.5 shows the first five terms of this geometric sequence. Now try Exercise 11. Example 3 Finding a Term of a Geometric Sequence Find the 15th term of the geometric sequence whose first term is 20 and whose common ratio is 1.05. Solution a15 a1r n1 201.05151 39.599 Formula for geometric sequence Substitute 20 for a1, 1.05 for r, and 15 for n. Use a calculator. Now try Exercise 27. 333202_0903.qxd 12/5/05 11:32 AM Page 665 Section 9.3 Geometric Sequences and Series 665 Example 4 Finding a Term of a Geometric Sequence Find the 12th term of the geometric sequence 5, 15, 45, . . . . Solution The common ratio of this sequence is r 15 5 3. a1 an Because the first term is a1r n1 53121 5177,147 885,735. a12 5, you can determine the 12th term n 12 to be Formula for geometric sequence Substitute 5 for a1, 3 for r, and 12 for n. Use a calculator. Simplify. Now try Exercise 35. If you know any two terms of a geometric sequence, you can use that infor- mation to find a formula for the th term of the sequence. n Example 5 Finding a Term of a Geometric Sequence The fourth term of a geometric sequence is 125, and the 10th term is Find the 14th term. (Assume that the terms of the sequence are positive.) 12564. Solution The 10th term is related to the fourth term by the equation r is the common Remember that ratio of consecutive terms of a geometric sequence. So, in Example 5, a10 a1r 9 a1 r r r r 6 a1 a2 a4 a3 a1 a3 a2 a4r 6. a10 a4r 6. a10 Because 12564 Multiply 4th term by r 104. and a4 125, you can solve for as follows. r r 6 125 64 1 64 1 2 125r6 Substitute 125 64 for a10 and 125 for a4. r 6 r Divide each side by 125. Take the sixth root of each side. You can obtain the 14th term by multiplying the 10th term by r 4. a14 4 a10r 4 1 125 2 64 125 1024 Multiply the 10th term by r1410. Substitute 125 64 for a10 and 1 2 for r. Simplify. Now try Exercise 41. 333202_0903.qxd 12/5/05 11:32 AM Page 666 666 Chapter 9 Sequences, Series, and Probability The Sum of a Finite Geometric Sequence The formula for the sum of a finite geometric sequence is as follows. The Sum of a Finite Geometric Sequence The sum of the finite geometric sequence a1, a1r, a1r 2, a1r 3, a1r 4, . . . , a1r n1 n is given by Sn r 1 with common ratio i1 a1 r i1 a11 r n 1 r . For a proof of the sum of a finite geometric sequence, see Proofs in Mathematics on page 723. Example 6 Finding the Sum of a Finite Geometric Sequence Find the sum 12 i1 40.3i1. Solution By writing out a few terms, you have 12 i1 40.3i1 40.30 40.31 40.32 . . . 40.311. n 12, Now, because sum of a finite geometric sequence to obtain r 0.3, 4, and a1 you can apply the formula for the Sn a11 r n 1 r 12 i1 40.3i1 41 0.312 1 0.3 5.714. Formula for the sum of a sequence Substitute 4 for a1, 0.3 for r, and 12 for n. Use a calculator. Now try Exercise 57. When using the formula for the sum of a finite geometric sequence, be careful to check that the sum is of the form n i1 a1 r i1. Exponent for r is i 1. If the sum is not of this form, you must adjust the formula. For instance, if the sum in Example 6 were 12 i1 40.3i, then you would evaluate the sum as follows. 12 i1 40.3i 40.3 40.32 40.33 . . . 40.312 40.3 40.30.3 40.30.32 . . . 40.30.311 40.31 0.312 1 0.3 40.3, r 0.3, n 12 1.714. a1 333202_0903.qxd 12/5/05 11:32 AM Page 667 Exploration Use a graphing utility to graph and What for 2, 3, 5. x → ? happens as Use a graphing utility to graph y 1 r x 1 r r 1.5, for happens as x → ? 2, and 3. What Section 9.3 Geometric Sequences and Series 667 Geometric Series The summation of the terms of an infinite geometric sequence is called an infinite geometric series or simply a geometric series. r, The formula for the sum of a finite geometric sequence can, depending on the value of be extended to produce a formula for the sum of an infinite geometric series. Specifically, if the common ratio has the property that it r n can be shown that increases without bound. Consequently, becomes arbitrarily close to zero as r < 1, n r a11 r n 1 r a11 0 1 r as n . This result is summarized as follows. The Sum of an Infinite Geometric Series the infinite geometric series If a1r a1r2 a1r3 . . . a1r n1 . . . r < 1, a1 has the sum S i0 a1r i a1 1 r . Note that if r ≥ 1, the series does not have a sum. Example 7 Finding the Sum of an Infinite Geometric Series Find each sum. 40.6n 1 n1 3 0.3 0.03 0.003 . . . a. b. Solution a. n1 40.6n 1 4 40.6 40.62 40.63 . . . 40.6n 1 . . . 4 1 0.6 10 a 1 1 r b. 3 0.3 0.03 0.003 . . . 3 30.1 30.12 30.13 . . . 3 1 0.1 a 1 1 r 10 3 3.33 Now try Exercise 79. 333202_0903.qxd 12/5/05 11:32 AM Page 668 668 Chapter 9 Sequences, Series, and Probability Application Example 8 Increasing Annuity Recall from Section 3.1 that the formula for compound interest is A P1 r n nt . A deposit of $50 is made on the first day of each month in a savings account that pays 6% compounded monthly. What is the balance at the end of 2 years? (This type of savings plan is called an increasing annuity.) Solution The first deposit will gain interest for 24 months, and its balance will be P, 0.06 is the interest So, in Example 8, $50 is the principal r, rate 12 is the number of n, and compoundings per year 2 is the time in years. If you substitute these values into the formula, you obtain t A 501 0.06 12 501 0.06 12 122 24 . A24 24 501 0.06 12 501.00524. The second deposit will gain interest for 23 months, and its balance will be A23 23 501 0.06 12 501.00523. The last deposit will gain interest for only 1 month, and its balance will be A1 501 0.06 12 1 501.005. The total balance in the annuity will be the sum of the balances of the 24 deposits. Using the formula for the sum of a finite geometric sequence, with A1 501.005 r 1.005, and S24 501.0051 1.00524 1 1.005 you have 501.005 for Substitute and 24 for n. r, 1.005 for A1, $1277.96. Simplify. Now try Exercise 107. W RITING ABOUT MATHEMATICS An Experiment You will need a piece of string or yarn, a pair of scissors, and a tape measure. Measure out any length of string at least 5 feet long. Double over the string and cut it in half. Take one of the resulting halves, double it over, and cut it in half. Continue this process until you are no longer able to cut a length of string in half. How many cuts were you able to make? Construct a sequence of the resulting string lengths after each cut, starting with the original length of the string. Find a formula for the nth term of this sequence. How many cuts could you theoretically make? Discuss why you were not able to make that many cuts. 333202_0903.qxd 12/5/05 11:32 AM Page 669 Section 9.3 Geometric Sequences and Series 669 9.3 Exercises VOCABULARY CHECK: Fill in the blanks. 1. A sequence is called a ________ sequence if the ratios between consecutive terms are the same. This ratio is called the ________ ratio. 2. The th term of a geometric sequence has the form ________. n 3. The formula for the sum of a finite geometric sequence is given by ________. 4. The sum of the terms of an infinite geometric sequence is called a ________ ________. 5. The formula for the sum of an infinite geometric series is given by ________. PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. In Exercises 1–10, determine whether the sequence is geometric. If so, find the common ratio. In Exercises 35– 42, find the indicated geometric sequence. n th term of the 1. 3. 5. 7. 9. 5, 15, 45, 135,. . . . . 3, 12, 21, 30,. 1, 1 4, 1 2, 1 8,. . . 1 8, 1 4, 1 . . 2, 1,. 3, 1 2, 1 1, 1 4,. . . 2. 4. 6. 8. 10. 3, 12, 48, 192,. . . 36, 27, 18, 9,. . . 5, 1, 0.2, 0.04,. 9, 6, 4, 8 3,. 1 5, 2 9, 4 . 11,. . 7, 3 . . . . In Exercises 11–20, write the first five terms of the geometric sequence. 11. 13. 15. 17. a1 a1 a1 a1 19. a1 2, r 3 1, r 1 2 5, r 1 10 1, r e 2, r x 4 12. 14. 16. 18. 6, r 2 1, r 1 3 6, r 1 4 3, r 5 a1 a1 a1 a1 20. a1 5, r 2x In Exercises 21–26, write the first five terms of the geometric sequence. Determine the common ratio and write the th term of the sequence as a function of n 21. 23. 25. a1 a1 a1 64, ak1 7, ak1 6, ak1 1 2ak 2a k 3 2ak 22. 24. 26. n. 81, ak1 5, ak1 48, ak1 a1 a1 a1 1 3ak 2ak 1 2 ak In Exercises 27–34, write an expression for the th term of the geometric sequence. Then find the indicated term. 2, n 8 27. 28. n 5, r 3 64, r 1 a1 a1 30. 4, n
10 2, n 10 4, r 1 6, r 1 3, n 12 100, r ex, n 9 1, r 3, n 8 500, r 1.02, n 40 1000, r 1.005, n 60 a1 a1 a1 a1 a1 a1 29. 31. 32. 33. 34. 35. 9th term: 7, 21, 63, . . . 36. 7th term: 3, 36, 432, . . . 37. 10th term: 5, 30, 180, . . . 40. 1st term: 39. 3rd term: 38. 22nd term: 4, 8, 16, . . . 27 a1 4 a2 a4 a3 16, 3, a5 18, 16 a5 3 , a4 3 64 2 3 41. 6th term: 42. 7th term: a7 64 27 In Exercises 43– 46, match the geometric sequence with its graph. [The graphs are labeled (a), (b), (c), and (d).] (a) an (b) an 20 16 12 8 4 −4 (c) an 18 12 6 −2 −12 −18 43. 45. an an 750 600 450 300 150 −2 (d) an 600 400 200 −200 −400 −600 n n 2 4 6 8 10 2 8 10 n n 2 4 6 8 10 2 8 10 182 183 3 2 n1 n1 44. an 46. an 182 183 3 n1 n1 2 333202_0903.qxd 12/5/05 11:32 AM Page 670 670 Chapter 9 Sequences, Series, and Probability 47. In Exercises 47–52, use a graphing utility to graph the first 10 terms of the sequence. 120.75n1 120.4n1 21.3n1 101.5n1 201.25n1 101.2n1 51. 49. 50. 48. 52. an an an an an an In Exercises 53–72, find the sum of the finite geometric sequence. 9 2 n1 n1 10 53. 54. 5 2 n1 53 2 8 n1 n1 i1 21 4 10 i1 161 2 12 i1 i1 53 5 n 40 n0 101 5 n 20 n0 6 n0 5001.04n n1 55. 57. 59. 61. 63. 65. 67. 69. 71. 9 n1 2n1 641 2 7 i1 i1 321 4 6 i1 i1 20 33 2 n0 15 24 3 n0 n n 5 n0 3001.06n 21 4 n 40 n0 81 4 10 i1 51 3 10 i1 i1 i1 56. 58. 60. 62. 64. 66. 68. 70. 72. 4 n n0 n0 41 0.4n n0 8 6 9 2 1 9 1 3 30.9n . . . 27 8 1 3 . . . 83. 85. 87. 89. 91. n n0 n0 1 10 40.2n n0 9 6 4 8 3 125 25 6 36 100.2n . . . 5 6 . . . 84. 86. 88. 90. 92. In Exercises 93–96, find the rational number representation of the repeating decimal. 93. 0.36 95. 0.318 94. 0.297 96. 1.38 Graphical Reasoning In Exercises 97 and 98, use a graphing utility to graph the function. Identify the horizontal asymptote of the graph and determine its relationship to the sum. 97. 98. f x 61 0.5x 1 0.5, f x 21 0.8x 1 0.8, n 0 n 0 n n 61 2 24 5 102 3 50 n0 n1 Model It 81 2 i 25 i0 152 3 100 i1 i1 99. Data Analysis: Population The table shows the of China (in millions) from 1998 through population 2004. an (Source: U.S. Census Bureau) In Exercises 73–78, use summation notation to write the sum. 73. 74. 75. 76. 77. 78. 5 15 45 . . . 3645 7 14 28 . . . 896 2 1 1 8 2 15 3 3 5 . . . 1 2048 . . 3 625 . 0.1 0.4 1.6 . . . 102.4 32 24 18 . . . 10.125 In Exercises 79–92, find the sum of the infinite geometric series. Year 1998 1999 2000 2001 2002 2003 2004 Population, an 1250.4 1260.1 1268.9 1276.9 1284.3 1291.5 1298.8 (a) Use the exponential regression feature of a graphing utility to find a geometric sequence that models the data. Let represent the year, with n 8 corresponding to 1998. n 79. 81. n0 n0 n 1 2 1 2 n 80. 82. n0 n0 22 3 n 22 3 n (b) Use the sequence from part (a) to describe the rate at which the population of China is growing. 333202_0903.qxd 12/5/05 11:32 AM Page 671 Model It (co n t i n u e d ) (c) Use the sequence from part (a) to predict the population of China in 2010. The U.S. Census Bureau predicts the population of China will be 1374.6 million in 2010. How does this value compare with your prediction? (d) Use the sequence from part (a) to determine when the population of China will reach 1.32 billion. 100. Compound Interest A principal of $1000 is invested at 6% interest. Find the amount after 10 years if the interest (b) semiannually, (a) annually, is compounded (c) quarterly, (d) monthly, and (e) daily. 101. Compound Interest A principal of $2500 is invested at 2% interest. Find the amount after 20 years if the interest (b) semiannually, (a) annually, is compounded (c) quarterly, (d) monthly, and (e) daily. 102. Depreciation A tool and die company buys a machine for $135,000 and it depreciates at a rate of 30% per year. (In other words, at the end of each year the depreciated value is 70% of what it was at the beginning of the year.) Find the depreciated value of the machine after 5 full years. 103. Annuities A deposit of $100 is made at the beginning of each month in an account that pays 6%, compounded monthly. The balance in the account at the end of 5 years is A 1001 0.06 12 . . . 1001 0.06 12 60 . 1 A Find A. 104. Annuities A deposit of $50 is made at the beginning of each month in an account that pays 8%, compounded monthly. The balance in the account at the end of 5 years is A 501 0.08 12 . . . 501 0.08 12 60 . 1 A Find A. 105. Annuities A deposit of r, P dollars is made at the beginning of each month in an account earning an annual after interest rate t years is A P1 r 12 P1 r 12 compounded monthly. The balance . . . 2 A Show that the balance is A P1 r 12 12t 11 12 r P1 r 12 12t . . Section 9.3 Geometric Sequences and Series 671 106. Annuities A deposit of compounded continuously. The balance P dollars is made at the beginning of each month in an account earning an annual r, A interest rate A Per12 Pe 2r12 . . . Pe12tr12. t years is after Show that the balance is A Per12er t 1 . er12 1 Annuities In Exercises 107–110, consider making monthly deposits of dollars in a savings account earning an annual P interest rate Use the results of Exercises 105 and 106 to find the balance years if the interest is A compounded (a) monthly and (b) continuously. after r. t 107. 108. 109. 110. P $50, r 7%, t 20 years P $75, r 3%, t 25 years P $100, r 10%, t 40 years P $20, r 6%, t 50 years P r, 111. Annuities Consider an initial deposit of dollars in an compounded account earning an annual interest rate W monthly. At the end of each month, a withdrawal of t dollars will occur and the account will be depleted in years. The amount of the initial deposit required is 2 W1 r P W1 r 12 12 W1 r 12 . . . 12t 1 . Show that the initial deposit is 12t. P W12 r 1 1 r 12 112. Annuities Determine the amount required in a retirement account for an individual who retires at age 65 and wants an income of $2000 from the account each month for 20 years. Use the result of Exercise 111 and assume that the account earns 9% compounded monthly. Multiplier Effect In Exercises 113–116, use the following information. A tax rebate has been given to property owners by the state government with the anticipation that of the each property owner spends approximately rebate, and in turn each recipient of this amount spends p% of what they receive, and so on. Economists refer to this exchange of money and its circulation within the economy as the “multiplier effect.” The multiplier effect operates on the idea that the expenditures of one individual become the income of another individual. For the given tax rebate, find the total amount put back into the state’s economy, if this effect continues without end. p% Tax rebate 113. $400 114. $250 115. $600 116. $450 p% 75% 80% 72.5% 77.5% 333202_0903.qxd 12/5/05 11:32 AM Page 672 672 Chapter 9 Sequences, Series, and Probability 117. Geometry The sides of a square are 16 inches in length. A new square is formed by connecting the midpoints of the sides of the original square, and two of the resulting triangles are shaded (see figure). If this process is repeated five more times, determine the total area of the shaded region. 118. Sales The annual sales an (in millions of dollars) for Urban Outfitters for 1994 through 2003 can be approximated by the model an 54.6e0.172n, n 4, 5, . . . , 13 n 4 n represents the year, with where corresponding to 1994. Use this model and the formula for the sum of a finite geometric sequence to approximate the total sales earned during this 10-year period. (Source: Urban Outfitters Inc.) 119. Salary An investment firm has a job opening with a salary of $30,000 for the first year. Suppose that during the next 39 years, there is a 5% raise each year. Find the total compensation over the 40-year period. 120. Distance A ball is dropped from a height of 16 feet. Each time it drops h feet, it rebounds 0.81h feet. (a) Find the total vertical distance traveled by the ball. (b) The ball takes the following times (in seconds) for each fall. s1 s2 s3 s4 16t 2 16, 16t 2 160.81, 16t 2 160.812, 16t 2 160.813, .. . 16t 2 160.81n1, sn 0 if t 1 0 if t 0.9 0 if t 0.9 2 0 if t 0.93 .. . 0 if t 0.9n1 s1 s2 s3 s4 sn s2, the ball takes the same amount of Beginning with time to bounce up as it does to fall, and so the total time elapsed before it comes to rest is t 1 2 0.9n . n 1 122. You can find the n th term of a geometric sequence by multiplying its common ratio by the first term of the sequence raised to the n 1 th power. 123. Writing Write a brief paragraph explaining why the terms of a geometric sequence decrease in magnitude when 1 < r < 1. 124. Find two different geometric series with sums of 4. Skills Review the function for 125. and gx x 2 1. In Exercises 125–128, evaluate f x 3x 1 gx 1 f x 1 f gx 1 g f x 1 126. 127. 128. In Exercises 129–132, completely factor the expression. 129. 130. 131. 132. 9x3 64x x2 4x 63 6x2 13x 5 16x2 4x 4 In Exercises 133–138, perform the indicated operation(s) and simplify. xx 3 x 3 2xx 7 6xx 2 6x 3 10 2x 23 x 2 x 2 133. 134. 135. 136. 137. 138 3x 1x 4 Find this total time. Synthesis True or False? In Exercises 121 and 122, determine whether the statement is true or false. Justify your answer. 121. A sequence is geometric if the ratios of consecutive differences of consecutive terms are the same. 139. Make a Decision To work an extended application analyzing the amounts spent on research and development in the United States from 1980 to 2003, visit this text’s website at college.hmco.com. (Data Source: U.S. Census Bureau) 333202_0904.qxd 12/5/05 11:35 AM Page 673 9.4 Mathematical Induction Section 9.4 Mathematical Induction 673 What you should learn • Use mathematical induction to prove statements involving a positive integer n. • Recognize patterns and write the th term of a sequence. • Find the sums of powers of n integers. • Find finite differences of sequences. Why you should learn it Finite differences can be used to determine what type of model can be used to represent a sequence. For instance, in Exercise 61 on page 682,
you will use finite differences to find a model that represents the number of individual income tax returns filed in the United States from 1998 to 2003. Introduction In this section, you will study a form of mathematical proof called mathematical induction. It is important that you see clearly the logical need for it, so take a closer look at the problem discussed in Example 5 in Section 9.2. S1 S2 S3 S4 S5 1 12 1 3 22 1 3 5 32 1 3 5 7 42 1 3 5 7 9 52 Judging from the pattern formed by these first five sums, it appears that the sum of the first odd integers is n 1 3 5 7 9 . . . 2n 1 n2. Sn Although this particular formula is valid, it is important for you to see that recognizing a pattern and then simply jumping to the conclusion that the pattern n is not a logically valid method of proof. There are must be true for all values of many examples in which a pattern appears to be developing for small values of n and then at some point the pattern fails. One of the most famous cases of this was the conjecture by the French mathematician Pierre de Fermat (1601–1665), who speculated that all numbers of the form 22n 1, n 0, 1, 2, . . . n 0, 1, 2, 3, and 4, the conjecture is true. are prime. For Fn Mario Tama/Getty Images F0 F1 F2 F3 F4 3 5 17 257 65,537 The size of the next Fermat number is so great that it was difficult for Fermat to determine whether it was prime or not. However, another later found the well-known mathematician, Leonhard Euler (1707–1783), factorization 4,294,967,297 F5 F5 4,294,967,297 6416,700,417 which proved that F5 is not prime and therefore Fermat’s conjecture was false. Just because a rule, pattern, or formula seems to work for several values of you cannot simply decide that it is valid for all values of without going n, through a legitimate proof. Mathematical induction is one method of proof. n 333202_0904.qxd 12/5/05 11:35 AM Page 674 674 Chapter 9 Sequences, Series, and Probability It is important to recognize that in order to prove a statement by induction, both parts of the Principle of Mathematical Induction are necessary. The Principle of Mathematical Induction n. be a statement involving the positive integer Let Pn If 1. P1 is true, and 2. for every positive integer k, the truth of Pk implies the truth of Pk1 then the statement must be true for all positive integers n. Pn To apply the Principle of Mathematical Induction, you need to be able to Pk1, To determine Pk1 Pk. determine the statement substitute the quantity k 1 for a given statement k Pk. in the statement for Example 1 A Preliminary Example Find the statement for each given statement Pk. Pk1 k 2k 12 4 a. Pk : Sk b. c. d. 1 5 9 . . . 4k 1 3 4k 3 Pk : Sk Pk : k 3 < 5k2 Pk : 3k ≥ 2k 1 Solution a. Pk1 : Sk1 k 12k 1 12 4 k 12k 22 4 Replace by k k 1. Simplify. b. Pk1 : Sk1 1 5 9 . . . 4k 1 1 3 4k 1 3 1 5 9 . . . 4k 3 4k 1 c. Pk1: k 1 3 < 5k 12 k 4 < 5k2 2k 1 d. Pk1 : 3k1 ≥ 2k 1 1 3k1 ≥ 2k 3 Now try Exercise 1. A well-known illustration used to explain why the Principle of Mathematical Induction works is the unending line of dominoes shown in Figure 9.6. If the line actually contains infinitely many dominoes, it is clear that you could not knock the entire line down by knocking down only one domino at a time. However, suppose it were true that each domino would knock down the next one as it fell. Then you could knock them all down simply by pushing the first one and starting a chain reaction. Mathematical induction works in the same way. If the truth of Pk is true, the chain reaction proceeds as implies follows: implies the truth of and so on. implies implies and if Pk1 P1 P3, P3 P2, P2 P4, P1 FIGURE 9.6 333202_0904.qxd 12/5/05 11:35 AM Page 675 Section 9.4 Mathematical Induction 675 When using mathematical induction to prove a summation formula (such as the one in Example 2), it is helpful to think of Sk1 as Sk1 Sk ak1 where ak1 is the k 1 th term of the original sum. Example 2 Using Mathematical Induction Use mathematical induction to prove the following formula. Sn 1 3 5 7 . . . 2n 1 n2 Solution Mathematical induction consists of two distinct parts. First, you must show that the formula is true when n 1, 1 12. the formula is valid, because 1. When n 1. S1 The second part of mathematical induction has two steps. The first step is to assume that the formula is valid for some integer The second step is to use this assumption to prove that the formula is valid for the next integer, k 1. k. 2. Assuming that the formula Sk 1 3 5 7 . . . 2k 1 k2 is true, you must show that the formula Sk1 k 12 is true. Sk1 1 3 5 7 . . . 2k 1 2k 1 1 1 3 5 7 . . . 2k 1 2k 2 1 2k 1 Sk k2 2k 1 k 12 Group terms to form Replace Sk. by k 2. Sk Combining the results of parts (1) and (2), you can conclude by mathematical induction that the formula is valid for all positive integer values of n. Now try Exercise 5. k 1 positive integers but is true for all values of It occasionally happens that a statement involving natural numbers is not true for the first In these instances, you use a slight variation of the Principle of Mathematical Induction This variation is called the extended P1. in which you verify principle of mathematical induction. To see the validity of this, note from Figure dominoes can be knocked down by knocking over 9.6 that all but the first k the th domino. This suggests that you can prove a statement to be true for n ≥ k In Exercises 17–22 of this section, you are asked to apply this extension of mathematical induction. by showing that is true and that rather than implies n ≥ k. k 1 Pk1. Pn Pk Pk Pk 333202_0904.qxd 12/5/05 11:35 AM Page 676 676 Chapter 9 Sequences, Series, and Probability Example 3 Using Mathematical Induction Use mathematical induction to prove the formula 12 22 32 42 . . . n2 Sn nn 12n 1 6 for all integers n ≥ 1. Solution 1. When n 1, 12 123 . S1 6 the formula is valid, because 2. Assuming that 12 22 32 42 . . . k2 Sk kk 12k 1 6 k2 ak you must show that Sk1 k 1k 1 12k 1 1 6 k 1k 22k 3 6 . To do this, write the following. Sk1 ak1 Sk 12 22 32 42 . . . k 2 k 12 kk 12k 1 6 k 12 kk 12k 1 6k 12 6 k 1k2k 1 6k 1 6 k 12k 2 7k 6 6 k 1k 22k 3 6 Substitute for Sk. By assumption Combine fractions. Factor. Simplify. Sk implies Sk1. Combining the results of parts (1) and (2), you can conclude by mathematical induction that the formula is valid for all integers n ≥ 1. Now try Exercise 11. Remember that when adding rational expressions, you must first find the least common denominator (LCD). In Example 3, the LCD is 6. When proving a formula using mathematical induction, the only statement that you need to verify is As a check, however, it is a good idea to try verifying some of the other statements. For instance, in Example 3, try verifying P2 and P3. P1. 333202_0904.qxd 12/5/05 11:35 AM Page 677 To check a result that you have proved by mathematical induction, it helps to list the n. statement for several values of For instance, in Example 4, you could list 1 < 21 2, 2 < 23 8, 5 < 25 32, 2 < 22 4, 4 < 24 16, 6 < 26 64, From this list, your intuition confirms that the statement n < 2n is reasonable. Section 9.4 Mathematical Induction 677 Example 4 Proving an Inequality by Mathematical Induction Prove that n < 2n for all positive integers n. Solution 1. For n 1 and n 2, the statement is true because 1 < 21 and 2 < 22. 2. Assuming that k < 2k you need to show that k 1 < 2k1. For n k, you have 2k1 22k > 2k 2k. 2k k k > k 1 Because 2k1 > 2k > k 1 or By assumption for all it follows that k > 1, k 1 < 2k1. Combining the results of parts (1) and (2), you can conclude by mathematical n ≥ 1. induction that for all integers n < 2n Now try Exercise 17. Example 5 Proving Factors by Mathematical Induction Prove that 3 is a factor of 4n 1 for all positive integers n. Solution 1. For n 1, 41 1 3. the statement is true because So, 3 is a factor. 2. Assuming that 3 is a factor of 4k 1, you must show that 3 is a factor of 4k1 1. To do this, write the following. 4k. Subtract and add 4k1 1 4k1 4k 4k 1 4k4 1 4k 1 4k 3 4k 1 4k 3 it follows that 3 is Combining the results of parts (1) and (2), you can for all positive Because 3 is a factor of 4k1 1. a factor of conclude by mathematical induction that 3 is a factor of integers and 3 is also a factor of Regroup terms. 4k 1, 4n 1 Simplify. n. Now try Exercise 29. Pattern Recognition Although choosing a formula on the basis of a few observations does not guarantee the validity of the formula, pattern recognition is important. Once you have a pattern or formula that you think works, you can try using mathematical induction to prove your formula. 333202_0904.qxd 12/5/05 11:35 AM Page 678 678 Chapter 9 Sequences, Series, and Probability Finding a Formula for the nth Term of a Sequence To find a formula for the th term of a sequence, consider these guidelines. n 1. Calculate the first several terms of the sequence. It is often a good idea to write the terms in both simplified and factored forms. n 2. Try to find a recognizable pattern for the terms and write a formula for the th term of the sequence. This is your hypothesis or conjecture. You might try computing one or two more terms in the sequence to test your hypothesis. 3. Use mathematical induction to prove your hypothesis. Example 6 Finding a Formula for a Finite Sum Find a formula for the finite sum and prove its validity. 2 3 1 1 1 1 1 2 Solution Begin by writing out the first few sums. 3 4 4 5 . . . 1 nn 1 S1 S2 S3 S4 12 1 2 1 3 4 48 60 From this sequence, it appears that the formula for the th sum is k Sk kk 1 k k 1 . To prove the validity of this hypothesis, use mathematical induction. Note that so you can begin by assuming you have already verified the formula for that the formula is valid for and trying to show that it is valid for n k 1. n 1, n k Sk1 kk 1 1 k 1k 2 1 k 1k 2 k k 1 kk 2 1 k 1k 2 By assumption k 2 2k 1 k 1k 2 k 12 k 1k 2 k 1 k 2 So, by mathematical induction, you can conclude that the hypothesis is valid. Now try Exercise 35. 333202_0904.qxd
12/5/05 11:35 AM Page 679 Section 9.4 Mathematical Induction 679 Sums of Powers of Integers The formula in Example 3 is one of a collection of useful summation formulas. n This and other formulas dealing with the sums of various powers of the first positive integers are as follows. Sums of Powers of Integers 1. 2. 3. 4. 1 2 3 4 . . . n nn 1 2 12 22 32 42 . . . n2 nn 12n 1 6 13 23 33 43 . . . n3 n2n 12 4 14 24 34 44 . . . n4 nn 12n 13n2 3n 1 30 5. 15 25 35 45 . . . n5 n2n 122n2 2n 1 12 Example 7 Finding a Sum of Powers of Integers Find each sum. a. 7 i1 i3 13 23 33 43 53 63 73 b. 4 i1 6i 4i 2 Solution a. Using the formula for the sum of the cubes of the first positive integers, you n obtain 7 i1 i3 13 23 33 43 53 63 73 4964 4 784. Formula 3 b. 4 i1 727 12 4 6i 4i2 4 64 i1 6i 4 i 44 i1 4i2 i2 i1 i1 644 1 2 444 18 1 6 Formula 1 and 2 610 430 60 120 60 Now try Exercise 47. 333202_0904.qxd 12/5/05 11:35 AM Page 680 680 Chapter 9 Sequences, Series, and Probability For a linear model, the first differences should be the same nonzero number. For a quadratic model, the second differences are the same nonzero number. Finite Differences The first differences of a sequence are found by subtracting consecutive terms. The second differences are found by subtracting consecutive first differences. are as The first and second differences of the sequence follows. 3, 5, 8, 12, 17, 23, . . . n: an: 1 3 First differences: 2 Second differences: 2 5 1 3 8 1 3 4 12 5 17 6 23 4 5 6 1 1 For this sequence, the second differences are all the same. When this happens, the sequence has a perfect quadratic model. If the first differences are all the same, the sequence has a linear model. That is, it is arithmetic. Example 8 Finding a Quadratic Model Find the quadratic model for the sequence 3, 5, 8, 12, 17, 23, . . . . Solution You know from the second differences shown above that the model is quadratic and has the form an an2 bn c. By substituting 1, 2, and 3 for tions in three variables. n, you can obtain a system of three linear equa- a12 b1 c 3 a22 b2 c 5 a32 b3 c 8 a1 a2 a3 Substitute 1 for n. Substitute 2 for n. Substitute 3 for n. You now have a system of three equations in a, b, and c. a b c 3 4a 2b c 5 9a 3b c 8 Equation 1 Equation 2 Equation 3 Using the techniques discussed in Chapter 7, you can find the solution to be So, the quadratic model is c 2. and a 1 2, 1 2 b 1 2, n2 1 2 an n 2. Try checking the values of a1, a2, and a3. Now try Exercise 57. 333202_0904.qxd 12/5/05 11:35 AM Page 681 Section 9.4 Mathematical Induction 681 9.4 Exercises VOCABULARY CHECK: Fill in the blanks. 1. The first step in proving a formula by ________ ________ is to show that the formula is true when n 1. 2. The ________ differences of a sequence are found by subtracting consecutive terms. 3. A sequence is an ________ sequence if the first differences are all the same nonzero number. 4. If the ________ differences of a sequence are all the same nonzero number, then the sequence has a perfect quadratic model. PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. In Exercises 1–4, find Pk1 for the given Pk. 1. Pk 3. Pk 5 kk 1 k 2k 1 2 4 2. Pk 1 2k 2 4. Pk k 3 2k 1 In Exercises 5–16, use mathematical induction to prove the formula for every positive integer n. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 2 4 6 8 . . . 2n nn 1 3 7 11 15 . . . 4n 1 n2n 1 2 7 12 17 . . . 5n 3 n 2 1 4 7 10 . . . 3n 2 n 2 3n 1 5n 1 1 2 22 23 . . . 2n1 2n 1 21 3 32 33 . . . 3n1 3n 1 1 2 3 4 . . . n nn 1 2 13 23 33 43 . . . n3 n2n 1 2 4 i5 n2n 122n2 2n 1 12 i 4 nn 12n 13n2 3n 1 30 ii 1 nn 1n 2 n i1 n i1 n i1 n i1 1 2i 12i 1 3 n 2n 1 In Exercises 17–22, prove the inequality for the indicated integer values of n. 17. 19. n! > 2n 18. n 4 3 > n, n ≥ 2 n1 n < x x , y y 1 an ≥ na, 2n2 > n 12, 20. 21. 22. n ≥ 1 and 0 < x < y and a > 0 n ≥ 1 n ≥ 3 In Exercises 23–34, use mathematical induction to prove n. the property for all positive integers n a b abn an bn an bn 24. 23. 25. If 0, x2 x1 x1 x 2 x3 . . . xn 0, . . . , xn 1 x1 1x2 0, then 1 . . . xn 1. 1x3 26. If x1 > 0, x2 > 0, . . . , xn > 0, ln ln x2 . . . xn lnx1x2 x1 then . . . ln xn . 27. Generalized Distributive Law: xy1 . . . yn a bin and xy2 . . . xyn are complex conjugates for all 28. xy1 y2 a bin n ≥ 1. 29. A factor of 30. A factor of 31. A factor of 32. A factor of 33. A factor of 34. A factor of is 3. is 3. is 2. n3 3n2 2n n3 n 3 n4 n 4 22n1 1 24n2 1 is 5. 22n1 32n1 is 3. is 5. In Exercises 35– 40, find a formula for the sum of the first terms of the sequence. n 35. 37. 39. 40. 1, 5, 9, 13, . . . 1, 9 100, 729 10, 81 1000 40 24 12 , . . . , 36. 38. 25, 22, 19, 16, . . . 3, 9 4 , 81 8 , . . . 2, 27 1 2nn 1 1n 2, . . . 333202_0904.qxd 12/5/05 11:35 AM Page 682 682 Chapter 9 Sequences, Series, and Probability In Exercises 41–50, find the sum using the formulas for the sums of powers of integers. Model It (co n t i n u e d ) 41. 43. 45. 47. 49. 15 n n1 6 n1 n2 5 n1 n4 6 n1 n2 n 6 i1 6i 8i 3 42. 44. 46. 48. 50. 30 n n1 10 n1 n3 8 n1 n5 20 n1 n3 n 3 1 2 j 1 2 j 2 10 j1 (a) Find the first differences of the data shown in the table. (b) Use your results from part (a) to determine whether a linear model can be used to approximate the data. n If so, find a model algebraically. Let represent the year, with corresponding to 1998. n 8 (c) Use the regression feature of a graphing utility to find a linear model for the data. Compare this model with the one from part (b). (d) Use the models found in parts (b) and (c) to estimate the number of individual tax returns filed in 2008. How do these values compare? In Exercises 51–56, write the first six terms of the sequence beginning with the given term. Then calculate the first and second differences of the sequence. State whether the sequence has a linear model, a quadratic model, or neither. Synthesis 51. 53. 55. a1 an a1 an a0 an 0 an1 3 an1 2 an1 2 3 n 2 52. 54. 56. a1 an a2 an a0 an 2 an1 3 2an1 0 an1 n In Exercises 57–60, find a quadratic model for the sequence with the indicated terms. 57. 58. 59. 60. a0 a0 a0 a0 3, a1 7, a1 3, a2 3, a2 3, a4 6, a3 1, a4 0, a6 15 10 9 36 Model It 61. Data Analysis: Tax Returns The table shows the (in millions) of individual tax returns filed in (Source: number the United States from 1998 to 2003. Internal Revenue Service) an Year 1998 1999 2000 2001 2002 2003 Number of returns, an 120.3 122.5 124.9 127.1 129.4 130.3 62. Writing In your own words, explain what is meant by a proof by mathematical induction. True or False? statement is true or false. Justify your answer. In Exercises 63–66, determine whether the 63. If the statement P1 P7 imply that the statement true for all positive integers is true but the true statement is true, then n. Pn P6 does not is not necessarily 64. If the statement Pk is true and Pk implies Pk1, then P1 is also true. 65. If the second differences of a sequence are all zero, then the sequence is arithmetic. 66. A sequence with n terms has n 1 second differences. Skills Review In Exercises 67–70, find the product. 67. 69. 2x2 12 5 4x3 68. 70. 2x y2 2x 4y3 In Exercises 71–74, (a) state the domain of the function, (b) identify all intercepts, (c) find any vertical and horizontal asymptotes, and (d) plot additional solution points as needed to sketch the graph of the rational function. 71. 72. 73. 74. f x x x 3 gx x2 x2 4 ht t 7 t f x 5 x 1 x 333202_0905.qxd 12/5/05 11:37 AM Page 683 9.5 The Binomial Theorem Section 9.5 The Binomial Theorem 683 What you should learn • Use the Binomial Theorem to calculate binomial coefficients. • Use Pascal’s Triangle to calculate binomial coefficients. • Use binomial coefficients to write binomial expansions. Why you should learn it You can use binomial coefficients to model and solve real-life problems. For instance, in Exercise 80 on page 690, you will use binomial coefficients to write the expansion of a model that represents the amounts of child support collected in the U. S. Binomial Coefficients Recall that a binomial is a polynomial that has two terms. In this section, you will study a formula that gives a quick method of raising a binomial to a power. To begin, look at the expansion of for several values of x yn n. x y0 1 x y1 x y x y2 x 2 2xy y 2 x y3 x3 3x 2y 3xy 2 y3 x y4 x4 4x 3y 6x 2y 2 4xy 3 y4 x y5 x 5 5x 4y 10x 3y 2 10x 2y 3 5xy4 y 5 There are several observations you can make about these expansions. n 1 terms. 1. In each expansion, there are 2. In each expansion, y x by 1 in successive terms, whereas the powers of and have symmetrical roles. The powers of decrease y increase by 1. x 3. The sum of the powers of each term is n. For instance, in the expansion of x y5, the sum of the powers of each term is 5. 4 1 5 3 2 5 x y5 x 5 5x4y1 10x3y 2 10x 2y 3 5x1y4 y 5 4. The coefficients increase and then decrease in a symmetric pattern. The coefficients of a binomial expansion are called binomial coefficients. To © Vince Streano/Corbis find them, you can use the Binomial Theorem. The Binomial Theorem In the expansion of x yn x yn x n nxn1y . . . xnr y r the coefficient of is nCr x n1y r . . . nxyn1 yn nCr . n! n r!r! n r The symbol is often used in place of nCr to denote binomial coefficients. For a proof of the Binomial Theorem, see Proofs in Mathematics on page 724. 333202_0905.qxd 12/5/05 11:37 AM Page 684 684 Chapter 9 Sequences, Series, and Probability Te c h n o l o g y Most graphing calculators are programmed to evaluate nC r. Consult the user’s guide for your calculator and then evaluate 8C5. You should get an answer of 56. Example 1 Finding Binomial Coefficients Find each binomial coefficient. b. 10 3 c. 7C0 d. 8 8 a. 8C2 Solution a. b. 8C2 10 3 c. 7C0 8! 6! 2! 10! 7! 3! 7! 7! 0! 1 8 7 2 1 8 7 6! 6! 2! 10 9 8 7! 7! 3! 8 8 d. 8! 0! 8! 1 28 10 9 8 3 2 1 120 Now try Exercise 1. r 0 r n, and When as in parts (a) and (b) above, there is a simple pattern for evaluating binomial coefficients that works because there will always be factor
ial terms that divide out from the expression. 2 factors 3 factors 8C2 8 7 2 1 and 10 3 10 9 8 3 2 1 2 factors 3 factors Example 2 Finding Binomial Coefficients Find each binomial coefficient. b. 7 4 c. 12C1 d. 12 11 a. 7C3 Solution a. b. c. d. 7 6 5 35 7C3 12 12 1 12! 1! 11! 12C1 12 11 35 12 11! 1! 11! 12 1 12 Now try Exercise 7. It is not a coincidence that the results in parts (a) and (b) of Example 2 are the same and that the results in parts (c) and (d) are the same. In general, it is true that nCr nCnr. This shows the symmetric property of binomial coefficients that was identified earlier. 333202_0905.qxd 12/5/05 11:37 AM Page 685 Exploration Complete the table and describe the result. n 9 7 12 6 10 r 5 1 4 0 7 nCr nCnr What characteristic of Pascal’s Triangle is illustrated by this table? Section 9.5 The Binomial Theorem 685 Pascal’s Triangle There is a convenient way to remember the pattern for binomial coefficients. By arranging the coefficients in a triangular pattern, you obtain the following array, which is called Pascal’s Triangle. This triangle is named after the famous French mathematician Blaise Pascal (1623–1662). 1 2 6 20 1 3 10 35 1 4 15 1 4 15 1 3 10 35 1 5 21 1 6 1 6 1 5 21 1 7 1 1 1 7 1 4 6 10 1 15 6 21 The first and last numbers in each row of Pascal’s Triangle are 1. Every other number in each row is formed by adding the two numbers immediately above the number. Pascal noticed that numbers in this triangle are precisely the same numbers that are the coefficients of binomial expansions, as follows. 0th row x y0 1 x y1 1x 1y x y2 1x 2 2xy 1y 2 x y3 1x3 3x 2y 3xy 2 1y3 x y4 1x4 4x3y 6x 2y 2 4xy 3 1y4 x y5 1x5 5x4y 10x3y 2 10x 2y 3 5xy4 1y 5 x y6 1x6 6x5y 15x4y 2 20x3y 3 15x 2y4 6xy5 1y6 x y7 1x7 7x 6y 21x5y 2 35x4y 3 35x3y4 21x 2y5 7xy6 1y7 3rd row 2nd row 1st row The top row in Pascal’s Triangle is called the zeroth row because it Similarly, the next row is corresponds to the binomial expansion called the first row because it corresponds to the binomial expansion x y1 1x 1y. In general, the nth row in Pascal’s Triangle gives the x yn . coefficients of x y0 1. Example 3 Using Pascal’s Triangle Use the seventh row of Pascal’s Triangle to find the binomial coefficients. 8C0, 8C1, 8C2, 8C3, 8C4, 8C5, 8C6, 8C7, 8C8 Solution 1 7 21 35 35 21 7 1 1 8C0 8 8C1 28 8C2 56 8C3 70 8C4 56 8C5 28 8C6 8 8C7 1 8C8 Now try Exercise 11. 333202_0905.qxd 12/5/05 11:37 AM Page 686 686 Chapter 9 Sequences, Series, and Probability Historical Note Precious Mirror “Pascal’s”Triangle and forms of the Binomial Theorem were known in Eastern cultures prior to the Western “discovery” of the theorem. A Chinese text entitled Precious Mirror contains a triangle of binomial expansions through the eighth power. Binomial Expansions As mentioned at the beginning of this section, when you write out the coefficients for a binomial that is raised to a power, you are expanding a binomial. The formulas for binomial coefficients give you an easy way to expand binomials, as demonstrated in the next four examples. Example 4 Expanding a Binomial Write the expansion for the expression x 13. Solution The binomial coefficients from the third row of Pascal’s Triangle are 1, 3, 3, 1. So, the expansion is as follows. x 13 1x3 3x 21 3x12 113 x3 3x 2 3x 1 Now try Exercise 15. To expand binomials representing differences rather than sums, you alternate signs. Here are two examples. x 13 x3 3x 2 3x 1 x 14 x4 4x3 6x 2 4x 1 Example 5 Expanding a Binomial Write the expansion for each expression. a. 2x 34 b. x 2y4 Solution The binomial coefficients from the fourth row of Pascal’s Triangle are 1, 4, 6, 4, 1. Therefore, the expansions are as follows. a. 2x 34 12x4 42x33 62x232 42x33 134 16x4 96x3 216x 2 216x 81 b. x 2y4 1x4 4x32y 6x22y2 4x2y3 12y4 x 4 8x3y 24x 2y2 32xy3 16y4 Now try Exercise 19. 333202_0905.qxd 12/5/05 11:37 AM Page 687 Section 9.5 The Binomial Theorem 687 Te c h n o l o g y You can use a graphing utility to check the expansion in Example 6. Graph the original binomial expression and the expansion in the same viewing window. The graphs should coincide as shown below. 200 Example 6 Expanding a Binomial Write the expansion for x2 43. Solution Use the third row of Pascal’s Triangle, as follows. x2 43 1x23 3x224 3x242 143 x 6 12x 4 48x2 64 Now try Exercise 29. −5 5 Sometimes you will need to find a specific term in a binomial expansion. Instead of writing out the entire expansion, you can use the fact that, from the Binomial Theorem, the r 1th term is nCr xnr yr. −100 Example 7 Finding a Term in a Binomial Expansion a 2b8. a. Find the sixth term of b. Find the coefficient of the term a6b5 in the expansion of 3a 2b11. Solution a. Remember that the formula is for the is one less than the number of the term you are looking for. So, to find the sixth term in this binomial expansion, use as shown. term, so n 8, r 5, r r 1th 8C5a 852b5 56 a3 2b5 x 3a, n 11, r 5, b. In this case, x a, and 5625a3b5 y 2b. and y 2b, 1792a3b5. Substitute these values to obtain nCr x nr y r 11C5 3a62b5 462729a632b5 10,777,536a6b5. So, the coefficient is 10,777,536. Now try Exercise 41. W RITING ABOUT MATHEMATICS Error Analysis You are a math instructor and receive the following solutions from one of your students on a quiz. Find the error(s) in each solution. Discuss ways that your student could avoid the error(s) in the future. a. Find the second term in the expansion of 2x 3y5. 52x43y 2 720x 4y 2 b. Find the fourth term in the expansion of 2 x27y4 9003.75x 2y 4 1 6C4 2 x 7y6 1 . 333202_0905.qxd 12/5/05 11:37 AM Page 688 688 Chapter 9 Sequences, Series, and Probability 9.5 Exercises VOCABULARY CHECK: Fill in the blanks. 1. The coefficients of a binomial expansion are called ________ ________. 2. To find binomial coefficients, you can use the ________ ________ or ________ ________. 3. The notation used to denote a binomial coefficient is ________ or ________. 4. When you write out the coefficients for a binomial that is raised to a power, you are ________ a ________. PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. In Exercises 1–10, calculate the binomial coefficient. 1. 3. 5. 7. 9. 5C3 12C0 20C15 10 4 100 98 2. 4. 6. 8. 10. 8C6 20C20 12C5 10 6 100 2 In Exercises 11–14, evaluate using Pascal’s Triangle. 8 5 7C4 11. 13. 8 7 6C3 12. 14. In Exercises 15–34, use the Binomial Theorem to expand and simplify the expression. 15. 17. 19. 21. 23. 25. 27. 29. 31. 33. 34. x 14 a 64 y 43 x y5 r 3s6 3a 4b5 2x y3 x 2 y24 y5 1 x 2x 34 5x 3 2 3x 15 4x 13 16. 18. 20. 22. 24. 26. 28. 30. 32. x 16 a 55 y 25 c d3 x 2y4 2x 5y5 7a b3 x 2 y 26 2y6 1 x In Exercises 35–38, expand the binomial by using Pascal’s Triangle to determine the coefficients. 35. 37. 2t s5 x 2y5 36. 38. 3 2z4 2v 36 In Exercises 39– 46, find the specified expansion of the binomial. n th term in the 39. 41. 43. 45. x y10, x 6y5, 4x 3y9, 10x 3y12, n 4 n 3 n 8 n 9 40. 42. 44. 46. x y6, x 10z7, 5a 6b5, 7x 2y15, n 7 n 4 n 5 n 7 In Exercises 47–54, find the coefficient a of the term in the expansion of the binomial. Binomial x 312 x 2 312 x 2y10 4x y10 3x 2y9 2x 3y8 x 2 y10 z 2 t10 47. 48. 49. 50. 51. 52. 53. 54. Term ax5 ax8 ax8y 2 ax 2y8 ax4y5 ax 6y 2 ax8y 6 az4t8 In Exercises 55–58, use the Binomial Theorem to expand and simplify the expression. 55. 56. 57. 58. x 34 2t 13 x 23 y133 u35 25 In Exercises 59–62, expand the expression in the difference quotient and simplify. f x h f x h 59. f x x3 61. f x x Difference quotient 60. 62. f x x4 f x 1 x 333202_0905.qxd 12/5/05 11:37 AM Page 689 In Exercises 63–68, use the Binomial Theorem to expand the complex number. Simplify your result. 63. 65. 67. 1 i4 2 3i6 1 2 3 2 64. 66. 68. 2 i5 5 93 5 3 i4 i3 Section 9.5 The Binomial Theorem 689 78. To find the probability that the sales representative in Exercise 77 makes four sales if the probability of a sale with any one customer is evaluate the term 1 2, 1 2 41 2 4 8C4 in the expansion of 1 2 8 . 1 2 Approximation In Exercises 69–72, use the Binomial Theorem to approximate the quantity accurate to three decimal places. For example, in Exercise 69, use the expansion 1.028 1 0.028 1 80.02 280.022 . . . . 69. 71. 1.028 2.9912 70. 72. 2.00510 1.989 Model It 79. Data Analysis: Water Consumption The table f t shows the per capita consumption of bottled water (in gallons) in the United States from 1990 through (Source: Economic Research Service, U.S. 2003. Department of Agriculture) Year Consumption, f t Graphical Reasoning In Exercises 73 and 74, use a graphf ing utility to graph and in the same viewing window. What is the relationship between the two graphs? Use the Binomial Theorem to write the polynomial function in standard form. g g 73. 74. f x x3 4x, f x x4 4x 2 1, gx f x 4 gx f x 3 n In Exercises 75–78, consider Probability independent trials of an experiment in which each trial has two possible outcomes: “success” or “failure.” The probability of a success on each trial is and the probability of a failure is q 1 p. in the successes expansion of in the n In this context, the term trials of the experiment. gives the probability of nC k p k q n k p qn p, k 75. A fair coin is tossed seven times. To find the probability of obtaining four heads, evaluate the term 1 2 4 1 2 3 7C4 in the expansion of 1 2 7 . 1 2 1 4. 76. The probability of a baseball player getting a hit during any given time at bat is To find the probability that the player gets three hits during the next 10 times at bat, evaluate the term 1 33 7 10C3 4 4 in the expansion of 1 4 10 . 3 4 1 3. 77. The probability of a sales representative making a sale with any one customer is The sales representative makes eight contacts a day. To find the probability of making four sales, evaluate the term 4 42 3 8C4 1 3 in the expansion of 1 3 8 . 2 3 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 8.0 8.0 9.7 10.3 11.3 12.1 13.0 13.9 15.0 16.4 17.4 18.8 20.7 22.0 (a) Use the regression feature of a graphi
ng utility to find a cubic model for the data. Let represent the year, with corresponding to 1990. t 0 t (b) Use a graphing utility to plot the data and the model in the same viewing window. (c) You want to adjust the model so that corresponds to 2000 rather than 1990. To do this, you 10 units to the left to obtain shift the graph of gt f t 10. in standard form. f Write gt t 0 (d) Use a graphing utility to graph g in the same viewing window as f. (e) Use both models to estimate the per capita consumption of bottled water in 2008. Do you obtain the same answer? (f) Describe the overall trend in the data. What factors do you think may have contributed to the increase in the per capita consumption of bottled water? 333202_0905.qxd 12/5/05 11:37 AM Page 690 690 Chapter 9 Sequences, Series, and Probability 80. Child Support The amounts (in billions of dollars) of child support collected in the United States from 1990 to 2002 can be approximated by the model f t 0.031t 2 0.82t 6.1, 0 ≤ t ≤ 12 f t represents the year, with corresponding to (Source: U.S. Department of Health t 0 t where 1990 (see figure). and Human Services) t 92. The sum of the numbers in the th row of Pascal’s Triangle n 88. Graphical Reasoning Which two functions have identical graphs, and why? Use a graphing utility to graph the functions in the given order and in the same viewing window. Compare the graphs. (a) (b) (c) (d) (e) f x 1 x3 gx 1 x3 hx 1 3x 3x 2 x3 kx 1 3x 3x 2 x 3 px 1 3x 3x 2 x 3 Proof integers and where In Exercises 89–92, prove the property for all r 0 ≤ r ≤ n. n 89. 90. 91. nCr nC0 n1Cr nCn r nC1 nCr . . . ± nCn 0 nC2 nCr 1 is 2n. Skills Review In Exercises 93–96, the graph of and use the graph to write an equation for the graph of is shown. Graph f g. y gx f x x2 y 6 5 4 3 2 1 −1 1 32 4 5 6 952 −1 1 2 3 x x 94. f x x2 y 4 3 2 x 1 2 −3 −2 −1 −2 −3 96. f x x y 1 −1 1 2 3 4 x −4 −5 f(t ( 27 24 21 18 15 12 10 11 12 13 Year (0 ↔ 1990) (a) You want to adjust the model so that corresponds to 2000 rather than 1990. To do this, you shift the graph gt f t 10. f of Write 10 units to the left and obtain gt in standard form. t 0 (b) Use a graphing utility to graph f and g in the same 93. viewing window. (c) Use the graphs to estimate when the child support col- lections will exceed $30 billion. Synthesis True or False? statement is true or false. Justify your answer. In Exercises 81– 83, determine whether the 81. The Binomial Theorem could be used to produce each row of Pascal’s Triangle. 82. A binomial that represents a difference cannot always be accurately expanded using the Binomial Theorem. 83. The x10 x2 312 -term and the x14 -term of the expansion of have identical coefficients. 84. Writing In your own words, explain how to form the rows of Pascal’s Triangle. 85. Form rows 8–10 of Pascal’s Triangle. 86. Think About It How many terms are in the expansion of x yn? 87. Think About It How do the expansions of x yn differ? x yn and In Exercises 97 and 98, find the inverse of the matrix. 97. 6 5 5 4 98. 1.2 2 2.3 4 333202_0906.qxd 12/5/05 11:39 AM Page 691 9.6 Counting Principles Section 9.6 Counting Principles 691 What you should learn • Solve simple counting problems. • Use the Fundamental Counting Principle to solve counting problems. • Use permutations to solve counting problems. • Use combinations to solve counting problems. Why you should learn it You can use counting principles to solve counting problems that occur in real life. For instance, in Exercise 65 on page 700, you are asked to use counting principles to determine the number of possible ways of selecting the winning numbers in the Powerball lottery. © Michael Simpson/FPG/Getty Images Simple Counting Problems This section and Section 9.7 present a brief introduction to some of the basic counting principles and their application to probability. In Section 9.7, you will see that much of probability has to do with counting the number of ways an event can occur. The following two examples describe simple counting problems. Example 1 Selecting Pairs of Numbers at Random Eight pieces of paper are numbered from 1 to 8 and placed in a box. One piece of paper is drawn from the box, its number is written down, and the piece of paper is replaced in the box. Then, a second piece of paper is drawn from the box, and its number is written down. Finally, the two numbers are added together. How many different ways can a sum of 12 be obtained? Solution To solve this problem, count the different ways that a sum of 12 can be obtained using two numbers from 1 to 8. First number Second number From this list, you can see that a sum of 12 can occur in five different ways. Now try Exercise 5. Example 2 Selecting Pairs of Numbers at Random Eight pieces of paper are numbered from 1 to 8 and placed in a box. Two pieces of paper are drawn from the box at the same time, and the numbers on the pieces of paper are written down and totaled. How many different ways can a sum of 12 be obtained? Solution To solve this problem, count the different ways that a sum of 12 can be obtained using two different numbers from 1 to 8. First number Second number 4 8 5 7 7 5 8 4 So, a sum of 12 can be obtained in four different ways. Now try Exercise 7. The difference between the counting problems in Examples 1 and 2 can be described by saying that the random selection in Example 1 occurs with replacement, whereas the random selection in Example 2 occurs without replacement, which eliminates the possibility of choosing two 6’s. 333202_0906.qxd 12/5/05 11:39 AM Page 692 692 Chapter 9 Sequences, Series, and Probability The Fundamental Counting Principle Examples 1 and 2 describe simple counting problems in which you can list each possible way that an event can occur. When it is possible, this is always the best way to solve a counting problem. However, some events can occur in so many different ways that it is not feasible to write out the entire list. In such cases, you must rely on formulas and counting principles. The most important of these is the Fundamental Counting Principle. Fundamental Counting Principle E1 E1 be two events. The first event Let and E2 m2 ways. After can occur in has occurred, of ways that the two events can occur is m1 m2. E2 E1 m1 can occur in different ways. The number different The Fundamental Counting Principle can be extended to three or more can events. For instance, the number of ways that three events occur is E1, E2, m1 m2 m3. and E3 Example 3 Using the Fundamental Counting Principle How many different pairs of letters from the English alphabet are possible? Solution There are two events in this situation. The first event is the choice of the first letter, and the second event is the choice of the second letter. Because the English alphabet contains 26 letters, it follows that the number of two-letter pairs is 26 26 676. Now try Exercise 13. Example 4 Using the Fundamental Counting Principle Telephone numbers in the United States currently have 10 digits. The first three are the area code and the next seven are the local telephone number. How many different telephone numbers are possible within each area code? (Note that at this time, a local telephone number cannot begin with 0 or 1.) Solution Because the first digit of a local telephone number cannot be 0 or 1, there are only eight choices for the first digit. For each of the other six digits, there are 10 choices. Area Code Local Number 8 10 10 10 10 10 10 So, the number of local telephone numbers that are possible within each area code is 8 10 10 10 10 10 10 8,000,000. Now try Exercise 19. 333202_0906.qxd 12/5/05 11:39 AM Page 693 Section 9.6 Counting Principles 693 Permutations One important application of the Fundamental Counting Principle is in determining the number of ways that elements can be arranged (in order). An ordering of elements is called a permutation of the elements. n n Definition of Permutation A permutation of different elements is an ordering of the elements such that one element is first, one is second, one is third, and so on. n Example 5 Finding the Number of Permutations of n Elements How many permutations are possible for the letters A, B, C, D, E, and F? Solution Consider the following reasoning. First position: Any of the six letters Second position: Any of the remaining five letters Third position: Any of the remaining four letters Fourth position: Any of the remaining three letters Fifth position: Any of the remaining two letters Sixth position: The one remaining letter So, the numbers of choices for the six positions are as follows. Permutations of six letters 6 5 4 3 2 1 The total number of permutations of the six letters is 6! 6 5 4 3 2 1 720. Now try Exercise 39. Number of Permutations of n Elements The number of permutations of elements is !. n In other words, there are n! different ways that elements can be ordered. n 333202_0906.qxd 12/5/05 11:39 AM Page 694 694 Chapter 9 Sequences, Series, and Probability Eleven thoroughbred racehorses hold the title of Triple Crown winner for winning the Kentucky Derby, the Preakness, and the Belmont Stakes in the same year. Forty-nine horses have won two out of the three races. Example 6 Counting Horse Race Finishes Eight horses are running in a race. In how many different ways can these horses come in first, second, and third? (Assume that there are no ties.) Solution Here are the different possibilities. Win (first position): Eight choices Place (second position): Seven choices Show (third position): Six choices Using the Fundamental Counting Principle, multiply these three numbers together to obtain the following. Different orders of horses 8 7 6 So, there are 8 7 6 336 different orders. Now try Exercise 43. It is useful, on occasion, to order a subset of a collection of elements rather than the entire collection. For example, you might want to choose and order r elements. Such an ordering is called a permutation of n elements taken r at a time. elements out of a
collection of n Te c h n o l o g y Most graphing calculators are programmed to evaluate Consult the user’s guide for your calculator and then evaluate 8P5. You should get an answer of 6720. nPr. Permutations of n Elements Taken r at a Time The number of permutations of elements taken at a time is n r nPr n! n r! nn 1n 2 . . . n r 1. Using this formula, you can rework Example 6 to find that the number of permutations of eight horses taken three at a time is 8P3 8! 8 3! 8! 5! 8 7 6 5! 5! 336 which is the same answer obtained in the example. 333202_0906.qxd 12/5/05 11:39 AM Page 695 Section 9.6 Counting Principles 695 Remember that for permutations, order is important. So, if you are looking at the possible permutations of the letters A, B, C, and D taken three at a time, the permutations (A, B, D) and (B, A, D) are counted as different because the order of the elements is different. Suppose, however, that you are asked to find the possible permutations of the letters A, A, B, and C. The total number of permutations of the four letters would be However, not all of these arrangements would be distinguishable because there are two A’s in the list. To find the number of distinguishable permutations, you can use the following formula. 4!. 4P4 Distinguishable Permutations Suppose a set of objects has n3 n n1 of a third kind, and so on, with n n1 . . . nk . n2 n 3 of one kind of object, n2 of a second kind, Then the number of distinguishable permutations of the objects is n n! n1! n2! n3! . . . nk ! . Example 7 Distinguishable Permutations In how many distinguishable ways can the letters in BANANA be written? Solution This word has six letters, of which three are A’s, two are N’s, and one is a B. So, the number of distinguishable ways the letters can be written is n! n1! n2! n3! 6! 3! 2! 1! 6 5 4 3! 3! 2! 60. The 60 different distinguishable permutations are as follows. AAABNN AANABN ABAANN ANAABN ANBAAN BAAANN BNAAAN NAABAN NABNAA NBANAA AAANBN AANANB ABANAN ANAANB ANBANA BAANAN BNAANA NAABNA NANAAB NBNAAA AAANNB AANBAN ABANNA ANABAN ANBNAA BAANNA BNANAA NAANAB NANABA NNAAAB AABANN AANBNA ABNAAN ANABNA ANNAAB BANAAN BNNAAA NAANBA NANBAA NNAABA AABNAN AANNAB ABNANA ANANAB ANNABA BANANA NAAABN NABAAN NBAAAN NNABAA AABNNA AANNBA ABNNAA ANANBA ANNBAA BANNAA NAAANB NABANA NBAANA NNBAAA Now try Exercise 45. 333202_0906.qxd 12/5/05 11:39 AM Page 696 696 Chapter 9 Sequences, Series, and Probability Combinations When you count the number of possible permutations of a set of elements, order is important. As a final topic in this section, you will look at a method of selecting subsets of a larger set in which order is not important. Such subsets are called combinations of n elements taken r at a time. For instance, the combinations A, B, C and B, A, C are equivalent because both sets contain the same three elements, and the order in which the elements are listed is not important. So, you would count only one of the two sets. A common example of how a combination occurs is a card game in which the player is free to reorder the cards after they have been dealt. Example 8 Combinations of n Elements Taken r at a Time In how many different ways can three letters be chosen from the letters A, B, C, D, and E? (The order of the three letters is not important.) Solution The following subsets represent the different combinations of three letters that can be chosen from the five letters. A, B, C A, B, E A, C, E B, C, D B, D, E A, B, D A, C, D A, D, E B, C, E C, D, E From this list, you can conclude that there are 10 different ways that three letters can be chosen from five letters. Now try Exercise 55. Combinations of n Elements Taken r at a Time The number of combinations of elements taken at a time is n r nCr n! n r!r! which is equivalent to nCr nPr r! . Note that the formula for is the same one given for binomial coefficients. To see how this formula is used, solve the counting problem in Example 8. In that problem, you are asked to find the number of combinations of five elements taken three at a time. So, and the number of combinations is n 5, r 3, nCr 5C3 5! 2!3! 2 5 4 3! 2 1 3! 10 which is the same answer obtained in Example 8. 333202_0906.qxd 12/5/05 11:39 AM Page 697 A A A A Example 9 Counting Card Hands Section 9.6 Counting Principles 697 2 3 4 5 6 7 8 9 10 10 10 10 J Q K FIGURE 9.7 cards Standard deck of playing A standard poker hand consists of five cards dealt from a deck of 52 (see Figure 9.7). How many different poker hands are possible? (After the cards are dealt, the player may reorder them, and so order is not important.) Solution You can find the number of different poker hands by using the formula for the number of combinations of 52 elements taken five at a time, as follows. 52C5 52! 52 5!5! 52! 47!5! 52 51 50 49 48 47! 5 4 3 2 1 47! 2,598,960 Now try Exercise 63. Example 10 Forming a Team You are forming a 12-member swim team from 10 girls and 15 boys. The team must consist of five girls and seven boys. How many different 12-member teams are possible? Solution There are boys. By the Fundamental Counting Principal, there are choosing five girls and seven boys. ways of choosing five girls. The are 15C7 10C5 ways of choosing seven ways of 10C5 15C7 10C5 15C7 15! 8! 7! 10! 5! 5! 252 6435 1,621,620 So, there are 1,621,620 12-member swim teams possible. Now try Exercise 65. When solving problems involving counting principles, you need to be able to distinguish among the various counting principles in order to determine which is necessary to solve the problem correctly. To do this, ask yourself the following questions. 1. Is the order of the elements important? Permutation 2. Are the chosen elements a subset of a larger set in which order is not important? Combination 3. Does the problem involve two or more separate events? Fundamental Counting Principle 333202_0906.qxd 12/5/05 11:39 AM Page 698 698 Chapter 9 Sequences, Series, and Probability 9.6 Exercises VOCABULARY CHECK: Fill in the blanks. 1. The ________ ________ ________ states that if there are ways for one event to occur and ways m2 m1 m2 m1 ways for both events to occur. for a second event to occur, there are 2. An ordering of elements is called a ________ of the elements. n 3. The number of permutations of elements taken at a time is given by the formula ________. n r 4. The number of ________ ________ of objects is given by n n! n1!n2!n3! . . . nk! . 5. When selecting subsets of a larger set in which order is not important, you are finding the number of ________ of elements taken at a time. n r PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. Random Selection In Exercises 1– 8, determine the number of ways a computer can randomly generate one or more such integers from 1 through 12. 1. An odd integer 3. A prime integer 2. An even integer 4. An integer that is greater than 9 5. An integer that is divisible by 4 6. An integer that is divisible by 3 7. Two distinct integers whose sum is 9 8. Two distinct integers whose sum is 8 9. Entertainment Systems A customer can choose one of three amplifiers, one of two compact disc players, and one of five speaker models for an entertainment system. Determine the number of possible system configurations. 10. Job Applicants A college needs two additional faculty members: a chemist and a statistician. In how many ways can these positions be filled if there are five applicants for the chemistry position and three applicants for the statistics position? 11. Course Schedule A college student is preparing a course schedule for the next semester. The student may select one of two mathematics courses, one of three science courses, and one of five courses from the social sciences and humanities. How many schedules are possible? 12. Aircraft Boarding Eight people are boarding an aircraft. Two have tickets for first class and board before those in the economy class. In how many ways can the eight people board the aircraft? 13. True-False Exam In how many ways can a six-question true-false exam be answered? (Assume that no questions are omitted.) 14. True-False Exam In how many ways can a 12-question true-false exam be answered? (Assume that no questions are omitted.) 15. License Plate Numbers In the state of Pennsylvania, each standard automobile license plate number consists of three letters followed by a four-digit number. How many distinct license plate numbers can be formed in Pennsylvania? 16. License Plate Numbers In a certain state, each automobile license plate number consists of two letters followed by a four-digit number. To avoid confusion between “O” and “zero” and between “I” and “one,” the letters “O” and “I” are not used. How many distinct license plate numbers can be formed in this state? 17. Three-Digit Numbers How many three-digit numbers can be formed under each condition? (a) The leading digit cannot be zero. (b) The leading digit cannot be zero and no repetition of digits is allowed. (c) The leading digit cannot be zero and the number must be a multiple of 5. (d) The number is at least 400. 18. Four-Digit Numbers How many four-digit numbers can be formed under each condition? (a) The leading digit cannot be zero. (b) The leading digit cannot be zero and no repetition of digits is allowed. (c) The leading digit cannot be zero and the number must be less than 5000. (d) The leading digit cannot be zero and the number must be even. 19. Combination Lock A combination lock will open when the right choice of three numbers (from 1 to 40, inclusive) is selected. How many different lock combinations are possible? 333202_0906.qxd 12/5/05 11:39 AM Page 699 20. Combination Lock A combination lock will open when the right choice of three numbers (from 1 to 50, inclusive) is selected. How many different lock combinations are possible? 21. Concert Seats Four couples have reserved seats in a row for a concert. In how many different ways can they be seated if (a) there are
no seating restrictions? (b) the two members of each couple wish to sit together? 22. Single File In how many orders can four girls and four boys walk through a doorway single file if (a) there are no restrictions? (b) the girls walk through before the boys? In Exercises 23–28, evaluate n Pr . 23. 25. 27. 4P4 8P3 5P4 24. 26. 28. 5 P5 20 P2 7P4 In Exercises 29 and 30, solve for n. 29. 14 nP3 n2P4 30. nP5 18 n2P4 In Exercises 31–36, evaluate using a graphing utility. 31. 33. 35. 20 P5 100 P3 20C5 32. 34. 36. 100 P5 10 P8 10C7 37. Posing for a Photograph In how many ways can five children posing for a photograph line up in a row? 38. Riding in a Car In how many ways can six people sit in a six-passenger car? 39. Choosing Officers From a pool of 12 candidates, the offices of president, vice-president, secretary, and treasurer will be filled. In how many different ways can the offices be filled? 40. Assembly Line Production There are four processes involved in assembling a product, and these processes can be performed in any order. The management wants to test each order to determine which is the least time-consuming. How many different orders will have to be tested? In Exercises 41–44, find the number of distinguishable permutations of the group of letters. 41. A, A, G, E, E, E, M 42. B, B, B, T, T, T, T, T 43. A, L, G, E, B, R, A 44. M, I, S, S, I, S, S, I, P, P, I 45. Write all permutations of the letters A, B, C, and D. 46. Write all permutations of the letters A, B, C, and D if the letters B and C must remain between the letters A and D. Section 9.6 Counting Principles 699 47. Batting Order A baseball coach is creating a nine-player batting order by selecting from a team of 15 players. How many different batting orders are possible? 48. Athletics Six sprinters have qualified for the finals in the 100-meter dash at the NCAA national track meet. In how many ways can the sprinters come in first, second, and third? (Assume there are no ties.) 49. Jury Selection From a group of 40 people, a jury of 12 people is to be selected. In how many different ways can the jury be selected? 50. Committee Members As of January 2005, the U.S. Senate Committee on Indian Affairs had 14 members. Assuming party affiliation was not a factor in selection, how many different committees were possible from the 100 U.S. senators? 51. Write all possible selections of two letters that can be formed from the letters A, B, C, D, E, and F. (The order of the two letters is not important.) 52. Forming an Experimental Group In order to conduct an experiment, five students are randomly selected from a class of 20. How many different groups of five students are possible? 53. Lottery Choices In the Massachusetts Mass Cash game, a player chooses five distinct numbers from 1 to 35. In how many ways can a player select the five numbers? 54. Lottery Choices In the Louisiana Lotto game, a player chooses six distinct numbers from 1 to 40. In how many ways can a player select the six numbers? 55. Defective Units A shipment of 10 microwave ovens contains three defective units. In how many ways can a vending company purchase four of these units and receive (a) all good units, (b) two good units, and (c) at least two good units? 56. Interpersonal Relationships The complexity of interpersonal relationships increases dramatically as the size of a group increases. Determine the numbers of different two-person relationships in groups of people of sizes (a) 3, (b) 8, (c) 12, and (d) 20. 57. Poker Hand You are dealt five cards from an ordinary deck of 52 playing cards. In how many ways can you get (a) a full house and (b) a five-card combination containing two jacks and three aces? (A full house consists of three of one kind and two of another. For example, A-A-A-5-5 and K-K-K-10-10 are full houses.) 58. Job Applicants A toy manufacturer interviews eight people for four openings in the research and development department of the company. Three of the eight people are women. If all eight are qualified, in how many ways can the employer fill the four positions if (a) the selection is random and (b) exactly two selections are women? 333202_0906.qxd 12/5/05 11:39 AM Page 700 700 Chapter 9 Sequences, Series, and Probability 59. Forming a Committee A six-member research committee at a local college is to be formed having one administrator, three faculty members, and two students. There are seven administrators, 12 faculty members, and 20 students in contention for the committee. How many six-member committees are possible? 60. Law Enforcement A police department uses computer imaging to create digital photographs of alleged perpetrators from eyewitness accounts. One software package contains 195 hairlines, 99 sets of eyes and eyebrows, 89 noses, 105 mouths, and 74 chins and cheek structures. (a) Find the possible number of different faces that the software could create. (b) A eyewitness can clearly recall the hairline and eyes and eyebrows of a suspect. How many different faces can be produced with this information? Geometry In Exercises 61–64, find the number of diagonals of the polygon. (A line segment connecting any two nonadjacent vertices is called a diagonal of the polygon.) (d) Number of two-scoop ice cream cones created from 31 different flavors Synthesis True or False? the statement is true or false. Justify your answer. In Exercises 67 and 68, determine whether 67. The number of letter pairs that can be formed in any order from any of the first 13 letters in the alphabet (A–M) is an example of a permutation. 68. The number of permutations of n elements can be deter- mined by using the Fundamental Counting Principle. 69. What is the relationship between nCr and nCnr ? 70. Without calculating the numbers, determine which of the following is greater. Explain. (a) The number of combinations of 10 elements taken six at a time (b) The number of permutations of 10 elements taken six at a time 61. Pentagon 63. Octagon 62. Hexagon 64. Decagon (10 sides) Proof In Exercises 71– 74, prove the identity. Model It 65. Lottery Powerball is a lottery game that is operated by the Multi-State Lottery Association and is played in 27 states, Washington D.C., and the U.S. Virgin Islands. The game is played by drawing five white balls out of a drum of 53 white balls (numbered 1–53) and one red powerball out of a drum of 42 red balls (numbered 1– 42). The jackpot is won by matching all five white balls in any order and the red powerball. (a) Find the possible number of winning Powerball numbers. (b) Find the possible number of winning Powerball numbers if the jackpot is won by matching all five white balls in order and the red power ball. (c) Compare the results of part (a) with a state lottery in which a jackpot is won by matching six balls from a drum of 53 balls. 66. Permutations or Combinations? Decide whether each scenario should be counted using permutations or combinations. Explain your reasoning. (a) Number of ways 10 people can line up in a row for con- cert tickets 71. n Pn 1 n Pn 73. nCn 1 nC1 72. nCn 74. nCr nC0 n Pr r! 75. Think About It Can your calculator evaluate 100P80? If not, explain why. 76. Writing Explain in words the meaning of n Pr . Skills Review In Exercises 77– 80, evaluate the function at each specified value of the independent variable and simplify. 77. 78. 79. (b) f 0 f 3 f x 3x2 8 (a) gx x 3 2 g7 (a) f x x 5 6 (a) f 5 g3 (b) (b) f 1 80. f x x2 2x 5, x2 2, (b) f 1 (a) f 4 (c) f 5 (c) gx 1 (c) x ≤ 4 x > 4 (c) f 11 f 20 In Exercises 81– 84, solve the equation. Round your answer to two decimal places, if necessary. (b) Number of different arrangements of three types of flowers from an array of 20 types (c) Number of three-digit pin numbers for a debit card 81. x 3 x 6 83. log2 x 3 5 82. 4 t 3 2t 1 84. e x3 16 333202_0907.qxd 12/5/05 11:41 AM Page 701 9.7 Probability What you should learn • Find the probabilities of events. • Find the probabilities of mutually exclusive events. • Find the probabilities of independent events. • Find the probability of the complement of an event. Why you should learn it Probability applies to many games of chance. For instance, in Exercise 55, on page 712, you will calculate probabilities that relate to the game of roulette. Section 9.7 Probability 701 The Probability of an Event Any happening for which the result is uncertain is called an experiment. The possible results of the experiment are outcomes, the set of all possible outcomes of the experiment is the sample space of the experiment, and any subcollection of a sample space is an event. For instance, when a six-sided die is tossed, the sample space can be represented by the numbers 1 through 6. For this experiment, each of the outcomes is equally likely. To describe sample spaces in such a way that each outcome is equally likely, you must sometimes distinguish between or among various outcomes in ways that appear artificial. Example 1 illustrates such a situation. Example 1 Finding a Sample Space Find the sample space for each of the following. a. One coin is tossed. b. Two coins are tossed. c. Three coins are tossed. Solution a. Because the coin will land either heads up (denoted by H ) or tails up (denoted Hank de Lespinasse/The Image Bank by ), the sample space is T S H, T . b. Because either coin can land heads up or tails up, the possible outcomes are as follows. HH HT TH T T heads up on both coins heads up on first coin and tails up on second coin tails up on first coin and heads up on second coin tails up on both coins So, the sample space is S HH, HT, TH, TT . Note that this list distinguishes between the two cases though these two outcomes appear to be similar. HT and TH, even c. Following the notation of part (b), the sample space is S HHH, HHT, HTH, HTT, THH, THT, TTH, TTT . Note that this list distinguishes among the cases HHT, HTH, and THH, and among the cases HTT, THT, and TTH. Now try Exercise 1. 333202_0907.qxd 12/5/05 11:41 AM Page 702 702 Chapter 9 Sequ
ences, Series, and Probability Exploration Toss two coins 100 times and write down the number of heads that occur on each toss (0, 1, or 2). How many times did two heads occur? How many times would you expect two heads to occur if you did the experiment 1000 times? Increasing likelihood of occurrence 0.0 0.5 1.0 Impossible event (cannot occur) The occurrence of the event is just as likely as it is unlikely. Certain event (must occur) FIGURE 9.8 You can write a probability as a fraction, decimal, or percent. For instance, in Example 2(a), the probability of getting two heads can be written as or 25%. 1 4, 0.25, To calculate the probability of an event, count the number of outcomes in the is denoted by nS. The event and in the sample space. The number of outcomes in event nE, E probability that event will occur is given by and the number of outcomes in the sample space S nEnS. is denoted by E The Probability of an Event If an event has nS nE E equally likely outcomes, the probability of event PE nE nS . E is equally likely outcomes and its sample space has S Because the number of outcomes in an event must be less than or equal to the number of outcomes in the sample space, the probability of an event must be a number between 0 and 1. That is, 0 ≤ PE ≤ 1 as indicated in Figure 9.8. If impossible event. If event. PE 1, PE 0, event E E event must occur, and cannot occur, and E is called an is called a certain E Example 2 Finding the Probability of an Event a. Two coins are tossed. What is the probability that both land heads up? b. A card is drawn from a standard deck of playing cards. What is the probability that it is an ace? Solution a. Following the procedure in Example 1(b), let E HH and S HH, HT, TH, TT . The probability of getting two heads is PE nE nS 1 4 . b. Because there are 52 cards in a standard deck of playing cards and there are four aces (one in each suit), the probability of drawing an ace is PE nE nS 4 52 1 13 . Now try Exercise 11. 333202_0907.qxd 12/5/05 11:41 AM Page 703 Section 9.7 Probability 703 Example 3 Finding the Probability of an Event Two six-sided dice are tossed. What is the probability that the total of the two dice is 7? (See Figure 9.9.) Solution Because there are six possible outcomes on each die, you can use the Fundamental Counting Principle to conclude that there are or 36 different outcomes when two dice are tossed. To find the probability of rolling a total of 7, you must first count the number of ways in which this can occur. 6 6 FIGURE 9.9 First die Second die You could have written out each sample space in Examples 2 and 3 and simply counted the outcomes in the desired events. For larger sample spaces, however, you should use the counting principles discussed in Section 9.6. So, a total of 7 can be rolled in six ways, which means that the probability of rolling a 7 is PE nE nS 6 36 1 6 . Now try Exercise 15. Example 4 Finding the Probability of an Event Twelve-sided dice, as shown in Figure 9.10, can be constructed (in the shape of regular dodecahedrons) such that each of the numbers from 1 to 6 appears twice on each die. Prove that these dice can be used in any game requiring ordinary six-sided dice without changing the probabilities of different outcomes. Solution For an ordinary six-sided die, each of the numbers 1, 2, 3, 4, 5, and 6 occurs only once, so the probability of any particular number coming up is PE nE nS 1 6 . For one of the 12-sided dice, each number occurs twice, so the probability of any particular number coming up is FIGURE 9.10 Now try Exercise 17. PE nE nS 2 12 1 6 . 333202_0907.qxd 12/5/05 11:41 AM Page 704 704 Chapter 9 Sequences, Series, and Probability Example 5 The Probability of Winning a Lottery In the Arizona state lottery, a player chooses six different numbers from 1 to 41. If these six numbers match the six numbers drawn (in any order) by the lottery commission, the player wins (or shares) the top prize. What is the probability of winning the top prize if the player buys one ticket? Solution To find the number of elements in the sample space, use the formula for the number of combinations of 41 elements taken six at a time. nS 41C6 41 40 39 38 37 36 6 5 4 3 2 1 4,496,388 If a person buys only one ticket, the probability of winning is PE nE nS 1 4,496,388 . Now try Exercise 21. Example 6 Random Selection The numbers of colleges and universities in various regions of the United States in 2003 are shown in Figure 9.11. One institution is selected at random. What is the probability that the institution is in one of the three southern regions? (Source: National Center for Education Statistics) Solution From the figure, the total number of colleges and universities is 4163. Because there are colleges and universities in the three southern regions, the probability that the institution is in one of these regions is 700 284 386 1370 PE nE nS 1370 4163 0.329. Mountain 274 West North Central 441 East North Central 630 Pacific 563 New England 261 Middle Atlantic 624 South Atlantic 700 West South Central 386 East South Central 284 FIGURE 9.11 Now try Exercise 33. 333202_0907.qxd 12/5/05 11:41 AM Page 705 Mutually Exclusive Events Section 9.7 Probability 705 B A and (from the same sample space) are mutually exclusive if A Two events B and have no outcomes in common. In the terminology of sets, the intersection and of PA B 0. is the empty set, which is written as B A B For instance, if two dice are tossed, the event of rolling a total of 6 and the event of rolling a total of 9 are mutually exclusive. To find the probability that one or the other of two mutually exclusive events will occur, you can add their individual probabilities. A Probability of the Union of Two Events B If occurring is given by and A are events in the same sample space, the probability of or A B PA B PA PB PA B. If A B are mutually exclusive, then and PA B PA PB. Example 7 The Probability of a Union of Events One card is selected from a standard deck of 52 playing cards. What is the probability that the card is either a heart or a face card? Hearts Solution Because the deck has 13 hearts, the probability of selecting a heart (event A ) is PA 13 52 . Similarly, because the deck has 12 face cards, the probability of selecting a face B card (event ) is PB 12 52 . Because three of the cards are hearts and face cards (see Figure 9.12), it follows that PA B 3 52 . Finally, applying the formula for the probability of the union of two events, you can conclude that the probability of selecting a heart or a face card is PA B PA PB PA B 13 52 12 52 3 52 22 52 0.423. Now try Exercise 45. 2♥ 4♥ 6♥ 8♥ A♥ 3♥ 5♥ 7♥ 9♥ 10♥ Face cards FIGURE 9.12 n(A ∩ B) = 3 K♣ K♥ Q♥ J♥ Q♣ K♦ J♣ J♦ Q♦ K♠ Q♠ J♠ 333202_0907.qxd 12/5/05 11:41 AM Page 706 706 Chapter 9 Sequences, Series, and Probability Example 8 Probability of Mutually Exclusive Events The personnel department of a company has compiled data on the numbers of employees who have been with the company for various periods of time. The results are shown in the table. Years of service Number of employees 0–4 5–9 10–14 15–19 20–24 25–29 30–34 35–39 40–44 157 89 74 63 42 38 37 21 8 If an employee is chosen at random, what is the probability that the employee has (a) 4 or fewer years of service and (b) 9 or fewer years of service? Solution a. To begin, add the number of employees to find that the total is 529. Next, let represent choosing an employee with 0 to 4 years of service. Then the event probability of choosing an employee who has 4 or fewer years of service is A PA 157 529 0.297. b. Let event represent choosing an employee with 5 to 9 years of service. Then B PB 89 529 . Because event can conclude that these two events are mutually exclusive and that from part (a) and event have no outcomes in common, you A B 89 529 PA B PA PB 157 529 246 529 0.465. So, the probability of choosing an employee who has 9 or fewer years of service is about 0.465. Now try Exercise 47. 333202_0907.qxd 12/5/05 11:41 AM Page 707 Section 9.7 Probability 707 Independent Events Two events are independent if the occurrence of one has no effect on the occurrence of the other. For instance, rolling a total of 12 with two six-sided dice has no effect on the outcome of future rolls of the dice. To find the probability that two independent events will occur, multiply the probabilities of each. Probability of Independent Events A and If occur is B are independent events, the probability that both A B and will PA and B PA PB. Example 9 Probability of Independent Events A random number generator on a computer selects three integers from 1 to 20. What is the probability that all three numbers are less than or equal to 5? Solution The probability of selecting a number from 1 to 5 is PA 5 20 1 4 . So, the probability that all three numbers are less than or equal to 5 is 1 4 1 4 PA PA PA 1 4 1 64 . Now try Exercise 48. Example 10 Probability of Independent Events In 2004, approximately 20% of the adult population of the United States got their news from the Internet every day. In a survey, 10 people were chosen at random from the adult population. What is the probability that all 10 got their news from the Internet every day? (Source: The Gallup Poll) Solution A represent choosing an adult who gets the news from the Internet every day. Let The probability of choosing an adult who got his or her news from the Internet every day is 0.20, the probability of choosing a second adult who got his or her news from the Internet every day is 0.20, and so on. Because these events are independent, you can conclude that the probability that all 10 people got their news from the Internet every day is PA10 0.2010 0.0000001. Now try Exercise 49. 333202_0907.qxd 12/5/05 11:41 AM Page 708 708 Chapter 9 Sequences, Series, and Probability Exploration You are in a class with 22 other people. What is the probability that at least two out of the 23 people will have a birthday on the same day of the year? The comple
ment of the probability that at least two people have the same birthday is the probability that all 23 birthdays are different. So, first find the probability that all 23 people have different birthdays and then find the complement. Now, determine the proba- bility that in a room with 50 people at least two people have the same birthday. The Complement of an Event The complement of an event space that are not in PA A 1 PA PA 1. The complement of event A A So, the probability of is the collection of all outcomes in the sample Because is denoted by are mutually exclusive, it follows that A A. and because and A. or is A A PA 1 PA. For instance, if the probability of winning a certain game is PA 1 4 the probability of losing the game is PA 1 1 4 3 4 . Probability of a Complement A Let be an event and let PA, the probability of the complement is PA 1 PA. A be its complement. If the probability of A is Example 11 Finding the Probability of a Complement A manufacturer has determined that a machine averages one faulty unit for every 1000 it produces. What is the probability that an order of 200 units will have one or more faulty units? Solution To solve this problem as stated, you would need to find the probabilities of having exactly one faulty unit, exactly two faulty units, exactly three faulty units, and so on. However, using complements, you can simply find the probability that all units are perfect and then subtract this value from 1. Because the probability that any given unit is perfect is 999/1000, the probability that all 200 units are perfect is 200 PA 999 1000 0.819. So, the probability that at least one unit is faulty is PA 1 PA 1 0.819. 0.181 Now try Exercise 51. 333202_0907.qxd 12/5/05 11:41 AM Page 709 Section 9.7 Probability 709 9.7 Exercises VOCABULARY CHECK: In Exercises 1–7, fill in the blanks. 1. An ________ is an event whose result is uncertain, and the possible results of the event are called ________. 2. The set of all possible outcomes of an experiment is called the ________ ________. 3. To determine the ________ of an event, you can use the formula where nE is the number of outcomes in the event and nS PE nE nS, is the number of outcomes in the sample space. 4. If PE 0, then E is an ________ event, and if PE 1, then E is a ________ event. 5. If two events from the same sample space have no outcomes in common, then the two events are ________ ________. 6. If the occurrence of one event has no effect on the occurrence of a second event, then the events are ________. 7. The ________ of an event A is the collection of all outcomes in the sample space that are not in A. 8. Match the probability formula with the correct probability name. (a) Probability of the union of two events (b) Probability of mutually exclusive events (c) Probability of independent events (d) Probability of a complement (i) (ii) (ii) (iv) PA B PA PB PA 1 PA PA B PA PB PA B PA and B PA PB PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. In Exercises 1–6, determine the sample space for the experiment. 1. A coin and a six-sided die are tossed. 2. A six-sided die is tossed twice and the sum of the points is recorded. Drawing a Card In Exercises 11–14, find the probability for the experiment of selecting one card from a standard deck of 52 playing cards. 11. The card is a face card. 12. The card is not a face card. 3. A taste tester has to rank three varieties of yogurt, A, B, 13. The card is a red face card. and C, according to preference. 4. Two marbles are selected from a bag containing two red marbles, two blue marbles, and one yellow marble. The color of each marble is recorded. 5. Two county supervisors are selected from five supervisors, A, B, C, D, and E, to study a recycling plan. 6. A sales representative makes presentations about a product in three homes per day. In each home, there may be a sale (denote by S) or there may be no sale (denote by F). 14. The card is a 6 or lower. (Aces are low.) Tossing a Die the experiment of tossing a six-sided die twice. In Exercises 15–20, find the probability for 15. The sum is 4. 16. The sum is at least 7. 17. The sum is less than 11. 18. The sum is 2, 3, or 12. 19. The sum is odd and no more than 7. 20. The sum is odd or prime. Tossing a Coin In Exercises 7–10, find the probability for the experiment of tossing a coin three times. Use S {HHH, HHT, HTH, H T T, THH, TH T, the sample space T TH, T T T}. Drawing Marbles In Exercises 21–24, find the probability for the experiment of drawing two marbles (without replacement) from a bag containing one green, two yellow, and three red marbles. 7. The probability of getting exactly one tail 21. Both marbles are red. 8. The probability of getting a head on the first toss 22. Both marbles are yellow. 9. The probability of getting at least one head 23. Neither marble is yellow. 10. The probability of getting at least two heads 24. The marbles are of different colors. 333202_0907.qxd 12/5/05 11:41 AM Page 710 710 Chapter 9 Sequences, Series, and Probability In Exercises 25–28, you are given the probability that an event will happen. Find the probability that the event will not happen. A person is selected at random from the sample. Find the probability that the described person is selected. (a) A person who doesn’t favor the amendment 26. PE 0.36 (b) A Republican 25. 27. 28. PE 0.7 PE 1 4 PE 2 3 In Exercises 29–32, you are given the probability that an event will not happen. Find the probability that the event will happen. 29. 30. 31. 32. PE 0.14 PE 0.92 PE 17 35 PE 61 100 33. Data Analysis A study of the effectiveness of a flu vaccine was conducted with a sample of 500 people. Some participants in the study were given no vaccine, some were given one injection, and some were given two injections. The results of the study are listed in the table. No vaccine One injection Two injections Total Flu No flu Total 7 149 156 2 52 54 13 277 290 22 478 500 A person is selected at random from the sample. Find the specified probability. (a) The person had two injections. (b) The person did not get the flu. (c) The person got the flu and had one injection. 34. Data Analysis One hundred college students were interviewed to determine their political party affiliations and whether they favored a balanced-budget amendment to the Constitution. The results of the study are listed in the table, represents Republican. where represents Democrat and D R Favor Not Favor Unsure Total D R Total 23 32 55 25 9 34 7 4 11 55 45 100 (c) A Democrat who favors the amendment 35. Graphical Reasoning The figure shows the results of a recent survey in which 1011 adults were asked to grade U.S. public schools. (Source: Phi Delta Kappa/Gallup Poll) Grading Public Schools A 2% Don’t know 7% D 12% C 52% B 24% Fail 3% (a) Estimate the number of adults who gave U.S. public schools a B. (b) An adult is selected at random. What is the probabilty that the adult will give the U.S. public schools an A? (c) An adult is selected at random. What is the probabilty the adult will give the U.S. public schools a C or a D? 36. Graphical Reasoning The figure shows the results of a survey in which auto racing fans listed their favorite type of racing. (Source: ESPN Sports Poll/TNS Sports) Favorite Type of Racing NHRA drag racing 13% Motorcycle 11% Other 11% Formula One 6% NASCAR 59% (a) What is the probability that an auto racing fan selected at random lists NASCAR racing as his or her favorite type of racing? (b) What is the probability that an auto racing fan selected at random lists Formula One or motorcycle racing as his or her favorite type of racing? (c) What is the probability that an auto racing fan selected at random does not list NHRA drag racing as his or her favorite type of racing? 333202_0907.qxd 12/5/05 11:41 AM Page 711 Section 9.7 Probability 711 44. Card Game The deck of a card game is made up of 108 cards. Twenty-five each are red, yellow, blue, and green, and eight are wild cards. Each player is randomly dealt a seven-card hand. (a) What is the probability that a hand will contain exactly two wild cards? (b) What is the probability that a hand will contain two wild cards, two red cards, and three blue cards? 45. Drawing a Card One card is selected at random from an ordinary deck of 52 playing cards. Find the probabilities that (a) the card is an even-numbered card, (b) the card is a heart or a diamond, and (c) the card is a nine or a face card. 46. Poker Hand Five cards are drawn from an ordinary deck of 52 playing cards. What is the probability that the hand drawn is a full house? (A full house is a hand that consists of two of one kind and three of another kind.) 47. Defective Units A shipment of 12 microwave ovens contains three defective units. A vending company has ordered four of these units, and because each is identically packaged, the selection will be random. What are the probabilities that (a) all four units are good, (b) exactly two units are good, and (c) at least two units are good? 48. Random Number Generator Two integers from 1 through 40 are chosen by a random number generator. What are the probabilities that (a) the numbers are both even, (b) one number is even and one is odd, (c) both numbers are less than 30, and (d) the same number is chosen twice? 49. Flexible Work Hours In a survey, people were asked if they would prefer to work flexible hours—even if it meant slower career advancement—so they could spend more time with their families. The results of the survey are shown in the figure. Three people from the survey were chosen at random. What is the probability that all three people would prefer flexible work hours? Flexible Work Hours Flexible hours 78% Don’t know 9% Rigid hours 13% 37. Alumni Association A college sends a survey to selected members of the class of 2006. Of the 1254 people who graduated that year, 672 are women, of whom 124 went on to graduate school. Of the 582 male graduates, 198 w
ent on to graduate school. An alumni member is selected at random. What are the probabilities that the person is (a) female, (b) male, and (c) female and did not attend graduate school? 38. Education In a high school graduating class of 202 students, 95 are on the honor roll. Of these, 71 are going on to college, and of the other 107 students, 53 are going on to college. A student is selected at random from the class. What are the probabilities that the person chosen is (a) going to college, (b) not going to college, and (c) on the honor roll, but not going to college? 39. Winning an Election Taylor, Moore, and Jenkins are candidates for public office. It is estimated that Moore and Jenkins have about the same probability of winning, and Taylor is believed to be twice as likely to win as either of the others. Find the probability of each candidate winning the election. 40. Winning an Election Three people have been nominated for president of a class. From a poll, it is estimated that the first candidate has a 37% chance of winning and the second candidate has a 44% chance of winning. What is the probability that the third candidate will win? In Exercises 41–52, the sample spaces are large and you should use the counting principles discussed in Section 9.6. 41. Preparing for a Test A class is given a list of 20 study problems, from which 10 will be part of an upcoming exam. A student knows how to solve 15 of the problems. Find the probabilities that the student will be able to answer (a) all 10 questions on the exam, (b) exactly eight questions on the exam, and (c) at least nine questions on the exam. 42. Payroll Mix-Up Five paychecks and envelopes are addressed to five different people. The paychecks are randomly inserted into the envelopes. What are the probabilities that (a) exactly one paycheck will be inserted in the correct envelope and (b) at least one paycheck will be inserted in the correct envelope? 43. Game Show On a game show, you are given five digits to arrange in the proper order to form the price of a car. If you are correct, you win the car. What is the probability of winning, given the following conditions? (a) You guess the position of each digit. (b) You know the first digit and guess the positions of the other digits. 333202_0907.qxd 12/5/05 11:41 AM Page 712 712 Chapter 9 Sequences, Series, and Probability 50. Consumer Awareness Suppose that the methods used by shoppers to pay for merchandise are as shown in the circle graph. Two shoppers are chosen at random. What is the probability that both shoppers paid for their purchases only in cash? How Shoppers Pay for Merchandise Mostly credit 7% Mostly cash 27% Half cash, half credit 30% Only credit 4% Only cash 32% 51. Backup System A space vehicle has an independent backup system for one of its communication networks. The probability that either system will function satisfactorily during a flight is 0.985. What are the probabilities that during a given flight (a) both systems function satisfactorily, functions (b) at satisfactorily, and (c) both systems fail? least one system 52. Backup Vehicle A fire company keeps two rescue vehicles. Because of the demand on the vehicles and the chance of mechanical failure, the probability that a specific vehicle is available when needed is 90%. The availability of one vehicle is independent of the availability of the other. Find the probabilities that (a) both vehicles are available at a given time, (b) neither vehicle is available at a given time, and (c) at least one vehicle is available at a given time. 53. A Boy or a Girl? Assume that the probability of the birth of a child of a particular sex is 50%. In a family with four children, what are the probabilities that (a) all the children are boys, (b) all the children are the same sex, and (c) there is at least one boy? 54. Geometry You and a friend agree to meet at your favorite fast-food restaurant between 5:00 and 6:00 P.M. The one who arrives first will wait 15 minutes for the other, and then will leave (see figure). What is the probability that the two of you will actually meet, assuming that your arrival times are random within the hour ( You meet You meet You don’t meet 60 45 30 15 Y ou arrive first Y our friend arrives first 45 30 15 Your arrival time (in minutes past 5:00 P.M.) 60 Model It 55. Roulette American roulette is a game in which a wheel turns on a spindle and is divided into 38 pockets. Thirty-six of the pockets are numbered 1–36, of which half are red and half are black. Two of the pockets are green and are numbered 0 and 00 (see figure). The dealer spins the wheel and a small ball in opposite directions. As the ball slows to a stop, it has an equal probability of landing in any of the numbered pockets. 10 25 29 12 8 2 7 00 1 19 1 3 13 36 20 35 14 2 0 8 2 11 30 26 9 (a) Find the probability of landing in the number 00 pocket. (b) Find the probability of landing in a red pocket. (c) Find the probability of landing in a green pocket or a black pocket. (d) Find the probability of landing in the number 14 pocket on two consecutive spins. (e) Find the probability of landing in a red pocket on three consecutive spins. (f) European roulette does not contain the 00 pocket. Repeat parts (a)–(e) for European roulette. How do the probabilities for European roulette compare with the probabilities for American roulette? 333202_0907.qxd 12/5/05 11:41 AM Page 713 56. Estimating is dropped onto a paper that contains a grid of squares units on a side (see figure). d A coin of diameter d (a) Find the probability that the coin covers a vertex of one of the squares on the grid. (b) Perform the experiment 100 times and use the results to approximate . Synthesis True or False? the statement is true or false. Justify your answer. In Exercises 57 and 58, determine whether 57. If A B and A ties, then are independent events with nonzero probabilican occur when occurs. B 58. Rolling a number less than 3 on a normal six-sided die has a probability of . The complement of this event is to roll a number greater than 3, and its probability is 1 3 1 2. 59. Pattern Recognition and Exploration Consider a group of people. n (a) Explain why the following pattern gives the probabili- ties that the people have distinct birthdays. n n 2: n 3: 365 365 365 365 364 365 364 365 365 364 3652 365 364 363 3653 363 365 (b) Use the pattern in part (a) to write an expression for the people have distinct birthdays. probability that n 4 (c) Let Pn be the probability that the people have distinct birthdays. Verify that this probability can be obtained recursively by n P1 1 and Pn 365 n 1 365 Pn1. (d) Explain why gives the probability that at least two people in a group of people have the same birthday. Qn 1 Pn n Section 9.7 Probability 713 (e) Use the results of parts (c) and (d) to complete the table. n Pn Qn 10 15 20 23 30 40 50 (f) How many people must be in a group so that the probability of at least two of them having the same birthday is greater than Explain. 1 2? 60. Think About It A weather forecast indicates that the probability of rain is 40%. What does this mean? Skills Review In Exercises 61–70, find all real solutions of the equation. 61. 62. 63. 64. 65. 66. 67. 68. 69. 70. 6x2 8 0 4x2 6x 12 0 x 3 x2 3x 0 x 5 x 3 2x 0 12 x 3 32 x 2x 2 x 5 3 2x 3 3 x 2 2 x 4 1 2x 3 1 4 x x 2 5 x 2 13 x2 2x In Exercises 71–74, sketch the graph of the solution set of the system of inequalities. 71. 72 5x 2y ≥ 10 x2 y ≥ 2 y ≥ x 4 74. x2 y2 ≤ 4 x y ≥ 2 73. 333202_090R.qxd 12/5/05 11:43 AM Page 714 714 Chapter 9 Sequences, Series, and Probability 9 Chapter Summary What did you learn? Section 9.1 Use sequence notation to write the terms of sequences (p. 642). Use factorial notation (p. 644). Use summation notation to write sums (p. 646). Find the sums of infinite series (p. 647). Use sequences and series to model and solve real-life problems (p. 648). Section 9.2 Recognize, write, and find the nth terms of arithmetic sequences (p. 653). Find nth partial sums of arithmetic sequences (p. 656). Use arithmetic sequences to model and solve real-life problems (p. 657). Section 9.3 Recognize, write, and find the nth terms of geometric sequences (p. 663). Find nth partial sums of geometric sequences (p. 666). Find sums of infinite geometric series (p. 667). Use geometric sequences to model and solve real-life problems (p. 668). Section 9.4 Use mathematical induction to prove statements involving a positive integer n (p. 673). Recognize patterns and write the nth term of a sequence (p. 677). Find the sums of powers of integers (p. 679). Find finite differences of sequences (p. 680). Section 9.5 Use the Binomial Theorem to calculate binomial coefficients (p. 683). Use Pascal’s Triangle to calculate binomial coefficients (p. 685). Use binomial coefficients to write binomial expansions (p. 686). Section 9.6 Solve simple counting problems (p. 691). Use the Fundamental Counting Principle to solve counting problems (p. 692). Use permutations to solve counting problems (p. 693). Use combinations to solve counting problems (p. 696). Section 9.7 Find the probabilities of events (p. 701). Find the probabilities of mutually exclusive events (p. 705). Find the probabilities of independent events (p. 707). Find the probability of the complement of an event (p. 708). Review Exercises 1–8 9–12 13–20 21–24 25, 26 27–40 41–46 47, 48 49–60 61–70 71–76 77, 78 79–82 83–86 87–90 91–94 95–98 99–102 103–108 109, 110 111, 112 113, 114 115, 116 117, 118 119, 120 121, 122 123, 124 333202_090R.qxd 12/5/05 11:43 AM Page 715 9 Review Exercises Review Exercises 715 In Exercises 1–4, write the first five terms of the 9.1 sequence. (Assume that begins with 1.) n 1. an 2. an 2 6 n 1n 5n 2n 1 3. an 72 n! 4. an nn 1 In Exercises 5–8, write an expression for the apparent th term of the sequence. (Assume that begins with 1.) n n 5. 6. 7. 2, 2, 2, 2, 2, . . . 1, 2, 7, 14, 23, . . . 4, 2, 4 3, 1, 4 5, . . . 8. 1, 1 2, 1 3, 1 4, 1 5, . . . 25. C
ompound Interest A deposit of $10,000 is made in an account that earns 8% interest compounded monthly. The n balance in the account after months is given by n , n 1, 2, 3, . . . 10,0001 0.08 12 An (a) Write the first 10 terms of this sequence. (b) Find the balance in this account after 10 years by find- ing the 120th term of the sequence. 26. Education The enrollment (in thousands) in Head Start programs in the United States from 1994 to 2002 can be approximated by the model an an 1.07n2 6.1n 693, n 4, 5, . . ., 12 n is the year, with n 4 where corresponding to 1994. Find the terms of this finite sequence. Use a graphing utility to construct a bar graph that represents the sequence. (Source: U.S. Administration for Children and Families) In Exercises 9–12, simplify the factorial expression. In Exercises 27–30, determine whether the sequence 9.2 is arithmetic. If so, find the common difference. 9. 5! 11. 3! 5! 6! 10. 12. 3! 2! 7! 6! 6! 8! In Exercises 13–18, find the sum. 13. 15. 17. 6 i1 5 6 j 2 2k3 4 j 1 10 k1 14. 16. 18. 5 k2 8 i1 4k i i 1 4 j 0 j 2 1 In Exercises 19 and 20, use sigma notation to write the sum. 19. 20. . . . 1 220 1 21 1 2 2 3 1 22 3 4 1 23 . . . 9 10 In Exercises 21–24, find the sum of the infinite series. 21. 23. i1 k1 5 10i 2 100 k 22. 24. i1 k2 3 10i 27. 5, 3, 1, 1 3 2, 2, 29. 1, 1, 5 2, 2, . . . 3, . . . 28. 0, 1, 3, 6, 10, . . . 6 9, . . . 30. 7 9, 8 9, 5 9, 9 9, In Exercises 31–34, write the first five terms of the arithmetic sequence. 31. 33. 34. a1 a1 a1 4, d 3 25, ak1 4.2, ak1 ak ak 3 0.4 32. a1 6, d 2 In Exercises 35–40, find a formula for sequence. an for the arithmetic 35. 37. 39. a1 a1 a2 7, y, 93, d 12 d 3y a6 65 36. 38. 40. a1 a1 a 7 25, 2x, 8, a13 d 3 d x 6 In Exercises 41–44, find the partial sum. 8 2j 3 10 42. 41. j1 20 3j j1 43. 3k 4 2 11 k1 44. 25 k1 3k 1 4 9 10 k 45. Find the sum of the first 100 positive multiples of 5. 46. Find the sum of the integers from 20 to 80 (inclusive). 333202_090R.qxd 12/5/05 11:43 AM Page 716 716 Chapter 9 Sequences, Series, and Probability 47. Job Offer The starting salary for an accountant is $34,000 with a guaranteed salary increase of $2250 per year. Determine (a) the salary during the fifth year and (b) the total compensation through 5 full years of employment. 48. Baling Hay In the first two trips baling hay around a large field, a farmer obtains 123 bales and 112 bales, respectively. Because each round gets shorter, the farmer estimates that the same pattern will continue. Estimate the total number of bales made if the farmer takes another six trips around the field. 75. k1 42 3 k1 76. k1 1.3 1 10 k1 77. Depreciation A paper manufacturer buys a machine for $120,000. During the next 5 years, it will depreciate at a rate of 30% per year. (That is, at the end of each year the depreciated value will be 70% of what it was at the beginning of the year.) (a) Find the formula for the th term of a geometric sequence that gives the value of the machine full years after it was purchased. t n In Exercises 49–52, determine whether the sequence 9.3 is geometric. If so, find the common ratio. (b) Find the depreciated value of the machine after 5 full years. 49. 51. 5, 10, 20, 40, . . . 1 3, 8 3, 2 3, . . . 3, 4 50. 52. 54, 18, 6, 2, . . . 1 4, 2 7, . . . 5, 3 6, 4 In Exercises 53–56, write the first five terms of the geometric sequence. 53. 55. a1 a1 4, r 1 4 9, a3 4 54. 56. a1 a1 2, r 2 2, a3 12 In Exercises 57–60, write an expression for the th term of the geometric sequence. Then find the 20th term of the sequence. n 57. 59. a1 a1 16, 8 a2 100, r 1.05 58. 60. a3 a1 6, 5, 1 a4 r 0.2 In Exercises 61–66, find the sum of the finite geometric sequence. 7 3i1 2i1 5 61. 62. i1 i1 63. 65. 4 i1 5 i1 i 1 2 2i1 64. 66. i1 1 3 6 i1 4 i1 63i In Exercises 67–70, use a graphing utility to find the sum of the finite geometric sequence. 67. 69. 10 i1 25 i1 103 5 i1 1001.06i1 68. 70. 15 i1 200.2i1 i1 86 5 20 i1 In Exercises 71–76, find the sum of the infinite geometric series. 71. 73. i1 i1 i1 7 8 0.1i1 72. 74. i1 i1 i1 1 3 0.5i1 78. Annuity You deposit $200 in an account at the beginning of each month for 10 years. The account pays 6% compounded monthly. What will your balance be at the end of 10 years? What would the balance be if the interest were compounded continuously? In Exercises 79–82, use mathematical induction to 79. 80. 9.4 n. prove the formula for every positive integer 3 5 7 . . . 2n 1 nn ari a1 rn 1 r a kd n 2 2 5 2 2a n 1d n1 n1 81. 82. i0 k0 n 3 In Exercises 83–86, find a formula for the sum of the first terms of the sequence. n 83. 9, 13, 17, 21, . . . 9 25, 85. 1, 27 125, . . . 3 5, 84. 68, 60, 52, 44, . . . 86. 12, 1, 1 12, 1 144, . . . In Exercises 87–90, find the sum using the formulas for the sums of powers of integers. 87. 89. 30 n1 7 n1 n n 4 n 88. 90. 10 n1 6 n1 n2 n 5 n2 In Exercises 91–94, write the first five terms of the sequence beginning with the given term. Then calculate the first and second differences of the sequence. State whether the sequence has a linear model, a quadratic model, or neither. 91. 93. a1 an a1 an 5 an1 16 an1 5 1 92. 94. a1 an a0 an 2n 3 an1 0 n an1 333202_090R.qxd 12/5/05 11:43 AM Page 717 In Exercises 95–98, use the Binomial Theorem to 9.5 calculate the binomial coefficient. 95. 97. 6C4 8C5 96. 98. 10C7 12C3 In Exercises 99–102, use Pascal’s Triangle to calculate the binomial coefficient. 101. 102. 100. 99. 9 4 5 3 7 3 8 6 In Exercises 103–108, use the Binomial Theorem to expand and simplify the expression. (Remember that i 1. ) 103. 104. 105. 106. 107. 108. x 44 x 36 a 3b5 3x y 27 5 2i 4 4 5i 3 Review Exercises 717 116. Menu Choices A local sub shop offers five different breads, seven different meats, three different cheeses, and six different vegetables. Find the total number of combinations of sandwiches possible. 9.7 117. Apparel A man has five pairs of socks, of which no two pairs are the same color. He randomly selects two socks from a drawer. What is the probability that he gets a matched pair? 118. Bookshelf Order A child returns a five-volume set of books to a bookshelf. The child is not able to read, and so cannot distinguish one volume from another. What is the probability that the books are shelved in the correct order? 119. Students by Class At a particular university, the numbers of students in the four classes are broken down by percents, as shown in the table. Class Percent Freshmen Sophomores Juniors Seniors 31 26 25 18 9.6 109. Numbers in a Hat Slips of paper numbered 1 through 14 are placed in a hat. In how many ways can you draw two numbers with replacement that total 12? 110. Home Theater Systems A customer in an electronics store can choose one of six speaker systems, one of five DVD players, and one of six plasma televisions to design a home theater system. How many systems can be designed? 111. Telephone Numbers The same three-digit prefix is used for all of the telephone numbers in a small town. How many different telephone numbers are possible by changing only the last four digits? 112. Course Schedule A college student is preparing a course schedule for the next semester. The student may select one of three mathematics courses, one of four science courses, and one of six history courses. How many schedules are possible? 113. Bike Race There are 10 bicyclists entered in a race. In how many different ways could the top three places be decided? 114. Jury Selection A group of potential jurors has been narrowed down to 32 people. In how many ways can a jury of 12 people be selected? 115. Apparel You have eight different suits to choose from to take on a trip. How many combinations of three suits could you take on your trip? A single student is picked randomly by lottery for a cash scholarship. What is the probability that the scholarship winner is (a) a junior or senior? (b) a freshman, sophomore, or junior? 120. Data Analysis A sample of college students, faculty, and administration were asked whether they favored a proposed increase in the annual activity fee to enhance student life on campus. The results of the study are listed in the table. Students Faculty Admin. Total Favor Oppose Total 237 163 400 37 38 75 18 7 25 292 208 500 A person is selected at random from the sample. Find each specified probability. (a) The person is not in favor of the proposal. (b) The person is a student. (c) The person is a faculty member and is in favor of the proposal. 333202_090R.qxd 12/5/05 11:43 AM Page 718 718 Chapter 9 Sequences, Series, and Probability 121. Tossing a Die A six-sided die is tossed three times. What is the probability of getting a 6 on each roll? 122. Tossing a Die A six-sided die is tossed six times. What is the probability that each side appears exactly once? 123. Drawing a Card You randomly select a card from a 52-card deck. What is the probability that the card is not a club? 124. Tossing a Coin Find the probability of obtaining at least one tail when a coin is tossed five times. Synthesis True or False? the statement is true or false. Justify your answer. In Exercises 125–129, determine whether n 2n 1 i3 5 i1 2i i1 125. 126. 127. 128. n 2! n! i 3 2i 5 5 i1 8 k1 3k 3 8 k1 k 6 j1 2 j 8 j3 2 j2 129. The value of nCr . 130. Think About It An infinite sequence is a function. What is always greater than the value of nPr is the domain of the function? 131. Think About It How do the two sequences differ? (a) an (b) an 1n n 1n1 n 132. Graphical Reasoning The graphs of two sequences are shown below. Identify each sequence as arithmetic or geometric. Explain your reasoning. (a) an 4 −2 −8 −12 −16 −20 2 86 10 n (b) an 100 80 60 40 20 −20 Graphical Reasoning In Exercises 135–138, match the sequence or sum of a sequence with its graph without doing any calculations. Explain your reasoning. [The graphs are labeled (a), (b), (c), and (d).] (b) 10 (d) 0 0 5 0 0 10 10 10 10 (a) 6 0 −4 (c) 5 0 0 135. 136. 137. an an an n1 2 41 41 n 41 2 2 k1 n1 k1 138. an n k1 41 2 k1 139. Population Growth Con
sider an idealized population with the characteristic that each member of the population produces one offspring at the end of every time period. If each member has a life span of three time periods and the population begins with 10 newborn members, then the following table shows the population during the first five time periods. Age Bracket 0–1 1–2 2–3 Total Time Period 1 10 2 10 10 10 20 3 20 10 10 40 4 40 20 10 70 5 70 40 20 130 2 864 10 n Sn Sn1 Sn2 Sn3, n > 3. The sequence for the total population has the property that 133. Writing Explain what is meant by a recursion formula. 134. Writing Explain why the terms of a geometric sequence decrease when 0 < r < 1. Find the total population during the next five time periods. 140. The probability of an event must be a real number in what interval? Is the interval open or closed? 333202_090R.qxd 12/5/05 11:43 AM Page 719 9 Chapter Test Chapter Test 719 Take this test as you would take a test in class. When you are finished, check your work against the answers given in the back of the book. 1. Write the first five terms of the sequence 1n 3n 2 2. Write an expression for the th term of the sequence. an n . n (Assume that begins with 1.) 3 1! , 4 2! , 5 3! , 6 4! , 7 5! , . . . 3. Find the next three terms of the series. Then find the fifth partial sum of the series. 6 17 28 39 . . . 4. The fifth term of an arithmetic sequence is 5.4, and the 12th term is 11.0. Find the n th term. 5. Write the first five terms of the sequence an 52n1. n (Assume that begins with 1.) i1 41 2 i . In Exercises 6 –8, find the sum. 6. 50 i1 2i 2 5. 7. 7 n1 8n 5 8. 9. Use mathematical induction to prove the formula. 5 10 15 . . . 5n 5nn 1 2 10. Use the Binomial Theorem to expand the expression x 2y4. 11. Find the coefficient of the term a3 b5 in the expansion of 2a 3b8. In Exercises 12 and 13, evaluate each expression. 12. (a) 13. (a) 9 P2 11C4 (b) (b) 70 P3 66C4 14. How many distinct license plates can be issued consisting of one letter followed by a three-digit number? 15. Eight people are going for a ride in a boat that seats eight people. The owner of the boat will drive, and only three of the remaining people are willing to ride in the two bow seats. How many seating arrangements are possible? 16. You attend a karaoke night and hope to hear your favorite song. The karaoke song book has 300 different songs (your favorite song is among the 300 songs). Assuming that the singers are equally likely to pick any song and no song is repeated, what is the probability that your favorite song is one of the 20 that you hear that night? 17. You are with seven of your friends at a party. Names of all of the 60 guests are placed in a hat and drawn randomly to award eight door prizes. Each guest is limited to one prize. What is the probability that you and your friends win all eight of the prizes? 18. The weather report calls for a 75% chance of snow. According to this report, what is the probability that it will not snow? 333202_090R.qxd 12/8/05 10:55 AM Page 720 720 Chapter 9 Sequences, Series, and Probability 9 Cumulative Test for Chapters 7–9 Take this test to review the material from earlier chapters. When you are finished, check your work against the answers given in the back of the book. In Exercises 1– 4, solve the system by the specified method. 1. Substitution y 2 y 2 3 x2 x 1 3. Elimination 2x 4y z x 2y 2z x 3y z 3 6 1 2. Elimination x 3y 2x 4y 1 0 4. Gauss-Jordan Elimination 7 5 3 x 3y 2z 2x y z 4x y z In Exercises 5 and 6, sketch the graph of the solution set of the system of inequalities. 5. 2x y ≥ x 3y ≤ 3 2 6. x y > 5x 2y < 6 10 7. Sketch the region determined by the constraints. Then find the minimum and z 3x 2y, maximum values, and where they occur, of the objective function subject to the indicated constraints. x 4y ≤ 20 2x y ≤ 12 x ≥ 0 y ≥ 0 8. A custom-blend bird seed is to be mixed from seed mixtures costing $0.75 per pound and $1.25 per pound. How many pounds of each seed mixture are used to make 200 pounds of custom-blend bird seed costing $0.95 per pound? x 2x 3x 2y y 3y z 2z 4z 9 9 7 SYSTEM FOR 10 AND 11 8 1 2 0 3 6 5 1 4 MATRIX FOR 16 9. Find the equation of the parabola 6, 4. 3, 1, and y ax2 bx c passing through the points 0, 4, In Exercises 10 and 11, use the system of equations at the left. 10. Write the augmented matrix corresponding to the system of equations. 11. Solve the system using the matrix found in Exercise 10 and Gauss-Jordan elimination. In Exercises 12–15, use the following matrices to find each of the following, if possible. A [ 4 1 A B A 2B 14. 12. 0 2], B [1 1 3 0] 13. 15. 2B AB 16. Find the determinant of the matrix at the left. 17. Find the inverse of the matrix (if it exists): 1 3 5 2 7 7 . 1 10 15 333202_090R.qxd 12/8/05 10:56 AM Page 721 Age group 14 17 18 24 2534 0.09 0.06 0.12 Gym Jogging Walking shoes shoes 0.09 0.10 0.25 shoes 0.03 0.05 0.12 MATRIX FOR 18 y 6 5 2 1 − ( 2, 3) (1, 5) (4, 1) −2 −1 1 2 3 4 x FIGURE FOR 21 Cumulative Test for Chapters 7–9 721 18. The percents (by age group) of the total amounts spent on three types of footwear in a recent year are shown in the matrix. The total amounts (in millions) spent by each age group on the three types of footwear were $442.20 (14–17 age group), $466.57(18–24 age group), and $1088.09 (25–34 age group). How many dollars worth of gym shoes, jogging shoes, and walking shoes were sold that year? (Source: National Sporting Goods Association) In Exercises 19 and 20, use Cramer’s Rule to solve the system of equations. 19. 20. 52 5 8x 3y 3x 5y 5x 4y 3z 3x 8y 7z 7x 5y 6z 7 9 53 21. Find the area of the triangle shown in the figure. 22. Write the first five terms of the sequence 1n1 2n 3 23. Write an expression for the th term of the sequence. an n n (assume that begins with 1). 2! 4 , 3! 5 , 4! 6 , 5! 7 , 6! 8 , . . . 24. Find the sum of the first 20 terms of the arithmetic sequence 8, 12, 16, 20, . . . . 25. The sixth term of an arithmetic sequence is 20.6, and the ninth term is 30.2. (a) Find the 20th term. (b) Find the th term. n 26. Write the first five terms of the sequence 32n1 an n (assume that begins with 1). 27. Find the sum: i0 1.3 1 10 i1 . 28. Use mathematical induction to prove the formula 3 7 11 15 . . . 4n 1 n2n 1. 29. Use the Binomial Theorem to expand and simplify z 34. In Exercises 30–33, evaluate the expression. 30. 7P3 31. 25P2 32. 8 4 33. 10C3 In Exercises 34 and 35, find the number of distinguishable permutations of the group of letters. 34. B, A, S, K, E, T, B, A, L, L 35. A, N, T, A, R, C, T, I, C, A 36. A personnel manager at a department store has 10 applicants to fill three different sales positions. In how many ways can this be done, assuming that all the applicants are qualified for any of the three positions? 37. On a game show, the digits 3, 4, and 5 must be arranged in the proper order to form the price of an appliance. If the digits are arranged correctly, the contestant wins the appliance. What is the probability of winning if the contestant knows that the price is at least $400? 333202_090R.qxd 12/5/05 11:43 AM Page 722 Proofs in Mathematics Properties of Sums (p. 647) 1. 2. 3. n i1 c cn, c is a constant. n i1 cai cn ai, i1 c is a constant. n i1 ai bi n i1 ai n i1 bi 4. n i1 ai bi n ai i1 n i1 bi Proof Each of these properties follows directly from the properties of real numbers. 1. n i1 c c c c . . . c cn n terms The Distributive Property is used in the proof of Property 2. 2. n i1 cai ca1 ca2 ca3 . . . can ca1 a2 a3 . . . an cn i1 ai The proof of Property 3 uses the Commutative and Associative Properties of Addition. 3. n i1 ai bi a1 a1 n i1 b1 a2 b2 a3 . . . an bn b3 b1 b2 b3 . . . bn a2 a3 . . . an ai n i1 bi The proof of Property 4 uses the Commutative and Associative Properties of Addition and the Distributive Property. 4. n i1 ai bi b1 a2 b2 a3 . . . an bn a2 a2 a3 a3 . . . an . . . an b2 b3 b2 b3 . . . bn . . . bn b3 b1 b1 a1 a1 a1 n i1 ai n i1 bi Infinite Series The study of infinite series was considered a novelty in the fourteenth century. Logician Richard Suiseth, whose nickname was Calculator, solved this problem. If throughout the first half of a given time interval a variation continues at a certain intensity; throughout the next quarter of the interval at double the intensity; throughout the following eighth at triple the intensity and so ad infinitum; The average intensity for the whole interval will be the intensity of the variation during the second subinterval (or double the intensity). This is the same as saying that the sum of the infinite series 2n . . . is 2. 722 333202_090R.qxd 12/5/05 11:43 AM Page 723 The Sum of a Finite Arithmetic Sequence The sum of a finite arithmetic sequence with terms is n (p. 656) Sn n 2 a1 an . Proof Begin by generating the terms of the arithmetic sequence in two ways. In the first way, repeatedly add a1 a1 to the first term to obtain an1 d a3 d a1 . . . an2 a2 a1 n 1d. Sn In the second way, repeatedly subtract from the th term to obtain . . . a3 d an 2d . . . a1 n a1 2d . . . an Sn, d a1 . . . a1 a2 an n 1d. the multiples of subtract out and you obtain an n terms Sn an an an2 an1 d an an If you add these two versions of an an an a1 2Sn 2Sn a1 na1 n 2 a1 Sn an . The Sum of a Finite Geometric Sequence The sum of the finite geometric sequence (p. 666) a1, a1r, a1r 2, a1r 3, a1r 4, . . . , a1r n1 n is given by Sn r 1 with common ratio i1 a1r i1 a11 r n 1 r . Proof Sn rSn a1r a1r 2 . . . a1rn2 a1r n1 a1 a1r a1r 2 a1r 3 . . . a1rn1 a1r n Subtracting the second equation from the first yields Multiply by r. a1 rSn Sn 1 r a1 Sn a1r n. 1 r n, So, and, because r 1, you have Sn a11 r n 1 r . 723 333202_090R.qxd 12/5/05 11:43 AM Page 724 The Binomial Theorem (p. 683) In the expansion of x yn x yn xn nx n1y . . . x nryr the coefficient of is nCr x nry r . . . nxy n1 y n nCr n! n r!r! . Proof The Binomial Theorem can be proved quite nicely using mathematical induction. The steps are straightforward but look a little messy, so only a
n outline of the proof is presented. n 1, x y1 x1 y1 and the formula is you have 1. If 1C0x 1C1y, valid. 2. Assuming that the formula is true for kCr k! k r!r! the coefficient of n k, kk 1k 2 . . . k r 1 r! . xkryr is To show that the formula is true for x k1ryr in the expansion of x yk1 x ykx y. n k 1, look at the coefficient of x k1ryr x k1ry r kCr1x k1ry r1y k! k 1 r!r 1! From the right-hand side, you can determine that the term involving is the sum of two products. kCr x kry rx k! k r!r! k 1 rk! k 1 r!r! k!k 1 r r k 1 r!r! k 1! k 1 r!r! k1Cr xk1ryr k!r k 1 r!r! x k1ry r x k1ry r x k1ry r So, by mathematical induction, the Binomial Theorem is valid for all positive integers n. 724 333202_090R.qxd 12/5/05 11:43 AM Page 725 P.S. Problem Solving This collection of thought-provoking and challenging exercises further explores and expands upon concepts learned in this chapter. 1. Let 1 and consider the sequence xn given by x0 1 2 xn xn1 1 xn1 , n 1, 2, . . . Use a graphing utility to compute the first 10 terms of the n sequence and make a conjecture about the value of approaches infinity. as xn 2. Consider the sequence n 1 n2 1 an . (a) Use a graphing utility to graph the first 10 terms of the sequence. (b) Use the graph from part (a) to estimate the value of an as approaches infinity. n (c) Complete the table. 1 10 100 1000 10,000 n an (d) Use the table from part (c) to determine (if possible) the value of an as approaches infinity. n 3. Consider the sequence 3 1n. an (a) Use a graphing utility to graph the first 10 terms of the sequence. (b) Use the graph from part (a) to describe the behavior of the graph of the sequence. (c) Complete the table. 1 10 101 1000 10,001 n an (d) Use the table from part (c) to determine (if possible) the value of an as approaches infinity. n 4. The following operations are performed on each term of an arithmetic sequence. Determine if the resulting sequence is arithmetic, and if so, state the common difference. (a) A constant C is added to each term. (b) Each term is multiplied by a nonzero constant C. (c) Each term is squared. 5. The following sequence of perfect squares is not arithmetic. 1, 4, 9, 16, 25, 36, 49, 64, 81, . . . However, you can form a related sequence that is arithmetic by finding the differences of consecutive terms. (a) Write the first eight terms of the related arithmetic n th term of this sequence described above. What is the sequence? (b) Describe how you can find an arithmetic sequence that is related to the following sequence of perfect cubes. 1, 8, 27, 64, 125, 216, 343, 512, 729, . . . (c) Write the first seven terms of the related sequence in part (b) and find the th term of the sequence. n (d) Describe how you can find the arithmetic sequence that is related to the following sequence of perfect fourth powers. 1, 16, 81, 256, 625, 1296, 2401, 4096, 6561, . . . (e) Write the first six terms of the related sequence in part (d) and find the th term of the sequence. n 6. Can the Greek hero Achilles, running at 20 feet per second, ever catch a tortoise, starting 20 feet ahead of Achilles and running at 10 feet per second? The Greek mathematician Zeno said no. When Achilles runs 20 feet, the tortoise will be 10 feet ahead. Then, when Achilles runs 10 feet, the tortoise will be 5 feet ahead. Achilles will keep cutting the distance in half but will never catch the tortoise. The table shows Zeno’s reasoning. From the table you can see that both the distances and the times required to achieve them form infinite geometric series. Using the table, show that both series have finite sums. What do these sums represent? Distance (in feet) Time (in seconds) 20 10 5 2.5 1.25 0.625 1 0.5 0.25 0.125 0.0625 0.03125 7. Recall that a fractal is a geometric figure that consists of a pattern that is repeated infinitely on a smaller and smaller scale. A well-known fractal is called the Sierpinski Triangle. In the first stage, the midpoints of the three sides are used to create the vertices of a new triangle, which is then removed, leaving three triangles. The first three stages are shown on the next page. Note that each remaining triangle is similar to the original triangle. Assume that the length of each side of the original triangle is one unit. 725 333202_090R.qxd 12/5/05 11:43 AM Page 726 Write a formula that describes the side length of the triangles n that will be generated in the th stage. Write a formula for the area of the triangles that will be generated in the th stage. n FIGURE FOR 7 8. You can define a sequence using a piecewise formula. The following is an example of a piecewise-defined sequence. a1 7, an an1 2 3an1 , if an1 is even 1, if an1 is odd (a) Write the first 10 terms of the sequence. 7. (b) Choose three different values for For each value of a1 other than a1, a1 find the first 10 terms of the sequence. What conclusions can you make about the behavior of this sequence? 9. The numbers 1, 5, 12, 22, 35, 51, are called pentagonal numbers because they represent the numbers of dots used to make pentagons, as shown below. Use mathematical induction to prove that the th pentagonal number Pn . . . n is given by n3n 1 2 . Pn 10. What conclusion can be drawn from the following infor- mation about the sequence of statements Pk implies (a) Pn? Pk1. are all true. (b) (c) (d) P3 is true and P1, P2, P3, . . . , P50 P1, P2, and P3 Pk1 imply that P2 P2k is true and f1, f2, . . . , fn, . . . is true. implies P2k2. 11. Let be the Fibonacci sequence. are all true, but the truth of Pk does not (a) Use mathematical induction to prove that f1 f2 . . . fn fn2 1. (b) Find the sum of the first 20 terms of the Fibonacci sequence. 726 12. The odds in favor of an event occurring are the ratio of the probability that the event will occur to the probability that the event will not occur. The reciprocal of this ratio represents the odds against the event occurring. (a) Six marbles in a bag are red. The odds against choosing a red marble are 4 to 1. How many marbles are in the bag? (b) A bag contains three blue marbles and seven yellow marbles. What are the odds in favor of choosing a blue marble? What are the odds against choosing a blue marble? (c) Write a formula for converting the odds in favor of an event to the probability of the event. (d) Write a formula for converting the probability of an event to the odds in favor of the event. 13. You are taking a test that contains only multiple choice questions (there are five choices for each question). You are on the last question and you know that the answer is not B or D, but you are not sure about answers A, C, and E. What is the probability that you will get the right answer if you take a guess? 14. A dart is thrown at the circular target shown below. The dart is equally likely to hit any point inside the target. What is the probability that it hits the region outside the triangle? 6 n V A p1, p2, . . ., pn. The expected value x1, x2, . . ., xn. ring are A and their values, 15. An event has possible outcomes, which have the values n The probabilities of the outcomes occurof an event is the sum of the products of the outcomes’ probabilities p2x2 (a) To win California’s Super Lotto Plus game, you must match five different numbers chosen from the numbers 1 to 47, plus one Mega number chosen from the numbers 1 to 27. You purchase a ticket for $1. If the jackpot for the next drawing is $12,000,000, what is the expected value for the ticket? . . . pnxn. V p1x1 (b) You are playing a dice game in which you need to score 60 points to win. On each turn, you roll two sixsided dice. Your score for the turn is 0 if the dice do not show the same number, and the product of the numbers on the dice if they do show the same number. What is the expected value for each turn? How many turns will it take on average to score 60 points? 1010 333202_1000.qxd 12/8/05 8:52 AM Page 727 Topics in Analytic Geometry 10.1 Lines 10.2 Introduction to Conics: Parabolas 10.3 Ellipses 10.4 Hyperbolas 10.5 Rotation of Conics 10.6 Parametric Equations 10.7 Polar Coordinates 10.8 Graphs of Polar Equations 10.9 Polar Equations of Conics The nine planets move about the sun in elliptical orbits. You can use the techniques presented in this chapter to determine the distances between the planets and the center of the sun AT I O N S Analytic geometry concepts have many real-life applications. The applications listed below represent a small sample of the applications in this chapter. • Inclined Plane, Exercise 56, page 734 • Satellite Orbit, Exercise 60, page 752 • Projectile Motion, Exercises 57 and 58, page 777 • Revenue, • LORAN, Exercise 59, page 741 Exercise 42, page 761 • Architecture, Exercise 57, page 751 • Running Path, Exercise 44, page 762 • Planetary Motion, Exercises 51–56, page 798 • Locating an Explosion, Exercise 40, page 802 727 333202_1001.qxd 12/8/05 8:54 AM Page 728 728 Chapter 10 Topics in Analytic Geometry 10.1 Lines What you should learn • Find the inclination of a line. • Find the angle between two lines. • Find the distance between a point and a line. Why you should learn it The inclination of a line can be used to measure heights indirectly. For instance, in Exercise 56 on page 734, the inclination of a line can be used to determine the change in elevation from the base to the top of the Johnstown Inclined Plane. Inclination of a Line In Section 1.3, you learned that the graph of the linear equation y mx b 0, b. is a nonvertical line with slope There, the slope of a line x. was described as the rate of change in with respect to In this section, you will look at the slope of a line in terms of the angle of inclination of the line. and -intercept y y m Every nonhorizontal line must intersect the -axis. The angle formed by such an intersection determines the inclination of the line, as specified in the following definition. x Definition of Inclination The inclination of a nonhorizontal line is the positive angle (less than measured counterclockwise from the
-axis to the line. (See Figure 10.1.) x ) y y y y = 0θ AP/Wide World Photos =θ π 2 x x θ x θ x Horizontal Line FIGURE 10.1 Vertical Line Acute Angle Obtuse Angle The inclination of a line is related to its slope in the following manner. Inclination and Slope If a nonvertical line has inclination and slope m, then m tan . The HM mathSpace® CD-ROM and Eduspace® for this text contain additional resources related to the concepts discussed in this chapter. For a proof of the relation between inclination and slope, see Proofs in Mathematics on page 806. 333202_1001.qxd 12/8/05 8:54 AM Page 729 y 3 1 FIGURE 10.2 2x + 3y = 6 θ ≈ 146.3° 1 2 3 y θ θ 2= − θ 1 θ θ 1 θ 2 FIGURE 10.3 x x Section 10.1 Lines 729 Example 1 Finding the Inclination of a Line Find the inclination of the line 2x 3y 6. Solution The slope of this line is equation tan 2 3 . m 2 3. So, its inclination is determined from the < < . This means that From Figure 10.2, it follows that arctan 2 3 2 0.588 0.588 2.554. The angle of inclination is about 2.554 radians or about 146.3. Now try Exercise 19. The Angle Between Two Lines Two distinct lines in a plane are either parallel or intersecting. If they intersect and are nonperpendicular, their intersection forms two pairs of opposite angles. One pair is acute and the other pair is obtuse. The smaller of these angles is called the angle between the two lines. As shown in Figure 10.3, you can use the inclinations of the two lines to find the angle between the two lines. If two lines have inclinations the angle between the and two lines is 1 and 2, 1 < 2, 1 < where 2 2 2 1. You can use the formula for the tangent of the difference of two angles tan tan 2 tan 2 1 tan 1 tan 1 1 tan 2 to obtain the formula for the angle between two lines. Angle Between Two Lines If two nonperpendicular lines have slopes two lines is tan m2 1 m1m2. m1 m1 and m2, the angle between the 333202_1001.qxd 12/8/05 8:54 AM Page 730 730 Chapter 10 Topics in Analytic Geometry y 4 2 1 3x + 4y − 12 = 0 θ ≈ 79.70° 2x − y − 4 = 0 Example 2 Finding the Angle Between Two Lines Find the angle between the two lines. Line 1: 2x y 4 0 Line 2: 3x 4y 12 0 Solution 2 The two lines have slopes of of the angle between the two lines is m1 tan m2 1 m1m2 m1 1 234 34 2 24 114 11 2 . Finally, you can conclude that the angle is and m2 3 4, respectively. So, the tangent 1 3 4 x arctan 11 2 1.391 radians 79.70 FIGURE 10.4 as shown in Figure 10.4. Now try Exercise 27. (x1, y1) d y (x2, y2) FIGURE 10.5 y 4 3 2 1 y = 2x + 1 (4, 1) −3 −2 1 2 3 4 5 −2 −3 −4 x x The Distance Between a Point and a Line Finding the distance between a line and a point not on the line is an application of perpendicular lines. This distance is defined as the length of the perpendicular line segment joining the point and the line, as shown in Figure 10.5. Distance Between a Point and a Line The distance between the point By1 A2 B2 C x1, y1 Ax1 d . and the line Ax By C 0 is Remember that the values of to the general equation of a line, between a point and a line, see Proofs in Mathematics on page 806. B,A, C Ax By C 0. in this distance formula correspond For a proof of the distance and Example 3 Finding the Distance Between a Point and a Line Find the distance between the point 4, 1 and the line y 2x 1. Solution The general form of the equation is 2x y 1 0. So, the distance between the point and the line is d 24 11 1 22 12 8 5 3.58 units. The line and the point are shown in Figure 10.6. FIGURE 10.6 Now try Exercise 39. 333202_1001.qxd 12/8/05 8:54 AM Page 731 y 6 5 4 2 1 B (0, 4) h C (5, 2) A (−3, 0) 1 2 3 4 5 x point −2 FIGURE 10.7 Section 10.1 Lines 731 Example 4 An Application of Two Distance Formulas Figure 10.7 shows a triangle with vertices A3, 0, B0, 4, and C5, 2. a. Find the altitude b. Find the area of the triangle. from vertex h B to side AC. Solution a. To find the altitude, use the formula for the distance between line AC and the 0, 4. Slope: The equation of line m 2 0 2 5 3 8 AC 1 4 is obtained as follows. Equation: x 3 y 0 1 4 4y x 3 x 4y 3 0 Point-slope form Multiply each side by 4. General form 0, 4 is So, the distance between this line and the point Altitude h 10 44 3 12 42 13 17 units. b. Using the formula for the distance between two points, you can find the length of the base AC to be b 5 32 2 02 82 22 68 217 units. Distance Formula Simplify. Simplify. Simplify. Finally, the area of the triangle in Figure 10.7 is A 1 2 1 2 bh Formula for the area of a triangle 217 13 17 13 square units. Substitute for b and h. Simplify. Now try Exercise 45. W RITING ABOUT MATHEMATICS Inclination and the Angle Between Two Lines Discuss why the inclination of a line but the angle between two lines cannot can be an angle that is larger than 2. be larger than Decide whether the following statement is true or false: “The inclination of a line is the angle between the line and the -axis.” Explain. 2, x 333202_1001.qxd 12/8/05 11:05 AM Page 732 732 Chapter 10 Topics in Analytic Geometry 10.1 Exercises The HM mathSpace® CD-ROM and Eduspace® for this text contain step-by-step solutions to all odd-numbered exercises. They also provide Tutorial Exercises for additional help. VOCABULARY CHECK: Fill in the blanks. 1. The ________ of a nonhorizontal line is the positive angle x- counterclockwise from the axis to the line. ) (less than measured 2. If a nonvertical line has inclination and slope m, m ________ . 3. If two nonperpendicular lines have slopes m1 and the angle between the two lines is tan ________ . 4. The distance between the point x1, y1 and the line Ax By C 0 is given by d ________ . PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. then m2, 1. 3. 5. 7. In Exercises 1–8, find the slope of the line with inclination . =θ π 6 x y y 2. 4. y y In Exercises 19–22, find the inclination degrees) of the line. (in radians and =θ π 4 x 19. 20. 21. 22. 6x 2y 8 0 4x 5y 9 0 5x 3y 0 x y 10 0 In Exercises 23–32, find the angle between the lines. (in radians and degrees) 23. 3x y 3 2x y 2 24. x 3y 2 x 2y 3 y π =θ 3 4 x π =θ 2 3 x radians 3 1.27 radians 6. 5 6 radians 8. 2.88 radians In Exercises 9–14, find the inclination degrees) of the line with a slope of m. (in radians and 9. 11. 13. m 1 m 1 m 3 4 10. 12. 14. m 2 m 2 m 5 2 In Exercises 15–18, find the inclination degrees) of the line passing through the points. (in radians and 15. 16. 17. 18. 6, 1, 10, 8 4, 3 12, 8, 2, 20, 10, 0 0, 100, 50, 0 y 2 1 −1 −3 −2 −1 25. 3x 2y 0 3x 2y 1 26. 2x 3y 22 4x 3y 24 y 2 1 θ x 1 2 −2 −1 −1 − 27. 29. x 2y 7 6x 2y 5 x 2y 8 x 2y 2 28. 30. 5x 2y 16 3x 5y 1 3x 5y 3 3x 5y 12 333202_1001.qxd 12/8/05 8:54 AM Page 733 31. 32. 0.05x 0.03y 0.21 0.07x 0.02y 0.16 0.02x 0.05y 0.19 0.03x 0.04y 0.52 Section 10.1 Lines 733 In Exercises 49 and 50, find the distance between the parallel lines. 49. x y 1 x y 5 50. 3x 4y 1 3x 4y 10 Angle Measurement In Exercises 33–36, find the slope of each side of the triangle and use the slopes to find the measures of the interior angles. y 4 33. y 34. y 6 4 2 (−3, 2) −4 −2 −2 36. − ( 3, 4) y 4 (1, 3) (2, 0) 2 4 x 6 4 2 (2, 1) (4, 4) (6, 2) x 2 4 6 35. y 4 2 (3, 2) x (1, 0) 4 (−4, −1) y 2 −2 −4 −4 −2 x 4 −2 −2 x 4 51. Road Grade A straight road rises with an inclination of 0.10 radian from the horizontal (see figure). Find the slope of the road and the change in elevation over a two-mile stretch of the road. − ( 2, 2) (2, 1) −4 −2 2 4 −2 x 2 mi 0.1 radian In Exercises 37–44, find the distance between the point and the line. Point 0, 0 0, 0 2, 3 2, 1 6, 2 10, 8 0, 8 4, 2 37. 38. 39. 40. 41. 42. 43. 44. Line 4x 3y 0 2x y 4 4x 3y 10 x y 2 x 1 0 y 4 0 6x y 0 x y 20 52. Road Grade A straight road rises with an inclination of 0.20 radian from the horizontal. Find the slope of the road and the change in elevation over a one-mile stretch of the road. 53. Pitch of a Roof A roof has a rise of 3 feet for every horizontal change of 5 feet (see figure). Find the inclination of the roof. 3 ft 5 ft In Exercises 45–48, the points represent the vertices of a ABC in the coordinate plane, triangle. (a) Draw triangle B AC, (b) find the altitude from vertex of the triangle to side and (c) find the area of the triangle. C 4, 0 C 5, 2 C 5 2, 0 C 6, 12 A 0, 0, A 0, 0, A 1 2, 1 A 4, 5, B 1, 4, B 4, 5, , B 3, 10, B 2, 3, 47. 48. 46. 45. 2 333202_1001.qxd 12/8/05 8:54 AM Page 734 734 Chapter 10 Topics in Analytic Geometry 54. Conveyor Design A moving conveyor is built so that it rises 1 meter for each 3 meters of horizontal travel. (a) Draw a diagram that gives a visual representation of the problem. (b) Find the inclination of the conveyor. (c) The conveyor runs between two floors in a factory. The distance between the floors is 5 meters. Find the length of the conveyor. 55. Truss Find the angles and shown in the drawing of the roof truss. α 36 ft 6 ft 6 ft β 9 ft 58. To find the angle between two lines whose angles of m1 are known, substitute , respectively, in the formula for the angle between and and for 1 2 1 2 inclination m2 and two lines. 59. Exploration Consider a line with slope m and -intercept y 0, 4. (a) Write the distance between the origin and the line as d a function of m. (b) Graph the function in part (a). (c) Find the slope that yields the maximum distance between the origin and the line. (d) Find the asymptote of the graph in part (b) and interpret its meaning in the context of the problem. 60. Exploration Consider a line with slope m and -intercept y 0, 4. (a) Write the distance d between the point m. line as a function of 3, 1 and the Model It 56. Inclined Plane The Johnstown Inclined Plane in Johnstown, Pennsylvania is an inclined railway that was designed to carry people to the hilltop community of Westmont. It also proved useful in carrying people and vehicles to safety during severe floods. The railway is 896.5 feet long with a 70.9% uphill grade (see figure). (b) Graph the function in part (a). (c) Find the slope that yields the maximum distance be
tween the point and the line. (d) Is it possible for the distance to be 0? If so, what is the slope of the line that yields a distance of 0? (e) Find the asymptote of the graph in part (b) and interpret its meaning in the context of the problem. Skills Review In Exercises 61– 66, find all -intercepts and -intercepts of the graph of the quadratic function. x y 896.5 ft θ Not drawn to scale (a) Find the inclination of the railway. (b) Find the change in elevation from the base to the top of the railway. (c) Using the origin of a rectangular coordinate system as the base of the inclined plane, find the equation of the line that models the railway track. (d) Sketch a graph of the equation you found in part (c). Synthesis 61. 62. 63. 64. 65. 66. f x x 72 f x x 92 f x x 52 5 f x x 112 12 f x x2 7x 1 f x x2 9x 22 In Exercises 67–72, write the quadratic function in standard form by completing the square. Identify the vertex of the function. 67. 69. 71. 72. f x 3x2 2x 16 f x 5x2 34x 7 f x 6x2 x 12 f x 8x2 34x 21 68. 70. f x 2x2 x 21 f x x2 8x 15 True or False? the statement is true or false. Justify your answer. In Exercises 57 and 58, determine whether 57. A line that has an inclination greater than 2 radians has a negative slope. In Exercises 73–76, graph the quadratic function. 73. 75. f x x 42 3 gx 2x2 3x 1 f x 6 x 12 74. 76. gx x2 6x 8 333202_1002.qxd 12/8/05 9:00 AM Page 735 10.2 Introduction to Conics: Parabolas Section 10.2 Introduction to Conics: Parabolas 735 What you should learn • Recognize a conic as the intersection of a plane and a double-napped cone. • Write equations of parabolas in standard form and graph parabolas. • Use the reflective property of parabolas to solve real-life problems. Why you should learn it Parabolas can be used to model and solve many types of real-life problems. For instance, in Exercise 62 on page 742, a parabola is used to model the cables of the Golden Gate Bridge. Cosmo Condina/Getty Images Conics Conic sections were discovered during the classical Greek period, 600 to 300 B.C. The early Greeks were concerned largely with the geometric properties of conics. It was not until the 17th century that the broad applicability of conics became apparent and played a prominent role in the early development of calculus. A conic section (or simply conic) is the intersection of a plane and a doublenapped cone. Notice in Figure 10.8 that in the formation of the four basic conics, the intersecting plane does not pass through the vertex of the cone. When the plane does pass through the vertex, the resulting figure is a degenerate conic, as shown in Figure 10.9. Circle FIGURE 10.8 Basic Conics Ellipse Parabola Hyperbola Point FIGURE 10.9 Degenerate Conics Line Two Intersecting Lines There are several ways to approach the study of conics. You could begin by defining conics in terms of the intersections of planes and cones, as the Greeks did, or you could define them algebraically, in terms of the general seconddegree equation Ax 2 Bxy Cy 2 Dx Ey F 0. However, you will study a third approach, in which each of the conics is defined as a locus (collection) of points satisfying a geometric property. For example, in Section 1.2, you learned that a circle is defined as the collection of all points x, y This leads to the standard form of the equation of a circle that are equidistant from a fixed point h, k. x h 2 y k 2 r 2. Equation of circle 333202_1002.qxd 12/8/05 9:00 AM Page 736 736 Chapter 10 Topics in Analytic Geometry Parabolas In Section 2.1, you learned that the graph of the quadratic function f x ax2 bx c is a parabola that opens upward or downward. The following definition of a parabola is more general in the sense that it is independent of the orientation of the parabola. Definition of Parabola A parabola is the set of all points a fixed line (directrix) and a fixed point (focus) not on the line. x, y in a plane that are equidistant from y d2 Focus Vertex d1 d2 d1 Directrix FIGURE 10.10 Parabola The midpoint between the focus and the directrix is called the vertex, and the line passing through the focus and the vertex is called the axis of the parabola. Note in Figure 10.10 that a parabola is symmetric with respect to its axis. Using the definition of a parabola, you can derive the following standard form of the equation of a parabola whose directrix is parallel to the -axis or to the -axis. y x Standard Equation of a Parabola The standard form of the equation of a parabola with vertex at follows. x h, k is as x h2 4py k, p 0 y k2 4px h, p 0 Vertical axis, directrix: y k p Horizontal axis, directrix: x h p The focus lies on the axis units (directed distance) from the vertex. If the the equation takes one of the following forms. vertex is at the origin p 0, 0, x 2 4py y 2 4px See Figure 10.11. Vertical axis Horizontal axis For a proof of the standard form of the equation of a parabola, see Proofs in Mathematics on page 807. Axis: =x h Focus Axis: x = h Directrix: y = k − p Vertex: (h, k) Directrix: p− =x h p > 0 p > 0 Vertex: )h k ( , Directrix: p− k =y Focus: (h, k + p) Axis: y = k = Focus: h p ( + , k ) Vertex: ( , )h k Directrix: x = h − p p < 0 Focus: (h + p, k) Axis: y = k Vertex: (h, k) (a) x h2 4py k p > 0 Vertical axis: (b) x h2 4py k Vertical axis: p < 0 (c) y k2 4px h Horizontal axis: p > 0 (d) y k2 4px h Horizontal axis: p < 0 FIGURE 10.11 333202_1002.qxd 12/8/05 9:00 AM Page 737 Te c h n o l o g y Use a graphing utility to confirm the equation found in Example 1. In order to graph the equation, you may have to use two separate equations: 8x y1 Upper part and 8x. y2 Lower part Section 10.2 Introduction to Conics: Parabolas 737 Example 1 Vertex at the Origin Find the standard equation of the parabola with vertex at the origin and focus 2, 0. Solution The axis of the parabola is horizontal, passing through in Figure 10.12. 0, 0 and 2, 0, as shown y 2 1 −1 −2 2 y x= 8 Focus (2, 0) 2 3 4 x Vertex 1 (0, 0) You may want to review the technique of completing the square found in Appendix A.5, which will be used to rewrite each of the conics in standard form. FIGURE 10.12 So, the standard form is equation is y 2 8x. y 2 4px, where h 0, k 0, and p 2. So, the Now try Exercise 33. Example 2 Finding the Focus of a Parabola Find the focus of the parabola given by y 1 2 x 2 x 1 2. Solution To find the focus, convert to standard form by completing the square. y − 2 Vertex ( 1, 1) 1 2 ) 1 ( 1,− Focus −3 −2 −1 x 1 y = − −21 x 2 x + 1 2 −1 −2 FIGURE 10.13 y 1 2 x 2 x 1 2 Write original equation. 2y x 2 2x 1 Multiply each side by –2. 1 2y x 2 2x 1 1 2y x 2 2x 1 2 2y x2 2x 1 2y 1 x 12 Comparing this equation with x h2 4p y k Add 1 to each side. Complete the square. Combine like terms. Standard form you can conclude that is negative, the parabola opens downward, as shown in Figure 10.13. So, the focus of the parabola is h, k p 1, 1 Because and . h 1, k 1, p 1 2. p 2 Now try Exercise 21. 333202_1002.qxd 12/8/05 9:00 AM Page 738 738 Chapter 10 Topics in Analytic Geometry y 8 6 4 (x − 2)2 = 12(y − 1) Focus (2, 4) Vertex (2, 1) −4 −2 2 4 6 8 x −2 −4 FIGURE 10.14 Light source at focus Example 3 Finding the Standard Equation of a Parabola Find the standard form of the equation of the parabola with vertex focus 2, 4. 2, 1 and Solution Because the axis of the parabola is vertical, passing through consider the equation 2, 1 and 2, 4, x h2 4p y k where h 2, k 1, and p 4 1 3. So, the standard form is x 22 12 y 1. You can obtain the more common quadratic form as follows. x 22 12 y 1 x2 4x 4 12y 12 x2 4x 16 12y 1 12 x 2 4x 16 y Write original equation. Multiply. Add 12 to each side. Divide each side by 12. The graph of this parabola is shown in Figure 10.14. Now try Exercise 45. Focus Axis Application Parabolic reflector: Light is reflected in parallel rays. FIGURE 10.15 Axis P α Focus α Tangent line FIGURE 10.16 A line segment that passes through the focus of a parabola and has endpoints on the parabola is called a focal chord. The specific focal chord perpendicular to the axis of the parabola is called the latus rectum. Parabolas occur in a wide variety of applications. For instance, a parabolic reflector can be formed by revolving a parabola around its axis. The resulting surface has the property that all incoming rays parallel to the axis are reflected through the focus of the parabola. This is the principle behind the construction of the parabolic mirrors used in reflecting telescopes. Conversely, the light rays emanating from the focus of a parabolic reflector used in a flashlight are all parallel to one another, as shown in Figure 10.15. A line is tangent to a parabola at a point on the parabola if the line intersects, but does not cross, the parabola at the point. Tangent lines to parabolas have special properties related to the use of parabolas in constructing reflective surfaces. Reflective Property of a Parabola The tangent line to a parabola at a point makes equal angles with the following two lines (see Figure 10.16). P 1. The line passing through P and the focus 2. The axis of the parabola 333202_1002.qxd 12/8/05 9:00 AM Page 739 y y = x2 1 d 2 ( 0, 1 4 ) α (1, 1) −1 d 1 α 1 x Section 10.2 Introduction to Conics: Parabolas 739 Example 4 Finding the Tangent Line at a Point on a Parabola Find the equation of the tangent line to the parabola given by 1, 1. y x 2 at the point Solution p 1 and the focus is For this parabola, 4 0, b can find the -intercept two sides of the isosceles triangle shown in Figure 10.17: 0, 1 , y as shown in Figure 10.17. You of the tangent line by equating the lengths of the 4 d1 1 4 b (0, b) and d2 FIGURE 10.17 1 02 . Note that is important because the distance must be positive. Setting The order of subtraction for the distance d1 rather than produces d2 d1 1 4 b Te c h n o l o g y Use a graphing utility to confirm the result of Example 4. By graphing x 2 y1 and 2x 1 y2 in the same viewing window, you should be able to see that the line touches the parabola at the point 1, 1.
1 4 b 5 4 b 1. So, the slope of the tangent line is m 1 1 1 0 2 and the equation of the tangent line in slope-intercept form is y 2x 1. Now try Exercise 55. W RITING ABOUT MATHEMATICS Television Antenna Dishes Cross sections of television antenna dishes are parabolic in shape. Use the figure shown to write a paragraph explaining why these dishes are parabolic. Amplifier Dish reflector Cable to radio or TV 333202_1002.qxd 12/8/05 9:00 AM Page 740 740 Chapter 10 Topics in Analytic Geometry 10.2 Exercises VOCABULARY CHECK: Fill in the blanks. 1. A ________ is the intersection of a plane and a double-napped cone. 2. A collection of points satisfying a geometric property can also be referred to as a ________ of points. x, y 3. A ________ is defined as the set of all points in a plane that are equidistant from a fixed line, called the ________, and a fixed point, called the ________, not on the line. 4. The line that passes through the focus and vertex of a parabola is called the ________ of the parabola. 5. The ________ of a parabola is the midpoint between the focus and the directrix. 6. A line segment that passes through the focus of a parabola and has endpoints on the parabola is called a ________ ________ . 7. A line is ________ to a parabola at a point on the parabola if the line intersects, but does not cross, the parabola at the point. PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. In Exercises 1– 4, describe in words how a plane could intersect with the double-napped cone shown to form the conic section. (e) y 4 (f) y 4 −6 −4 −2 x −4 −2 2 x −4 5. 7. 9. y 2 4x x 2 8y y 12 4x 3 6. 8. 10. x 2 2y y 2 12x x 32 2y 1 In Exercises 11–24, find the vertex, focus, and directrix of the parabola and sketch its graph. 11. 13. 15. 17. 18. 19. 21. 23. 24. y 1 2x 2 y 2 6x x 2 6y 0 x 1 2 8y 4y 2 2 x 2 2x 5 y 1 4 y 2 6y 8x 25 0 y 2 4y 4x 0 12. 14. 16. y 2x 2 y 2 3x x y 2 0 20. 22. 2 4y 1 x 1 2 y2 2y 33 x 1 4 In Exercises 25–28, find the vertex, focus, and directrix of the parabola. Use a graphing utility to graph the parabola. 26. 25. x2 4x 6y 2 0 x 2 2x 8y 9 0 y 2 x y 0 27. 28. y 2 4x 4 0 1. Circle 3. Parabola 2. Ellipse 4. Hyperbola In Exercises 5–10, match the equation with its graph. [The graphs are labeled (a), (b), (c), (d), (e), and (f).] (a) y (b) y 4 2 −2 (c) 2 6 y 2 −4 −6 −6 −4 −2 x x 6 4 2 −4 −2 2 4 x (d) y −4 2 −2 −4 x 4 333202_1002.qxd 12/8/05 9:00 AM Page 741 In Exercises 29–40, find the standard form of the equation of the parabola with the given characteristic(s) and vertex at the origin. 29. y 30. 6 (3, 6) 4 2 y 8 (−2, 6) −8 −4 4 x −4 −2 2 4 x −8 31. Focus: 32. Focus: 33. Focus: 34. Focus: 2 0, 3 5 2, 0 2, 0 0, 2 35. Directrix: 36. Directrix: 37. Directrix: 38. Directrix: y 1 y 3 x 2 x 3 39. Horizontal axis and passes through the point 40. Vertical axis and passes through the point 4, 6 3, 3 In Exercises 41–50, find the standard form of the equation of the parabola with the given characteristics. 41. y 42. y (2, 0) 2 (3, 1) (4, 0) 2 4 6 −2 − 4 43. y 8 − ( 4, 0) (0, 4) 4 8 4 2 (4.5, 4) (5, 3) 2 4 44. y 12 8 (0, 0) − 4 − 4 8 (3, −3) x x x x 3, 2 − 8 45. Vertex: 46. Vertex: 47. Vertex: 48. Vertex: 49. Focus: 50. Focus: 5, 2; 1, 2; 0, 4; 2, 1; 2, 2; 0, 0; focus: focus: directrix: 1, 0 y 2 directrix: directrix: directrix: x 1 x 2 y 8 Section 10.2 Introduction to Conics: Parabolas 741 In Exercises 51 and 52, change the equation of the parabola so that its graph matches the description. 51. 52. y 3 2 6x 1; y 1 2 2x 4; upper half of parabola lower half of parabola In Exercises 53 and 54, the equations of a parabola and a tangent line to the parabola are given. Use a graphing utility to graph both equations in the same viewing window. Determine the coordinates of the point of tangency. Parabola y2 8x 0 x2 12y 0 53. 54. Tangent Line x y 2 0 x y 3 0 In Exercises 55–58, find an equation of the tangent line to the parabola at the given point, and find the -intercept of the line. x 55. 56. 57. 58. x 2 2y, x 2 2y, y 2x 2, y 2x 2, 4, 8 3, 9 2 1, 2 2, 8 59. Revenue The revenue R (in dollars) generated by the sale of units of a patio furniture set is given by x x 1062 4 5 R 14,045. Use a graphing utility to graph the function and approximate the number of sales that will maximize revenue. 60. Revenue The revenue R (in dollars) generated by the sale of units of a digital camera is given by x x 1352 5 7 R 25,515. Use a graphing utility to graph the function and approximate the number of sales that will maximize revenue. 61. Satellite Antenna The receiver in a parabolic television dish antenna is 4.5 feet from the vertex and is located at the focus (see figure). Write an equation for a cross section of the reflector. (Assume that the dish is directed upward and the vertex is at the origin.) y Receiver 4.5 ft x 333202_1002.qxd 12/8/05 9:00 AM Page 742 742 Chapter 10 Topics in Analytic Geometry Model It 62. Suspension Bridge Each cable of the Golden Gate Bridge is suspended (in the shape of a parabola) between two towers that are 1280 meters apart. The top of each tower is 152 meters above the roadway. The cables touch the roadway midway between the towers. (a) Draw a sketch of the bridge. Locate the origin of a rectangular coordinate system at the center of the roadway. Label the coordinates of the known points. (b) Write an equation that models the cables. (c) Complete the table by finding the height of the suspension cables over the roadway at a distance of x meters from the center of the bridge. y Distance, x Height, y 0 250 400 500 1000 63. Road Design Roads are often designed with parabolic surfaces to allow rain to drain off. A particular road that is 32 feet wide is 0.4 foot higher in the center than it is on the sides (see figure). 32 ft 0.4 ft Not drawn to scale Cross section of road surface (a) Find an equation of the parabola that models the road surface. (Assume that the origin is at the center of the road.) (b) How far from the center of the road is the road surface 0.1 foot lower than in the middle? 64. Highway Design Highway engineers design a parabolic curve for an entrance ramp from a straight street to an interstate highway (see figure). Find an equation of the parabola. y 800 400 −400 −800 Interstate (1000, 800) 400 800 1200 1600 x (1000, −800) Street FIGURE FOR 64 65. Satellite Orbit A satellite in a 100-mile-high circular orbit around Earth has a velocity of approximately 17,500 miles per hour. If this velocity is multiplied by the satellite will have the minimum velocity necessary to escape Earth’s gravity and it will follow a parabolic path with the center of Earth as the focus (see figure). 2, Circular t orbi y 4100 miles Parabolic path x Not drawn to scale (a) Find the escape velocity of the satellite. (b) Find an equation of the parabolic path of the satellite (assume that the radius of Earth is 4000 miles). 66. Path of a Softball The path of a softball is modeled by 12.5 y 7.125 x 6.252 where the coordinates x 0 was thrown. x y are measured in feet, with corresponding to the position from which the ball and (a) Use a graphing utility to graph the trajectory of the softball. (b) Use the trace feature of the graphing utility to approximate the highest point and the range of the trajectory. feet per second at a height of Projectile Motion In Exercises 67 and 68, consider the path of a projectile projected horizontally with a velocity of feet, where the model for v the path is x2 v2 16 y s. s In this model (in which air resistance is disregarded), the height (in feet) of the projectile and distance (in feet) the projectile travels. is is the horizontal y x 333202_1002.qxd 12/8/05 9:00 AM Page 743 67. A ball is thrown from the top of a 75-foot tower with a (a) Find the area when p 2 and b 4. Section 10.2 Introduction to Conics: Parabolas 743 velocity of 32 feet per second. (a) Find the equation of the parabolic path. (b) How far does the ball travel horizontally before striking the ground? 68. A cargo plane is flying at an altitude of 30,000 feet and a speed of 540 miles per hour. A supply crate is dropped from the plane. How many feet will the crate travel horizontally before it hits the ground? Synthesis True or False? the statement is true or false. Justify your answer. In Exercises 69 and 70, determine whether 69. It is possible for a parabola to intersect its directrix. 70. If the vertex and focus of a parabola are on a horizontal line, then the directrix of the parabola is vertical. 71. Exploration Consider the parabola x 2 4py. (a) Use a graphing utility to graph the parabola for p 1, Describe the effect on the p 2, graph when p 3, p p 4. and increases. (b) Give a geometric explanation of why the area p approaches 0 as approaches 0. x1, y1 73. Exploration Let the parabola the parabola at the point is x 2 4py. be the coordinates of a point on The equation of the line tangent to y y1 x1 2p x x1 . What is the slope of the tangent line? 74. Writing In your own words, state the reflective property of a parabola. Skills Review 75. In Exercises 75–78, list the possible rational zeros of given by the Rational Zero Test. f x x3 2x2 2x 4 f x 2x3 4x2 3x 10 f x 2x5 x2 16 f x 3x3 12x 22 76. 77. 78. f (b) Locate the focus for each parabola in part (a). (c) For each parabola in part (a), find the length of the chord passing through the focus and parallel to the directrix (see figure). How can the length of this chord be determined directly from the standard form of the equation of the parabola? y Chord Focus 2 x = 4py x (d) Explain how the result of part (c) can be used as a sketching aid when graphing parabolas. 72. Geometry The area of the shaded region in the figure is A 8 3 p12 b32. y 2 x = 4py y = b x 79. Find a polynomial with real coefficients that has the zeros 3, 2 i, 2 i. 80. Find all the zeros of and f x 2x3 3x 2 50x 75 if one of the zeros is x 3 2. 81. Find all the zeros of the function gx 6x4 7x3 29x 2 28x 20 if two of the zeros are x ±2. 82.
Use a graphing utility to graph the function given by hx) 2x4 x3 19x 2 9x 9. Use the graph to approximate the zeros of h. In Exercises 83–90, use the information to solve the triangle. Round your answers to two decimal places. 85. 84. 83. 86. A 35, a 10, b 7 B 54, b 18, c 11 A 40, B 51, c 3 B 26, C 104, a 19 a 7, b 10, c 16 a 58, b 28, c 75 A 65, b 5, c 12 89. 90. B 71, a 21, c 29 88. 87. 333202_1003.qxd 12/8/05 9:01 AM Page 744 744 Chapter 10 Topics in Analytic Geometry 10.3 Ellipses What you should learn • Write equations of ellipses in standard form and graph ellipses. • Use properties of ellipses to model and solve real-life problems. • Find eccentricities of ellipses. Why you should learn it Ellipses can be used to model and solve many types of real-life problems. For instance, in Exercise 59 on page 751, an ellipse is used to model the orbit of Halley’s comet. Harvard College Observatory/ SPL/Photo Researchers, Inch k ( , c a 2 b2 + c 2 = 2a b2 + c 2 = a 2 FIGURE 10.21 Introduction The second type of conic is called an ellipse, and is defined as follows. Definition of Ellipse An ellipse is the set of all points distances from two distinct fixed points (foci) is constant. See Figure 10.18. in a plane, the sum of whose x, y (x, y) d 2 d 1 Focus Focus Major axis Vertex Center Minor axis Vertex d2 d1 FIGURE 10.18 is constant. FIGURE 10.19 The line through the foci intersects the ellipse at two points called vertices. The chord joining the vertices is the major axis, and its midpoint is the center of the ellipse. The chord perpendicular to the major axis at the center is the minor axis of the ellipse. See Figure 10.19. You can visualize the definition of an ellipse by imagining two thumbtacks placed at the foci, as shown in Figure 10.20. If the ends of a fixed length of string are fastened to the thumbtacks and the string is drawn taut with a pencil, the path traced by the pencil will be an ellipse. FIGURE 10.20 To derive the standard form of the equation of an ellipse, consider the ellipse foci, h ± a, k; vertices, h, k; in Figure 10.21 with the following points: center, h ± c, k. Note that the center is the midpoint of the segment joining the foci. 333202_1003.qxd 12/8/05 9:01 AM Page 745 Section 10.3 Ellipses 745 The sum of the distances from any point on the ellipse to the two foci is constant. Using a vertex point, this constant sum is a c a c 2a Length of major axis Consider the equation of the ellipse y k2 b2 x h2 a2 If you let tion can be rewritten as a b, then the equa- 1. x h2 y k2 a2 which is the standard form of the equation of a circle with r a (see Section 1.2). radius Geometrically, when for an ellipse, the major and minor axes are of equal length, and so the graph is a circle. a b or simply the length of the major axis. Now, if you let ellipse, the sum of the distances between That is, x, y x, y and the two foci must also be be any point on the 2a 2a. b2 a2 c 2, which implies that the Finally, in Figure 10.21, you can see that equation of the ellipse is b2x h2 a 2y k 2 a 2b 2 x h 2 a 2 y k2 b2 1. You would obtain a similar equation in the derivation by starting with a vertical major axis. Both results are summarized as follows. Standard Equation of an Ellipse The standard form of the equation of an ellipse, with center respectively, where major and minor axes of lengths and 2b, 2a h, k and 0 < b < a, is x h2 a 2 x h 2 b2 y k2 b2 y k2 a 2 1 1. Major axis is horizontal. Major axis is vertical. c The foci lie on the major axis, units from the center, with 0, 0, If the center is at the origin forms. c 2 a2 b2. the equation takes one of the following x 2 a2 y 2 b2 1 Major axis is horizontal. x 2 b2 y 2 a2 1 Major axis is vertical. Figure 10.22 shows both the horizontal and vertical orientations for an ellipse. y y (x − h)2 2a + (y − k)2 2b = 1 (x − h)2 2b + (y − k)2 2a = 1 (h, k) 2b (h, k) 2a 2a Major axis is horizontal. FIGURE 10.22 x 2b x Major axis is vertical. 333202_1003.qxd 12/8/05 9:01 AM Page 746 746 Chapter 10 Topics in Analytic Geometry y 4 3 b = 5 x (0, 1) (2, 1) (4, 1) −1 −1 −2 1 3 a = 3 FIGURE 10.23 ( x + 2 3) 22 + 2 ( 1) y − 12 = 1 (−5, 1) (−3, 2) (−1, 1) ( ) −3 − 3, 1 (−3, 1) ( )3, 1 −3 + y 4 3 2 1 −5 − 4 −3 (−3, 0) −1 x −1 Example 1 Finding the Standard Equation of an Ellipse Find the standard form of the equation of the ellipse having foci at 4, 1 and a major axis of length 6, as shown in Figure 10.23. 0, 1 and Solution Because the foci occur at the distance from the center to one of the foci is know that c 2 a2 b2, Now, from b a2 c2 32 22 5. a 3. 4, 1, 0, 1 and you have the center of the ellipse is c 2. Because 2, 1) 2a 6, and you Because the major axis is horizontal, the standard equation is x 2 2 32 y 12 52 1. This equation simplifies to y 12 5 x 22 9 1. Now try Exercise 49. Example 2 Sketching an Ellipse Sketch the ellipse given by x 2 4y 2 6x 8y 9 0. Solution Begin by writing the original equation in standard form. In the fourth step, note that 9 and 4 are added to both sides of the equation when completing the squares. x 2 4y 2 6x 8y 9 0 Write original equation. x 2 6x 4y 2 8y 9 x 2 6x 4y 2 2y 9 Group terms. Factor 4 out of y-terms. x 2 6x 9 4y 2 2y 1 9 9 41 x 3 2 4y 1 2 4 x 32 4 x 32 22 y 12 1 y 12 12 1 1 Divide each side by 4. Write in standard form. Write in completed square form. x a 2 22, h, k 3, 1. Because From this standard form, it follows that the center is the endpoints of the major axis lie two the denominator of the -term is units to the right and left of the center. Similarly, because the denominator of the y the endpoints of the minor axis lie one unit up and down from -term is So, the the center. Now, from foci of the ellipse are The ellipse is shown in Figure 10.24. c2 a2 b2, 3 3, 1 c 22 12 3. 3 3, 1. b 2 12, you have and FIGURE 10.24 Now try Exercise 25. 333202_1003.qxd 12/8/05 9:01 AM Page 747 Section 10.3 Ellipses 747 Example 3 Analyzing an Ellipse Find the center, vertices, and foci of the ellipse 4x 2 y 2 8x 4y 8 0. Solution By completing the square, you can write the original equation in standard form. 4x 2 y 2 8x 4y 8 0 4x 2 8x y 2 4y 8 4x 2 2x y 2 4y 8 Write original equation. Group terms. Factor 4 out of terms. (x − 1)2 22 + (y + 2)2 42 y = 1 Vertex Focus 2 (1, 2)− Center Focus Vertex (1, 2) x 4 (1, 6)− −( 1, 2 + 2 3 2 ( −4 −2 ( − − 1, 2 2 3 ( FIGURE 10.25 4x 2 2x 1 y 2 4y 4 8 41 4 4x 1 2 y 2 2 16 x 1 2 4 x 1 2 22 y 22 16 y 22 42 1 1 Divide each side by 16. Write in standard form. Write in completed square form. The major axis is vertical, where h 1, k 2, a 4, b 2, and c a2 b2 16 4 12 23. So, you have the following. Center: 1, 2 Vertices: 1, 6 1, 2 Foci: 1, 2 23 1, 2 23 The graph of the ellipse is shown in Figure 10.25. Now try Exercise 29. Te c h n o l o g y You can use a graphing utility to graph an ellipse by graphing the upper and lower portions in the same viewing window. For instance, to graph the ellipse in Example 3, first solve for to get y 2 41 y1 x 12 4 and 2 41 y2 x 12 4 . Use a viewing window in which the graph shown below. 6 ≤ x ≤ 9 and 7 ≤ y ≤ 3. You should obtain −6 3 −7 9 333202_1003.qxd 12/8/05 9:01 AM Page 748 748 Chapter 10 Topics in Analytic Geometry Application Ellipses have many practical and aesthetic uses. For instance, machine gears, supporting arches, and acoustic designs often involve elliptical shapes. The orbits of satellites and planets are also ellipses. Example 4 investigates the elliptical orbit of the moon about Earth. Example 4 An Application Involving an Elliptical Orbit The moon travels about Earth in an elliptical orbit with Earth at one focus, as shown in Figure 10.26. The major and minor axes of the orbit have lengths of 768,800 kilometers and 767,640 kilometers, respectively. Find the greatest and smallest distances (the apogee and perigee), respectively from Earth’s center to the moon’s center. Solution Because 2a 768,800 2b 767,640, you have a 384,400 and and b 383,820 Moon 767,640 km Earth 768,800 km Perigee Apogee FIGURE 10.26 which implies that c a 2 b2 384,4002 383,8202 21,108. So, the greatest distance between the center of Earth and the center of the moon is Note in Example 4 and Figure 10.26 that Earth is not the center of the moon’s orbit. a c 384,400 21,108 405,508 kilometers and the smallest distance is a c 384,400 21,108 363,292 kilometers. Now try Exercise 59. Eccentricity One of the reasons it was difficult for early astronomers to detect that the orbits of the planets are ellipses is that the foci of the planetary orbits are relatively close to their centers, and so the orbits are nearly circular. To measure the ovalness of an ellipse, you can use the concept of eccentricity. Definition of Eccentricity The eccentricity of an ellipse is given by the ratio e e c a . Note that 0 < e < 1 for every ellipse. 333202_1003.qxd 12/8/05 9:01 AM Page 749 Section 10.3 Ellipses 749 To see how this ratio is used to describe the shape of an ellipse, note that because the foci of an ellipse are located along the major axis between the vertices and the center, it follows that 0 < c < a. For an ellipse that is nearly circular, the foci are close to the center and the ratio ca is small, as shown in Figure 10.27. On the other hand, for an elongated ellipse, the foci are close to the vertices, and the ratio is close to 1, as shown in Figure 10.28. ca y y Foci Foci c e is small is close to 1. FIGURE 10.27 FIGURE 10.28 a a The orbit of the moon has an eccentricity of e 0.0549, and the eccentricities A S A N The time it takes Saturn to orbit the sun is equal to 29.4 Earth years. Venus: Mercury: of the nine planetary orbits are as follows. e 0.2056 e 0.0068 e 0.0167 e 0.0934 e 0.0484 Mars: Jupiter: Earth: Saturn: Uranus: Neptune: Pluto: e 0.0542 e 0.0472 e 0.0086 e 0.2488 W RITING ABOUT MATHEMATICS Ellipses and Circles a. Show that the equation of an ellipse can be written as x h2 a2 y k2 a21 e2 1. b. For the equation in part (a), let utility to graph the ellipse for e 0.1. a 4, e 0.95, h 1, and e 0.75, k 2, e 0.5, an
d use a graphing e 0.25, e and Discuss the changes in the shape of the ellipse as approaches 0. c. Make a conjecture about the shape of the graph in part (b) when e 0. What is the equation of this ellipse? What is another name for an ellipse with an eccentricity of 0? 333202_1003.qxd 12/8/05 9:01 AM Page 750 750 Chapter 10 Topics in Analytic Geometry 10.3 Exercises VOCABULARY CHECK: Fill in the blanks. x, y 1. An ________ is the set of all points in a plane, the sum of whose distances from two distinct fixed points, called ________, is constant. 2. The chord joining the vertices of an ellipse is called the ________ ________, and its midpoint is the ________ of the ellipse. 3. The chord perpendicular to the major axis at the center of the ellipse is called the ________ ________ of the ellipse. 4. The concept of ________ is used to measure the ovalness of an ellipse. PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. In Exercises 1–6, match the equation with its graph. [The graphs are labeled (a), (b), (c), (d), (e), and (f).] (a) y (b) In Exercises 7–30, identify the conic as a circle or an ellipse. Then find the center, radius, vertices, foci, and eccentricity of the conic (if applicable), and sketch its graph. x 2 4 −4 x 4 y 4 2 −4 y 6 2 −6 y 4 −4 x 4 6 x 4 2 −4 (c) (e) −4 −6 1. 3. 5. 6 25 4 x 2 2 16 x 2 2 9 (d) x 2 4 −4 y 4 −4 (f) x 2 −4 y 2 −2 −2 −6 1 1 y 1 2 1 y 22 4 1 2. 4 144 81 y 2 x2 9 9 y 2 x 2 28 64 x 42 12 x 52 94 x 32 254 y 32 16 1 y 12 1 y 12 254 1 y 2 x 2 16 25 y 2 x2 25 25 y 2 x 2 9 5 x 32 16 1 1 1 8. 10. 12. y 52 25 1 14. x2 49 y 12 49 1 16. 18. 1 x 22 y 42 14 9x 2 4y 2 36x 24y 36 0 9x 2 4y 2 54x 40y 37 0 x2 y2 2x 4y 31 0 x2 5y2 8x 30y 39 0 3x2 y2 18x 2y 8 0 6x2 2y2 18x 10y 2 0 x2 4y2 6x 20y 2 0 x2 y 2 4x 6y 3 0 9x 2 9y 2 18x 18y 14 0 16x 2 25y 2 32x 50y 16 0 9x 2 25y 2 36x 50y 60 0 16x2 16y 2 64x 32y 55 0 7. 9. 11. 13. 15. 17. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. In Exercises 31–34, use a graphing utility to graph the ellipse. Find the center, foci, and vertices. (Recall that it may be necessary to solve the equation for and obtain two equations.) y 31. 5x 2 3y 2 15 12x 2 20y 2 12x 40y 37 0 33. 34. 36x 2 9y 2 48x 36y 72 0 32. 3x 2 4y 2 12 333202_1003.qxd 12/8/05 9:01 AM Page 751 In Exercises 35–42, find the standard form of the equation of the ellipse with the given characteristics and center at the origin. 35. y 8 36. (0, 4) (2, 0) 8 4 (0, 4)− x − ( 2, 0) −8 −4 −8 y 4 −4 ( ) 0, 3 2 (2, 0) 4 ( 0, − ) 3 2 x (−2, 0) −4 37. Vertices: 38. Vertices: 39. Foci: 40. Foci: ±5, 0; ±2, 0; ±6, 0; 0, ±8; foci: ±2, 0 0, ±4 major axis of length 12 foci: major axis of length 8 41. Vertices: 4, 2 42. Major axis vertical; passes through the points passes through the point 0, ±5; 0, 4 2, 0 In Exercises 43–54, find the standard form of the equation of the ellipse with the given characteristics. 44. y 43. y (1, 3) 6 5 4 3 2 1 (2, 6) (3, 3) (2, 0) 1 432 5 6 x 45. y 46. − ( 2, 6) − ( 6, 3) 8 4 2 −6 −4 − ( 2, 0) (2, 3) x 2 4 4 3 2 1 −1 −2 −3 −4 y 1 −1 −1 −2 −3 −4 (4, 4) (7, 0) (1, 0) 6 2 4 5 3 (4, −4) x 8 x (2, 0) 2 1 (0, −1) 3 (2, −2) (4, −1) 47. Vertices: 0, 4, 4, 4; minor axis of length 2 48. Foci: 49. Foci: 50. Center: 51. Center: 52. Center: 0, 0, 4, 0; 0, 0, 0, 8; 2, 1; 0, 4; a 2c; 3, 2; a 3c; 0, 2, 4, 2; 53. Vertices: 2, 3, 2, 1 major axis of length 8 vertex: major axis of length 16 ; 2, 1 2 vertices: minor axis of length 2 4, 4, 4, 4 foci: 1, 2, 5, 2 endpoints of the minor axis: Section 10.3 Ellipses 751 54. Vertices: 5, 0, 5, 12; 1, 6, 9, 6 endpoints of the minor axis: 55. Find an equation of the ellipse with vertices eccentricity e 3 5. 56. Find an equation of the ellipse with vertices eccentricity e 1 2. ±5, 0 and 0, ±8 and 57. Architecture A semielliptical arch over a tunnel for a one-way road through a mountain has a major axis of 50 feet and a height at the center of 10 feet. (a) Draw a rectangular coordinate system on a sketch of the tunnel with the center of the road entering the tunnel at the origin. Identify the coordinates of the known points. (b) Find an equation of the semielliptical arch over the tunnel. and (c) You are driving a moving truck that has a width of 8 feet and a height of 9 feet. Will the moving truck clear the opening of the arch? 58. Architecture A fireplace arch is to be constructed in the shape of a semiellipse. The opening is to have a height of 2 feet at the center and a width of 6 feet along the base (see figure). The contractor draws the outline of the ellipse using tacks as described at the beginning of this section. Give the required positions of the tacks and the length of the string3 −2 −1 −2 Model It 59. Comet Orbit Halley’s comet has an elliptical orbit, with the sun at one focus. The eccentricity of the orbit is approximately 0.967. The length of the major axis of the orbit is approximately 35.88 astronomical units. (An astronomical unit is about 93 million miles.) (a) Find an equation of the orbit. Place the center of the orbit at the origin, and place the major axis on the x -axis. (b) Use a graphing utility to graph the equation of the orbit. (c) Find the greatest (aphelion) and smallest (perihelion) distances from the sun’s center to the comet’s center. 333202_1003.qxd 12/8/05 9:01 AM Page 752 752 Chapter 10 Topics in Analytic Geometry 60. Satellite Orbit The first artificial satellite to orbit Earth was Sputnik I (launched by the former Soviet Union in 1957). Its highest point above Earth’s surface was 947 kilometers, and its lowest point was 228 kilometers (see figure). The center of Earth was the focus of the elliptical orbit, and the radius of Earth is 6378 kilometers. Find the eccentricity of the orbit. Focus 228 km 947 km 61. Motion of a Pendulum The relation between the velocity (in radians per second) of a pendulum and its angular from the vertical can be modeled by a y 0 radian and 0, y displacement semiellipse. A 12-centimeter pendulum crests when the angular displacement is 0.2 radian. When the pendulum is at equilibrium the velocity is radians per second. 0.2 1.6 (a) Find an equation that models the motion of the pendulum. Place the center at the origin. (b) Graph the equation from part (a). (c) Which half of the ellipse models the motion of the pendulum? 62. Geometry A line segment through a focus of an ellipse with endpoints on the ellipse and perpendicular to the major axis is called a latus rectum of the ellipse. Therefore, an ellipse has two latera recta. Knowing the length of the latera recta is helpful in sketching an ellipse because it yields other points on the curve (see figure). Show that the length of each latus rectum is 2b2a. y Latera recta F1 F2 x Synthesis True or False? the statement is true or false. Justify your answer. In Exercises 67 and 68, determine whether 67. The graph of x2 4y4 4 0 is an ellipse. 68. It is easier to distinguish the graph of an ellipse from the graph of a circle if the eccentricity of the ellipse is large (close to 1). 69. Exploration Consider the ellipse x 2 a2 y 2 b2 1, a b 20. (a) The area of the ellipse is given by area of the ellipse as a function of A ab. a. Write the (b) Find the equation of an ellipse with an area of 264 square centimeters. (c) Complete the table using your equation from part (a), and make a conjecture about the shape of the ellipse with maximum area. 8 9 10 11 12 13 a A (d) Use a graphing utility to graph the area function and use the graph to support your conjecture in part (c). 70. Think About It At the beginning of this section it was noted that an ellipse can be drawn using two thumbtacks, a string of fixed length (greater than the distance between the two tacks), and a pencil. If the ends of the string are fastened at the tacks and the string is drawn taut with a pencil, the path traced by the pencil is an ellipse. (a) What is the length of the string in terms of a? (b) Explain why the path is an ellipse. Skills Review In Exercises 71–74, determine whether the sequence is arithmetic, geometric, or neither. 71. 80, 40, 20, 10, 5, . . . 5 2, 2,1 2, . . . 73. 3 2, 7 2, 1 72. 66, 55, 44, 33, 22, . . . 74. 1 4, 1 2, 1, 2, 4, . . . In Exercises 63– 66, sketch the graph of the ellipse, using latera recta (see Exercise 62). 1 y 2 x 2 16 9 5x 2 3y 2 15 63. 65. 1 y 2 x 2 1 4 9x 2 4y 2 36 64. 66. In Exercises 75–78, find the sum. 75. 77. 6 n0 10 n0 3n n 54 3 76. 6 n0 3n 78. 10 n1 n1 43 4 333202_1004.qxd 12/8/05 9:03 AM Page 753 10.4 Hyperbolas What you should learn • Write equations of hyperbolas in standard form. • Find asymptotes of and graph hyperbolas. • Use properties of hyperbolas to solve real-life problems. • Classify conics from their general equations. Why you should learn it Hyperbolas can be used to model and solve many types of real-life problems. For instance, in Exercise 42 on page 761, hyperbolas are used in long distance radio navigation for aircraft and ships. AP/Wide World Photos Section 10.4 Hyperbolas 753 Introduction The third type of conic is called a hyperbola. The definition of a hyperbola is similar to that of an ellipse. The difference is that for an ellipse the sum of the distances between the foci and a point on the ellipse is fixed, whereas for a hyperbola the difference of the distances between the foci and a point on the hyperbola is fixed. Definition of Hyperbola A hyperbola is the set of all points distances from two distinct fixed points (foci) is a positive constant. See Figure 10.29. in a plane, the difference of whose x, y x y ( , ) 2d 1d Focus Focus c a Branch Branch Vertex Center Vertex Transverse axis d 2 d− 1 is a positive constant. FIGURE 10.29 FIGURE 10.30 The graph of a hyperbola has two disconnected branches. The line through the two foci intersects the hyperbola at its two vertices. The line segment connecting the vertices is the transverse axis, and the midpoint of the transverse axis is the center of the hyperbola. See Figure 10.30. The development of the standard form of the
equation of a hyperbola is similar to that of an ellipse. Note in the definition below that and are related differently for hyperbolas than for ellipses. b, a, c Standard Equation of a Hyperbola The standard form of the equation of a hyperbola with center h, k is x h2 a 2 y k 2 a2 y k2 b2 x h2 b 2 1 1. Transverse axis is horizontal. Transverse axis is vertical. c 2 a2 b2. a The vertices are units from the center, and the foci are units from the center. Moreover, 0, 0, origin y 2 x 2 b2 a2 the equation takes one of the following forms. If the center of the hyperbola is at the Transverse axis is vertical. Transverse axis is horizontal. x 2 b2 y 2 a2 1 1 c 333202_1004.qxd 12/8/05 9:03 AM Page 754 754 Chapter 10 Topics in Analytic Geometry Figure 10.31 shows both the horizontal and vertical orientations for a hyperbola. 2 ) ( y − 2 ) k− 2 b = 1 ( x h Transverse axis is horizontal. FIGURE 10.31 2 ) ( x − 2 ) h− 2 b = 1 ( y k− ) Transverse axis is vertical. Example 1 Finding the Standard Equation of a Hyperbola Find the standard form of the equation of the hyperbola with foci 5, 2 and vertices 4, 2. 0, 2 and 1, 2 and When finding the standard form of the equation of any conic, it is helpful to sketch a graph of the conic with the given characteristics. Solution By the Midpoint Formula, the center of the hyperbola occurs at the point Furthermore, a 4 2 2, c 5 2 3 and it follows that and 2, 2. b c2 a2 32 22 9 4 5. So, the hyperbola has a horizontal transverse axis and the standard form of the equation is x 22 22 y 22 52 1. See Figure 10.32. This equation simplifies to y 22 5 x 22 4 1. (x − 2)2 2 2 − (y − 2)0, 2) (−1, 2) (4, 2) (2, 2) (5, 2) 1 2 3 4 −1 x FIGURE 10.32 Now try Exercise 27. 333202_1004.qxd 12/8/05 9:03 AM Page 755 Conjugate axis (h, k + b) Asy m ptote (h, k) (h, k − b) A s y m (h − a, k) (h + a, k) p t o t e FIGURE 10.33 Asymptotes of a Hyperbola Section 10.4 Hyperbolas 755 Each hyperbola has two asymptotes that intersect at the center of the hyperbola, as shown in Figure 10.33. The asymptotes pass through the vertices of a rectangle 2b by of dimensions is the conjugate joining and axis of the hyperbola. with its center at or h b, k and h b, k The line segment of length 2b, h, k b 2a h, k b h, k. Asymptotes of a Hyperbola The equations of the asymptotes of a hyperbola are . Transverse axis is horizontal. Transverse axis is vertical. Example 2 Using Asymptotes to Sketch a Hyperbola Sketch the hyperbola whose equation is 4x 2 y 2 16. Solution Divide each side of the original equation by 16, and rewrite the equation in standard form. y 2 42 Write in standard form. x 2 22 1 2, 0, 0, 4 a 2, 2, 0 0, 4. b 4, and the transverse axis is horiFrom this, you can conclude that and and the endpoints of the conzontal. So, the vertices occur at Using these four points, you are able to and jugate axis occur at sketch the rectangle shown in Figure 10.34. Now, from you have 25, 0 c 22 42 20 25. Finally, by drawing the asymptotes through the corners of this recand tangle, you can complete the sketch shown in Figure 10.35. Note that the asymptotes are So, the foci of the hyperbola are c2 a2 b2, 25, 0. y 2x. y 2x and y 8 6 (0, 4) − ( 2, 0) −6 −4 (2, 0) 4 6 x 2(− ) 5, 0 −6 −4 (0, 4)− −6 FIGURE 10.34 FIGURE 10.35 Now try Exercise 7. y 8 6 −6 2( ) 5, 0 4 2 x 22 6 − 2 y 42 x = 1 333202_1004.qxd 12/8/05 11:22 AM Page 756 756 Chapter 10 Topics in Analytic Geometry Example 3 Finding the Asymptotes of a Hyperbola Sketch the hyperbola given by of its asymptotes and the foci. 4x 2 3y 2 8x 16 0 and find the equations Solution 4x 2 3y 2 8x 16 0 4x2 8x 3y2 16 4x 2 2x 3y 2 16 4x 2 2x 1 3y 2 16 4 4x 12 3y 2 12 y2 4 x 12 32 x 12 3 y 2 22 1 1 Write original equation. Group terms. Factor 4 from x- terms. Add 4 to each side. Write in completed square form. Divide each side by 12. Write in standard form. 1, 0, has vertices From this equation you can conclude that the hyperbola has a vertical transverse 1, 2 and axis, centered at and has a conjugate 1 3, 0. To sketch the hyperbola, axis with endpoints draw a rectangle through these four points. The asymptotes are the lines passing through the corners of the rectangle. Using you can conclude that the equations of the asymptotes are 1 3, 0 1, 2, b 3, a 2 and and y 2 3 x 1 and y 2 3 x 1. Finally, you can determine the foci by using the equation have 1, 2 7. c 22 32 7, The hyperbola is shown in Figure 10.36. and the foci are c2 a2 b2. 1, 2 7 So, you and Now try Exercise 13. Te 7 −1, 2 + 4 3 (−1, 2) (−1, 0) 1 y2 22 − (x + 1)2 ( ) 3 2 −4 −3 −3 (−1, −2) ( ) −1, −2 − 7 FIGURE 10.36 You can use a graphing utility to graph a hyperbola by graphing the upper and lower portions in the same viewing window. For instance, to graph the hyperbola in Example 3, first solve for to get 21 y1 x 12 3 and 21 y2 y x 12 3 . Use a viewing window in which You should obtain the graph shown below. Notice that the graphing utility does not draw the asymptotes. However, if you trace along the branches, you will see that the values of the hyperbola approach the asymptotes. and . −9 6 −6 9 333202_1004.qxd 12/8/05 9:03 AM Page 757 y = 2x − 8 Example 4 Using Asymptotes to Find the Standard Equation Section 10.4 Hyperbolas 757 Find the standard form of the equation of the hyperbola having vertices and and having asymptotes 3, 5 3, 1 y 2x 8 and y 2x 4 y 2 (3, 1) −2 2 4 6 x −2 − 4 −6 (3, −5) FIGURE 10.37 as shown in Figure 10.37. Solution By the Midpoint Formula, the center of the hyperbola is a 3. hyperbola has a vertical transverse axis with you can determine the slopes of the asymptotes to be 3, 2. Furthermore, the From the original equations, y = −2x + 4 m1 2 a b a 3 and, because 2 a b and m2 2 a b you can conclude 2 3 b b 3 2 . So, the standard form of the equation is y 22 32 1. x 32 2 3 2 Now try Exercise 35. As with ellipses, the eccentricity of a hyperbola is e c a Eccentricity c > a, it follows that and because If the eccentricity is large, the branches of the hyperbola are nearly flat, as shown in Figure 10.38. If the eccentricity is close to 1, the branches of the hyperbola are more narrow, as shown in Figure 10.39. e > 1. y y e is large. e is close to 1. Vertex Focus x Focus Vertex FIGURE 10.38 FIGURE 10.39 333202_1004.qxd 12/8/05 9:03 AM Page 758 758 Chapter 10 Topics in Analytic Geometry Applications The following application was developed during World War II. It shows how the properties of hyperbolas can be used in radar and other detection systems. Example 5 An Application Involving Hyperbolas Two microphones, 1 mile apart, record an explosion. Microphone A receives the sound 2 seconds before microphone B. Where did the explosion occur? (Assume sound travels at 1100 feet per second.) y 3000 2000 0 0 2 2 B A x 2000 Solution Assuming sound travels at 1100 feet per second, you know that the explosion took place 2200 feet farther from B than from A, as shown in Figure 10.40. The locus of all points that are 2200 feet closer to A than to B is one branch of the hyperbola x2 a2 where y2 b2 1 2200 c a− c a− and 2 = 5280 c 2200 + 2( c a− ) = 5280 c 5280 2 2640 a 2200 2 1100. FIGURE 10.40 Hyperbolic orbit Vertex Sun p Elliptical orbit b2 c 2 a 2 26402 11002 5,759,600, So, the explosion occurred somewhere on the right branch of the hyperbola and you can conclude that x 2 1,210,000 y 2 5,759,600 1. Now try Exercise 41. Another interesting application of conic sections involves the orbits of comets in our solar system. Of the 610 comets identified prior to 1970, 245 have elliptical orbits, 295 have parabolic orbits, and 70 have hyperbolic orbits. The center of the sun is a focus of each of these orbits, and each orbit has a vertex at the point where the comet is closest to the sun, as shown in Figure 10.41. Undoubtedly, there have been many comets with parabolic or hyperbolic orbits that were not identified. We only get to see such comets once. Comets with elliptical orbits, such as Halley’s comet, are the only ones that remain in our solar system. is the distance between the vertex and the focus (in meters), and is the velocity of the comet at the vertex in (meters per second), then the type of orbit is determined as follows. If p v Parabolic orbit 1. Ellipse: 2. Parabola: 3. Hyperbola: v < 2GMp v 2GMp v > 2GMp FIGURE 10.41 In each of these relations, G 6.67 1011 gravitational constant). M 1.989 1030 kilograms (the mass of the sun) and cubic meter per kilogram-second squared (the universal 333202_1004.qxd 12/8/05 9:03 AM Page 759 Section 10.4 Hyperbolas 759 General Equations of Conics Classifying a Conic from Its General Equation Ax 2 Cy 2 Dx Ey F 0 The graph of is one of the following. 1. Circle: 2. Parabola: A C AC 0 A 0 or C 0, but not both. 3. Ellipse: AC > 0 A and have like signs. C 4. Hyperbola: AC < 0 A and have unlike signs. C The test above is valid if the graph is a conic. The test does not apply to equations such as x 2 y 2 1, whose graph is not a conic. Example 6 Classifying Conics from General Equations Classify the graph of each equation. a. b. c. d. 4x 2 9x y 5 0 4x 2 y 2 8x 6y 4 0 2x 2 4y 2 4x 12y 0 2x2 2y2 8x 12y 2 0 Solution a. For the equation 4x 2 9x y 5 0, you have AC 40 0. Parabola So, the graph is a parabola. b. For the equation 4x 2 y 2 8x 6y 4 0, you have AC 41 < 0. Hyperbola So, the graph is a hyperbola. c. For the equation 2x 2 4y 2 4x 12y 0, you have AC 24 > 0. Ellipse So, the graph is an ellipse. d. For the equation A C 2. 2x2 2y2 8x 12y 2 0, you have Circle So, the graph is a circle. Now try Exercise 49. W RITING ABOUT MATHEMATICS Sketching Conics Sketch each of the conics described in Example 6. Write a paragraph describing the procedures that allow you to sketch the conics efficiently Historical Note Caroline Herschel (1750–1848) was the first woman to be credited with detecting a new comet. During her long life, this English astronomer discovered a total of eight new comets. 333202_1004.qxd 12/8/05 9:03 AM Page 760 760 Chapter 10 Topics in Analytic Geometry 10.4 Exercises VOCABULARY CHECK: Fill in
the blanks. 1. A ________ is the set of all points x, y in a plane, the difference of whose distances from two distinct fixed points, called ________, is a positive constant. 2. The graph of a hyperbola has two disconnected parts called ________. 3. The line segment connecting the vertices of a hyperbola is called the ________ ________, and the midpoint of the line segment is the ________ of the hyperbola. 4. Each hyperbola has two ________ that intersect at the center of the hyperbola. 5. The general form of the equation of a conic is given by ________. PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. In Exercises 1–4, match the equation with its graph. [The graphs are labeled (a), (b), (c), and (d).] (a) (c) y 8 4 −8 4 8 −4 −8 y 8 −8 −4 4 8 −8 1 y 2 4 1 1. 3. x 2 y 2 25 9 x 1 2 16 (b) (d) x x y 8 x x −8 −4 4 8 −8 y 8 4 −4 4 8 −4 −8 2. 4. x 2 y 2 9 25 x 12 16 1 y 22 9 1 In Exercises 5–16, find the center, vertices, foci, and the equations of the asymptotes of the hyperbola, and sketch its graph using the asymptotes as an aid. 5. x 2 y 2 1 7. 9. x 2 y 2 81 25 x 12 4 1 y 22 1 1 6. 8. x 2 9 x 2 36 y 2 25 y 2 4 1 1 1 1 x 32 144 y 62 19 y 12 14 y 22 25 x 22 14 x 32 116 9x 2 y 2 36x 6y 18 0 x 2 9y 2 36y 72 0 x 2 9y 2 2x 54y 80 0 16y 2 x 2 2x 64y 63 0 1 10. 11. 12. 13. 14. 15. 16. In Exercises 17–20, find the center, vertices, foci, and the equations of the asymptotes of the hyperbola. Use a graphing utility to graph the hyperbola and its asymptotes. 17. 18. 19. 20. 2x 2 3y 2 6 6y 2 3x 2 18 9y 2 x 2 2x 54y 62 0 9x 2 y 2 54x 10y 55 0 In Exercises 21–26, find the standard form of the equation of the hyperbola with the given characteristics and center at the origin. 21. Vertices: 22. Vertices: 23. Vertices: 24. Vertices: 0, ±2; ±4, 0; ±1, 0; 0, ±3; foci: 0, ±4 ±6, 0 asymptotes: foci: asymptotes: y ±5x y ±3x 25. Foci: 26. Foci: 0, ±8; ±10, 0; asymptotes: asymptotes: y ±4x y ± 3 4x In Exercises 27–38, find the standard form of the equation of the hyperbola with the given characteristics. 27. Vertices: 28. Vertices: 2, 0, 6, 0; 2, 3, 2, 3; foci: 0, 0, 8, 0 foci: 2, 6, 2, 6 333202_1004.qxd 12/8/05 9:03 AM Page 761 Section 10.4 Hyperbolas 761 29. Vertices: 30. Vertices: 31. Vertices: 4, 1, 4, 9; 2, 1, 2, 1); 2, 3, 2, 3; passes through the point 2, 1, 2, 1; passes through the point 32. Vertices: foci: 4, 0, 4, 10 foci: 3, 1, 3, 1 0, 5 5, 4 33. Vertices: 0, 4, 0, 0; 34. Vertices: passes through the point 1, 2, 1, 2; passes through the point 5, 1 0, 5 asymptotes: y 4 x 35. Vertices: 36. Vertices: asymptotes: 37. Vertices: 38. Vertices: 1, 2, 3, 2; y x, 3, 0, 3, 6; y 6 x, 0, 2, 6, 2; y 2 3, 0, 3, 4; y 2 asymptotes: 3 x, y 4 2 3x asymptotes: 3 x, y 4 2 3x y x 39. Art A sculpture has a hyperbolic cross section (see figure). y (−2, 13) 16 (2, 13) (−1, 0) −3 −2 8 4 −4 −8 (1, 0) x 2 3 4 (−2, − 13) −16 (2, −13) (a) Write an equation that models the curved sides of the sculpture. (b) Each unit in the coordinate plane represents 1 foot. Find the width of the sculpture at a height of 5 feet. 40. Sound Location You and a friend live 4 miles apart (on the same “east-west” street) and are talking on the phone. You hear a clap of thunder from lightning in a storm, and 18 seconds later your friend hears the thunder. Find an equation that gives the possible places where the lightning could have occurred. (Assume that the coordinate system is measured in feet and that sound travels at 1100 feet per second.) 3300, 0 3300, 1100, 41. Sound Location Three listening stations located at 3300, 0, monitor an and explosion. The last two stations detect the explosion 1 second and 4 seconds after the first, respectively. Determine the coordinates of the explosion. (Assume that the coordinate system is measured in feet and that sound travels at 100 feet per second.) Model It 42. LORAN Long distance radio navigation for aircraft and ships uses synchronized pulses transmitted by widely separated transmitting stations. These pulses travel at the speed of light (186,000 miles per second). The difference in the times of arrival of these pulses at an aircraft or ship is constant on a hyperbola having the transmitting stations as foci. Assume that two stations, 300 miles apart, are positioned on the rectangular coordinate system at points with coordinates and 150, 0, and that a ship is traveling on a hyperbolic path with coordinates 150, 0 (see figure). x, 75 y 100 50 Station 2 −150 −50 50 −50 Not drawn to scale Station 1 x 150 Bay x (a) Find the -coordinate of the position of the ship if the time difference between the pulses from the transmitting stations is 1000 microseconds (0.001 second). (b) Determine the distance between the ship and station 1 when the ship reaches the shore. (c) The ship wants to enter a bay located between the two stations. The bay is 30 miles from station 1. What should the time difference be between the pulses? (d) The ship is 60 miles offshore when the time difference in part (c) is obtained. What is the position of the ship? 333202_1004.qxd 12/8/05 9:03 AM Page 762 762 Chapter 10 Topics in Analytic Geometry 43. Hyperbolic Mirror A hyperbolic mirror (used in some telescopes) has the property that a light ray directed at a focus will be reflected to the other focus. The focus of a hyperbolic mirror (see figure) has coordinates Find the vertex of the mirror if the mount at the top edge of the mirror has coordinates 24, 24. 24, 0. y (24, 24) x (24, 0) − ( 24, 0) 44. Running Path Let represent a water fountain located in a city park. Each day you run through the park along a path given by 0, 0 x 2 y 2 200x 52,500 0 where and are measured in meters. x y (a) What type of conic is your path? Explain your reasoning. (b) Write the equation of the path in standard form. Sketch a graph of the equation. (c) After you run, you walk to the water fountain. If you how far must you walk 100, 150, stop running at for a drink of water? In Exercises 45– 60, classify the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60. x 2 y 2 6x 4y 9 0 x 2 4y 2 6x 16y 21 0 4x 2 y 2 4x 3 0 y 2 6y 4x 21 0 y 2 4x 2 4x 2y 4 0 x 2 y 2 4x 6y 3 0 x 2 4x 8y 2 0 4x 2 y 2 8x 3 0 4x 2 3y 2 8x 24y 51 0 4y 2 2x 2 4y 8x 15 0 25x 2 10x 200y 119 0 4y 2 4x 2 24x 35 0 4x 2 16y 2 4x 32y 1 0 2y 2 2x 2y 1 0 100x 2 100y 2 100x 400y 409 0 4x 2 y 2 4x 2y 1 0 Synthesis True or False? the statement is true or false. Justify your answer. In Exercises 61 and 62, determine whether 61. In the standard form of the equation of a hyperbola, the the larger the eccentricity of the to a, b larger the ratio of hyperbola. 62. In the standard form of the equation of a hyperbola, the trivial solution of two intersecting lines occurs when b 0. 63. Consider a hyperbola centered at the origin with a horizontal transverse axis. Use the definition of a hyperbola to derive its standard form. 64. Writing Explain how the central rectangle of a hyperbola can be used to sketch its asymptotes. 65. Think About It Change the equation of the hyperbola so that its graph is the bottom half of the hyperbola. 9x 2 54x 4y 2 8y 41 0 66. Exploration A circle and a parabola can have 0, 1, 2, 3, or 4 points of intersection. Sketch the circle given by x 2 y 2 4. Discuss how this circle could intersect a parabola with an equation of the form Then C find the values of for each of the five cases described below. Use a graphing utility to verify your results. y x 2 C. (a) No points of intersection (b) One point of intersection (c) Two points of intersection (d) Three points of intersection (e) Four points of intersection Skills Review In Exercises 67–72, factor the polynomial completely. 67. 68. 69. 70. 71. 72. x3 16x x2 14x 49 2x3 24x2 72x 6x3 11x2 10x 16x3 54 4 x 4x 2 x3 In Exercises 73–76, sketch a graph of the function. Include two full periods. 73. 74. y 2 cos x 1 y sin x y tan 2x 75. 76. y 1 2 sec x 333202_1005.qxd 12/8/05 9:04 AM Page 763 10.5 Rotation of Conics Section 10.5 Rotation of Conics 763 What you should learn • Rotate the coordinate axes -term in to eliminate the equations of conics. • Use the discriminant to xy classify conics. Why you should learn it As illustrated in Exercises 7–18 on page 769, rotation of the coordinate axes can help you identify the graph of a general second-degree equation. Rotation In the preceding section, you learned that the equation of a conic with axes parallel to one of the coordinate axes has a standard form that can be written in the general form Ax 2 Cy 2 Dx Ey F 0. Horizontal or vertical axis In this section, you will study the equations of conics whose axes are rotated so x that they are not parallel to either the -axis or the -axis. The general equation for such conics contains an -term. xy y Ax 2 Bxy Cy 2 Dx Ey F 0 Equation in xy -plane xy -term, you can use a procedure called rotation of axes. The To eliminate this objective is to rotate the - and -axes until they are parallel to the axes of the conic. The rotated axes are denoted as the -axis, as shown in Figure 10.42. -axis and the FIGURE 10.42 After the rotation, the equation of the conic in the new form xy -plane will have the Ax2 Cy2 Dx Ey F 0. Equation in xy -plane -term, you can obtain a standard form by Because this equation has no completing the square. The following theorem identifies how much to rotate the axes to eliminate the -term and also the equations for determining the new coefficients xy A, C, D, E, and F. xy Rotation of Axes to Eliminate an xy-Term The general second-degree equation Ey F 0 can be rewritten as Ax 2 Bxy Cy 2 Dx Ax2 Cy2 Dx Ey F 0 by rotating the coordinate axes through an angle where , cot 2 A C B . The coefficients of the new equation are obtained by making the and y x sin y cos . substitutions x x cos y sin 333202_1005.qxd 12/8/05 9:04 AM Page 764 764 Chapter 10 Topics in Analytic Geometry Remember that the substitutions x x cos y sin and y x sin y
cos were developed to eliminate the xy -term in the rotated system. You can use this as a check on your work. In other words, if your final equation contains an xy -term, you know that you made a mistake. (x ′)2 2 ( 2( − (y ′)2 2 ( 2( = 1 y x ′ x 1 2 xy − 1 = 0 y ′ −2 −1 2 1 −1 -system: Vertices: xy In In xy-system: FIGURE 10.43 2, 0, 2, 0 1, 1, 1, 1 Example 1 Rotation of Axes for a Hyperbola Write the equation xy 1 0 in standard form. Solution Because A 0, B 1, and C 0, you have cot 2 A C B 0 2 2 4 which implies that 4 y sin y 1 2 4 x x cos x 1 2 x y 2 and 4 y cos y 1 2 4 y x sin x 1 2 x y . 2 -system is obtained by substituting these expressions in The equation in the the equation x y 2 xy xy 1 0. x y 2 1 0 1 0 x2 y2 2 x 2 22 y 2 2 2 1 Write in standard form. xy In the ± 2, 0, xy -system, this is a hyperbola centered at the origin with vertices at as shown in Figure 10.43. To find the coordinates of the vertices in the ± 2, 0 in the equations -system, substitute the coordinates x x y 2 and y x y . 2 1, 1 This substitution yields the vertices also that the asymptotes of the hyperbola have equations correspond to the original - and -axes. and x y in the 1, 1 xy y ±x, -system. Note which Now try Exercise 7. 333202_1005.qxd 12/8/05 9:04 AM Page 765 Section 10.5 Rotation of Conics 765 Example 2 Rotation of Axes for an Ellipse Sketch the graph of 7x 2 63xy 13y 2 16 0. Solution Because A 7, B 63, and C 13, you have cot 2 A C B 7 13 63 1 3 which implies that making the substitutions 6. The equation in the xy -system is obtained by 6 x x cos x3 y sin y 1 6 2 2 3x y 2 and 6 6 y cos y3 2 y x sin x1 2 x 3y 2 y ′ y 2 (x ′)2 22 + (y ′)2 12 = 1 x ′ in the original equation. So, you have 7x2 63 xy 13y2 16 0 2 73x y 2 13x 3y 2 16 0 2 633x y 2 x 3y 2 −2 −1 1 2 x which simplifies to −1 −2 7x2 − 6 3xy + 13y2 − 16 = 0 Vertices: xy In In xy-system: -system: ±2, 0, 0, ±1 3, 1, 3, 1 , , 4x 2 16y2 16 0 4x 2 16y 2 16 x 2 4 x 2 22 y2 1 y2 12 1 1. Write in standard form. This is the equation of an ellipse centered at the origin with vertices the -system, as shown in Figure 10.44. xy ±2, 0 in FIGURE 10.44 Now try Exercise 13. 333202_1005.qxd 12/8/05 9:04 AM Page 766 766 Chapter 10 Topics in Analytic Geometry Example 3 Rotation of Axes for a Parabola Sketch the graph of x 2 4xy 4y 2 55y 1 0. Solution Because A 1, B 4, cot 2 A C B and 1 4 4 C 4, you have 3 4 . Using this information, draw a right triangle as shown in Figure 10.45. From the figure, you can see that you can use the half-angle formulas in the forms To find the values of cos 2 3 5. cos , sin and sin 1 cos 2 2 and cos 1 cos 2 . 2 So, sin 1 cos 2 2 1 3 5 2 cos 1 cos . Consequently, you use the substitutions x x cos y sin y 1 x 2 5 5 2x y 5 y x sin y cos y 2 x 1 5 5 x 2y 5 . Substituting these expressions in the original equation, you have 2x y 2 5 42x y 5 x 2y 5 x 2 4xy 4y 2 55y 1 0 55x 2y 1 0 4x 2y 2 5 5 which simplifies as follows. 5y2 5x 10y 1 0 5y2 2y 5x 1 5y 1 2 5x 4 y 1 2 1x 4 5 The graph of this equation is a parabola with vertex to the -axis in the shown in Figure 10.46. -system, and because xy x Now try Exercise 17. Group terms. Write in completed square form. Write in standard form. 5, 1. 4 sin 15, Its axis is parallel 26.6, as 5 4 2θ 3 FIGURE 10.45 x2 − 4xy + 4y2 + 5 5y + 1 = 0 y y ′ x ′ θ ≈ 26.6° x 2 1 −1 −2 (y′ + 1)2 = (−1) x′ − 4 5 ( ) Vertex: xy In -system: In xy-system: FIGURE 10.46 5, 1 4 13 55 , 6 55 333202_1005.qxd 12/8/05 9:04 AM Page 767 Section 10.5 Rotation of Conics 767 Invariants Under Rotation In the rotation of axes theorem listed at the beginning of this section, note that the constant term is the same in both equations, Such quantities are invariant under rotation. The next theorem lists some other rotation invariants. F F. Rotation Invariants The rotation of the coordinate axes through an angle equation Ax 2 Bxy Cy 2 Dx Ey F 0 Ax 2 Cy 2 Dx Ey F 0 that transforms the into the form has the following rotation invariants. 1. F F A C A C 2. 3. B 2 4AC B 2 4AC term in the xyIf there is an equation of a conic, you should realize then that the conic is rotated. Before rotating the axes, you should use the discriminant to classify the conic. You can use the results of this theorem to classify the graph of a second-term in much the same way you do for a the invari- -term. Note that because B 0, xy xy degree equation with an second-degree equation without an ant reduces to B 2 4AC B 2 4AC 4AC. Discriminant This quantity is called the discriminant of the equation Ax 2 Bxy Cy 2 Dx Ey F 0. Now, from the classification procedure given in Section 10.4, you know that the sign of determines the type of graph for the equation AC Ax 2 Cy 2 Dx Ey F 0. Consequently, the sign of original equation, as given in the following classification. B 2 4AC will determine the type of graph for the Classification of Conics by the Discriminant The graph of the equation is, except in degenerate cases, determined by its discriminant as follows. Ax 2 Bxy Cy 2 Dx Ey F 0 1. Ellipse or circle: 2. Parabola: 3. Hyperbola: B 2 4AC < 0 B 2 4AC 0 B 2 4AC > 0 For example, in the general equation 3x2 7xy 5y2 6x 7y 15 0 you have A 3, B 7, and B2 4AC 72 435 49 60 11. So the discriminant is C 5. Because 11 < 0, the graph of the equation is an ellipse or a circle. 333202_1005.qxd 12/8/05 9:04 AM Page 768 768 Chapter 10 Topics in Analytic Geometry Example 4 Rotation and Graphing Utilities For each equation, classify the graph of the equation, use the Quadratic Formula and then use a graphing utility to graph the equation. to solve for y, a. c. 2x2 3xy 2y2 2x 0 3x2 8xy 4y2 7 0 b. x2 6xy 9y2 2y 1 0 Solution a. Because B2 4AC 9 16 < 0, the graph is a circle or an ellipse. Solve for as follows. y 2x2 3xy 2y2 2x 0 2y2 3xy 2x2 2x 0 Write original equation. ay2 by c 0 Quadratic form 3x ± 3x2 422x2 2x 22 y 3 Graph both of the equations to obtain the ellipse shown in Figure 10.47. y 3x ± x16 7x 4 −1 5 3x x16 7x 4 3x x16 7x 4 y1 y2 Top half of ellipse Bottom half of ellipse −1 FIGURE 10.47 4 0 0 FIGURE 10.48 −15 FIGURE 10.49 10 −10 b. Because B2 4AC 36 36 0, x2 6xy 9y2 2y 1 0 9y2 6x 2y x2 1 0 the graph is a parabola. Write original equation. Quadratic form ay2 by c 0 y 6x 2 ± 6x 22 49x2 1 29 Graphing both of the equations to obtain the parabola shown in Figure 10.48. c. Because B2 4AC 64 48 > 0, the graph is a hyperbola. 3x2 8xy 4y2 7 0 4y2 8xy 3x2 7 0 Write original equation. Quadratic form ay2 by c 0 y 8x ± 8x2 443x2 7 24 The graphs of these two equations yield the hyperbola shown in Figure 10.49. Now try Exercise 33. 6 15 W RITING ABOUT MATHEMATICS Classifying a Graph as a Hyperbola In Section 2.6, it was mentioned that the graph of is a hyperbola. Use the techniques in this section to verify this, and justify each step. Compare your results with those of another student. f x 1x 333202_1005.qxd 12/8/05 9:04 AM Page 769 10.5 Exercises Section 10.5 Rotation of Conics 769 VOCABULARY CHECK: Fill in the blanks. 1. The procedure used to eliminate the xy- term in a general second-degree equation is called ________ of ________. 2. After rotating the coordinate axes through an angle , the general second-degree equation in the new xy- plane will have the form ________. 3. Quantities that are equal in both the original equation of a conic and the equation of the rotated conic are ________ ________ ________. B 2 4AC 4. The quantity is called the ________ of the equation Ax 2 Bxy Cy 2 Dx Ey F 0. PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. xy degrees from the In Exercises 1–6, the -coordinate system has been -coordinate system. The rotated xy -coordinate system are coordinates of a point in the xy given. Find the coordinates of the point in the rotated coordinate system. 90, 0, 3 30, 1, 3 45, 2, 1 45, 3, 3 60, 3, 1 30, 2, 4 3. 1. 6. 5. 4. 2. 22. 23. 24. 25. 26. 40x 2 36xy 25y 2 52 32x 2 48xy 8y 2 50 24x2 18xy 12y2 34 4x 2 12xy 9y 2 413 12x 613 8y 91 6x2 4xy 8y2 55 10x 75 5y 80 In Exercises 27–32, match the graph with its equation. [The graphs are labeled (a), (b), (c), (d), (e), and (f).] (a) y y′ (b) y y ′ In Exercises 7–18, rotate the axes to eliminate the -term in the equation. Then write the equation in standard form. Sketch the graph of the resulting equation, showing both sets of axes. xy 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. xy 1 0 xy 2 0 x 2 2xy y 2 1 0 xy x 2y 3 0 xy 2y 4x 0 2x 2 3xy 2y 2 10 0 5x 2 6xy 5y 2 12 0 13x 2 63 xy 7y 2 16 0 3x 2 23 xy y 2 2x 23 y 0 16x 2 24xy 9y 2 60x 80y 100 0 9x 2 24xy 16y 2 90x 130y 0 9x 2 24xy 16y 2 80x 60y 0 −2 −3 y′ y 3 (c) −3 −2 3 2 x′ x 3 −3 (d) y y′ x′ x 3 x x′ x′ x 1 3 4 x′ −3 −4 y 4 2 In Exercises 19–26, use a graphing utility to graph the conic. Determine the angle through which the axes are rotated. Explain how you used the graphing utility to obtain the graph. (e) y y′ x′ (f) y′ 19. 20. 21. x 2 2xy y 2 20 x 2 4xy 2y 2 6 17x 2 32xy 7y 2 75 −4 −2 −2 −4 x −4 −2 2 4 x −2 −4 333202_1005.qxd 12/8/05 11:07 AM Page 770 770 Chapter 10 Topics in Analytic Geometry 27. 28. 29. 30. 31. 32. xy 2 0 x 2 2xy y 2 0 2x 2 3xy 2y 2 3 0 x 2 xy 3y 2 5 0 3x 2 2xy y 2 10 0 x 2 4xy 4y 2 10x 30 0 In Exercises 33– 40, (a) use the discriminant to classify the graph, (b) use the Quadratic Formula to solve for and (c) use a graphing utility to graph the equation. y, 33. 34. 35. 36. 37. 38. 39. 40. 16x 2 8xy y 2 10x 5y 0 x 2 4xy 2y 2 6 0 12x 2 6xy 7y 2 45 0 2x 2 4xy 5y 2 3x 4y 20 0 x 2 6xy 5y 2 4x 22 0 36x 2 60xy 25y 2 9y 0 x 2 4xy 4y 2 5x y 3 0 x 2 xy 4y 2 x y 4 0 In Exercises 41– 44, sketch (if possible) the graph of the degenerate conic. 41. 42. 43. 44. y 2 9x 2 0 x 2 y 2 2x 6y 10 0 x 2 2xy y 2 1 0 x 2 10xy y 2 0 53. 54. 55. 56. 57. 58. x 2 y2 4 0 3x y 2 0 4x 2 9y2 36y 0 x 2 9y 27 0 x 2 2y2 4x 6y 5 0 x y 4 0 x 2 2y2 4x 6y 5 0 x 2 4x y 4 0 xy x 2y 3 0 x 2 4y2 9 0 5x 2 2xy 5y 2 12 0 x y 1 0 Synthesis True or False? the statement is true or false. Justify your answer. In Exercises 59 and 60, determine whether 59. The graph of the equ
ation x2 xy ky2 6x 10 0 where k is any constant less than 1 4, is a hyperbola. 60. After a rotation of axes is used to eliminate the xy -term from an equation of the form Ax2 Bxy Cy2 Dx Ey F 0 the coefficients of the respectively. x2 - and y2 -terms remain A and C, In Exercises 45–58, find any points of intersection of the graphs algebraically and then verify using a graphing utility. 61. Show that the equation x 2 y 2 r 2 45. 46. 47. 48. 49. 50. 51. 52. x 2 y2 4x 6y 4 0 x 2 y 2 4x 6y 12 0 x 2 y 2 8x 20y 7 0 x 2 9y2 8x 4y 7 0 4x 2 y 2 16x 24y 16 0 4x 2 y2 40x 24y 208 0 x 2 4y2 20x 64y 172 0 16x 2 4y 2 320x 64y 1600 0 x 2 y 2 12x 16y 64 0 x 2 y 2 12x 16y 64 0 x 2 4y 2 2x 8y 1 0 x 2 2x 4y 1 0 16x 2 y 2 24y 80 0 16x 2 25y 2 400 0 16x 2 y 2 16y 128 0 y 2 48x 16y 32 0 is invariant under rotation of axes. 62. Find the lengths of the major and minor axes of the ellipse graphed in Exercise 14. Skills Review In Exercises 63–70, graph the function. 63. 65. 67. 69. f x x 3 gx 4 x2 ht t 23 3 f t t 5 1 64. 66. 68. 70. f x x 4 1 gx 3x 2 ht 1 t 43 2 f t 2t 3 In Exercises 71–74, find the area of the triangle. 71. 72. C 110, a 8, b 12 B 70, a 25, c 16 a 11, b 18, c 10 73. 74. a 23, b 35, c 27 333202_1006.qxd 12/8/05 9:05 AM Page 771 10.6 Parametric Equations Section 10.6 Parametric Equations 771 What you should learn • Evaluate sets of parametric equations for given values of the parameter. • Sketch curves that are represented by sets of parametric equations. • Rewrite sets of parametric equations as single rectangular equations by eliminating the parameter. • Find sets of parametric equations for graphs. Why you should learn it Parametric equations are useful for modeling the path of an object. For instance, in Exercise 59 on page 777, you will use a set of parametric equations to model the path of a baseball. Jed Jacobsohn/Getty Images Plane Curves x y. Up to this point you have been representing a graph by a single equation involving the two variables and In this section, you will study situations in which it is useful to introduce a third variable to represent a curve in the plane. To see the usefulness of this procedure, consider the path followed by an object that is propelled into the air at an angle of If the initial velocity of the object is 48 feet per second, it can be shown that the object follows the parabolic path y x2 72 Rectangular equation 45. x as shown in Figure 10.50. However, this equation does not tell the whole story. Although it does tell you where the object has been, it doesn’t tell you when the object was at a given point on the path. To determine this time, you can introduce a third variable , called a parameter. It is possible to write both and y to obtain the parametric equations x, y x t t as functions of x 242t y 16t 2 242t. Parametric equation for x Parametric equation for y From this set of equations you can determine that at time the point Similarly, at time 242, 242 16, and so on, as shown in Figure 10.50. 0, 0. t 1, t 0, the object is at the object is at the point Rectangular equation: − x2 72 + = y x Parametric equations: x t = 24 2 2 − = 16 + 24 2 y t t y 18 9 t = 3 2 4 (36, 18) (0, 0) t = 3 2 2 (72, 0) t = 0 9 18 27 36 45 54 63 72 81 x Curvilinear Motion: Two Variables for Position, One Variable for Time FIGURE 10.50 For this particular motion problem, t, and the resulting path is a plane curve. (Recall that a continuous function is one whose graph can be traced without lifting the pencil from the paper.) are continuous functions of and y x Definition of Plane Curve f If and are continuous functions of on an interval pairs The equations C. t g ft, gt x ft is a plane curve y gt and I, the set of ordered are parametric equations for C, and t is the parameter. 333202_1006.qxd 12/8/05 9:05 AM Page 772 772 Chapter 10 Topics in Analytic Geometry Sketching a Plane Curve When sketching a curve represented by a pair of parametric equations, you still plot points in the is determined from a value chosen for the parameter Plotting the resulting points in the order of increasing values of traces the curve in a specific direction. This is called the orientation of the curve. -plane. Each set of coordinates x, y xy t. t Example 1 Sketching a Curve Sketch the curve given by the parametric equations x t2 4 and y t 2 , 2 ≤ t ≤ 3. Solution Using values of shown in the table. t in the interval, the parametric equations yield the points x, y y y 6 4 2 −2 −4 6 4 2 −2 −1 FIGURE 10.51 FIGURE 10.52 x = t2 − 2 −2 ≤ t ≤ 3 x = 4t2 − 1 − 12 0 12 1 32 C By plotting these points in the order of increasing shown in Figure 10.51. Note that the arrows on the curve indicate its orientation t to 3. So, if a particle were moving on this curve, it would as increases from start at and then move along the curve to the point you obtain the curve 0, 1 5, 3 2 . t, 2 Now try Exercises 1(a) and (b). Note that the graph shown in Figure 10.51 does not define as a function of This points out one benefit of parametric equations—they can be used to x. represent graphs that are more general than graphs of functions. y It often happens that two different sets of parametric equations have the same graph. For example, the set of parametric equations x 4t2 4 and y t, 1 ≤ t ≤ 3 2 t has the same graph as the set given in Example 1. However, by comparing the values of in Figures 10.51 and 10.52, you see that this second graph is traced out more rapidly (considering as time) than the first graph. So, in applications, different parametric representations can be used to represent various speeds at which objects travel along a given path. t 333202_1006.qxd 12/8/05 9:05 AM Page 773 Section 10.6 Parametric Equations 773 Eliminating the Parameter Example 1 uses simple point plotting to sketch the curve. This tedious process can sometimes be simplified by finding a rectangular equation (in and ) that has the same graph. This process is called eliminating the parameter. y x Parametric equations x t2 4 y t2 Solve for t in one equation. t 2y Substitute in other equation. Rectangular equation x 2y2 4 x 4y2 4 Now you can recognize that the equation a horizontal axis and vertex 4, 0. x 4y2 4 represents a parabola with When converting equations from parametric to rectangular form, you may need to alter the domain of the rectangular equation so that its graph matches the graph of the parametric equations. Such a situation is demonstrated in Example 2. Example 2 Eliminating the Parameter Sketch the curve represented by the equations x 1 t 1 and y t t 1 by eliminating the parameter and adjusting the domain of the resulting rectangular equation. in the equation for produces x x2 1 t 1 Solution Solving for t x 1 t 1 which implies that t 1 x2 x2 . Now, substituting in the equation for 1 x2 x2 1 x2 x2 1 y, you obtain the rectangular equation 1 x2 x2 1 x2 x2 1 x2. x2 x2 1 From this rectangular equation, you can recognize that the curve is a parabola that opens downward and has its vertex at Also, this rectangular equation but from the parametric equation for you can see is defined for all values of that the curve is defined only when This implies that you should restrict the domain of t > 1. to positive values, as shown in Figure 10.53. 0, 1. x, x x Exploration Most graphing utilities have a parametric mode. If yours does, enter the parametric equations from Example 2. Over what values should you let vary to obtain the graph shown in Figure 10.53? t Parametric equations1 −2 −3 − t = 0.75 −2 − FIGURE 10.53 Now try Exercise 1(c). 333202_1006.qxd 12/8/05 9:05 AM Page 774 774 Chapter 10 Topics in Analytic Geometry It is not necessary for the parameter in a set of parametric equations to represent time. The next example uses an angle as the parameter. To eliminate the parameter in equations involving trigonometric functions, try using the identities sin2 cos2 1 sec2 tan2 1 as shown in Example 3. Example 3 Eliminating an Angle Parameter Sketch the curve represented by x 3 cos and y 4 sin , 0 ≤ ≤ 2 by eliminating the parameter. Solution Begin by solving for cos and sin in the equations. 3 2 1 −1 −2 −3 θ = π −4 −2 − cos x 3 and sin y 4 Solve for cos and sin . Use the identity sin2 cos2 1 to form an equation involving only and x y cos y = 4 sin θ θ cos2 sin2 1 y x 4 3 1 2 2 x2 9 y2 16 1 Pythagorean identity Substitute x 3 for cos and for sin . y 4 Rectangular equation 0, 0, From this rectangular equation, you can see that the graph is an ellipse centered as at shown in Figure 10.54. Note that the elliptic curve is traced out counterclockwise as varies from 0 to and minor axis of length with vertices 0, 4 2b 6, 0, 4 2. and FIGURE 10.54 Now try Exercise 13. In Examples 2 and 3, it is important to realize that eliminating the parameter is primarily an aid to curve sketching. If the parametric equations represent the path of a moving object, the graph alone is not sufficient to describe the object’s motion. You still need the parametric equations to tell you the position, direction, and speed at a given time. Finding Parametric Equations for a Graph You have been studying techniques for sketching the graph represented by a set of parametric equations. Now consider the reverse problem—that is, how can you find a set of parametric equations for a given graph or a given physical description? From the discussion following Example 1, you know that such a representation is not unique. That is, the equations 1 ≤ t ≤ 3 2 x 4t2 4 y t, and produced the same graph as the equations x t2 4 and y t 2 , 2 ≤ t ≤ 3. This is further demonstrated in Example 4. 333202_1006.qxd 12/8/05 9:05 AM Page 775 x = 1 − t y = 2t − 2 t = 0 x 2 −1 −2 −3 t = −1 t = 3 FIGURE 10.55 PD In Example 5, represents the arc of the circle between .D points and P Te c h n o l o g y Use a graphing utility in parametric mode to obtain a graph similar to Figure 10.56 by graphing the following equations. X1T Y1T T sin T 1 cos T Section 10.6 Parametric Equations 775 Example 4 Finding Parametric Equations for a Graph Find a set
of parametric equations to represent the graph of the following parameters. t 1 x t x b. a. y 1 x 2, using Solution a. Letting x t t x, you obtain the parametric equations and t 1 x, y 1 x 2 1 t 2. you obtain the parametric equations b. Letting x 1 t and y 1 x2 1 1 t2 2t t 2. In Figure 10.55, note how the resulting curve is oriented by the increasing values of For part (a), the curve would have the opposite orientation. t. Now try Exercise 37. Example 5 Parametric Equations for a Cycloid Describe the cycloid traced out by a point on the circumference of a circle of radius as the circle rolls along a straight line in a plane. P a Solution As the parameter, let be the measure of the circle’s rotation, and let the point P x, y 0, is at a maximum point is back on the -axis at and when 2a, 0. So, you have begin at the origin. When APC 180 . is at the origin; when P 2, , x P P a, 2a; From Figure 10.56, you can see that sin sin180 sinAPC AC a BD a cos cos180 cosAPC AP a BD a sin . and a. OD PD AP a cos which implies that x along the -axis, you know that DC a, you have Because the circle rolls BA Furthermore, because x OD BD a a sin So, the parametric equations are and x a sin y BA AP a a cos . y a1 cos . and Cycloid: x = a( − sin ), y = a(1 − cos ) θ θ (3 a, 2a) π θ y 2a a P = (x, y) π ( a, 2a2 a, 0) π 3 a π (4 a, 0) x FIGURE 10.56 Now try Exercise 63. 333202_1006.qxd 12/8/05 11:08 AM Page 776 776 Chapter 10 Topics in Analytic Geometry 10.6 Exercises VOCABULARY CHECK: Fill in the blanks. 1. If and are continuous functions of on an interval t x f t ________ ________ The equations and C. g f I, y gt the set of ordered pairs are ________ equations for f t, gt C, is a and t is the ________. 2. The ________ of a curve is the direction in which the curve is traced out for increasing values of the parameter. 3. The process of converting a set of parametric equations to a corresponding rectangular equation is called ________ the ________. PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. 1. Consider the parametric equations x t x (a) Create a table of - and -values using and t 0, y 3 t. 1, 2, 3, y and 4. (b) Plot the points x, y graph of the parametric equations. generated in part (a), and sketch a (c) Find the rectangular equation by eliminating the parameter. Sketch its graph. How do the graphs differ? 2. Consider the parametric equations x 4 cos2 and y 2 sin . (a) Create a table of 4, 4, (b) Plot the points x - and -values using 2. and x, y generated in part (a), and sketch a graph of the parametric equations. 2, 0, y (c) Find the rectangular equation by eliminating the parameter. Sketch its graph. How do the graphs differ? In Exercises 3–22, (a) sketch the curve represented by the parametric equations (indicate the orientation of the curve) and (b) eliminate the parameter and write the corresponding rectangular equation whose graph represents the curve. Adjust the domain of the resulting rectangular equation if necessary. 3. 5. 7. 9. 11. 13. x 3t 3 y 2t 1 x 1 4t 2t 1 y t 2 x 3 cos y 3 sin 4. 6. 8. 10. 12. 14. x 3 2t y 2 3t cos y 3 sin 15. 17. 19. 21. x 4 sin 2 y 2 cos 2 x 4 2 cos y 1 sin x et y e3t x t3 y 3 ln t 16. 18. 20. 22. x cos y 2 sin 2 x 4 2 cos y 2 3 sin x e2t y et x ln 2t y 2t 2 In Exercises 23 and 24, determine how the plane curves differ from each other. 23. (a) (c) 24. (a) (c) x t y 2t 1 x et y 2et 1 x t y t 2 1 x sin t y sin2 t 1 (b) (d) (b) (d) x cos y 2 cos 1 x et y 2et 1 x t 2 y t 4 1 x et y e2t 1 In Exercises 25 –28, eliminate the parameter and obtain the standard form of the rectangular equation. x2, y2 x1, y1 25. Line through , tx2 x1 x x1 x h r cos , 26. Circle: x h a cos , 27. Ellipse: and y y1 : t y2 y1 y k r sin y k b sin 28. Hyperbola: x h a sec , y k b tan 29. Line: passes through In Exercises 29–36, use the results of Exercises 25–28 to find a set of parametric equations for the line or conic. 0, 0 2, 3 radius: 4 30. Line: passes through 3, 2; 6, 3 6, 3 31. Circle: center: and and 333202_1006.qxd 12/8/05 9:05 AM Page 777 32. Circle: center: 3, 2; radius: 5 33. Ellipse: vertices: 34. Ellipse: vertices: ±4, 0; foci: 4, 7, 4, 3; ±3, 0 foci: 35. Hyperbola: vertices: 36. Hyperbola: vertices: (4, 5, 4, 1 ±4, 0; ±2, 0; foci: foci: ±5, 0 ±4, 0 Section 10.6 Parametric Equations 777 53. Lissajous curve: x 2 cos , y sin 2 54. Evolute of ellipse: 55. Involute of circle: 56. Serpentine curve: y 6 sin3 x 4 cos3 , cos sin x 1 2 sin cos y 1 2 x 1 2 cot , y 4 sin cos In Exercises 37– 44, find a set of parametric equations for t 2 x. the rectangular equation using (a) and (b) t x 37. 39. 41. 43. y 3x 2 y x 2 y x2 1 y 1 x 38. 40. 42. 44. x 3y 2 y x3 y 2 x y 1 2x v0 Projectile Motion A projectile is launched at a height of h feet above the ground at an angle of with the horizontal. The initial velocity is feet per second and the path of the projectile is modeled by the parametric equations x v0 cos t y h v0 sin t 16t 2. In Exercises 57 and 58, use a graphing utility to graph the paths of a projectile launched from ground level at each value of and For each case, use the graph to approximate the maximum height and the range of the projectile. and v0. In Exercises 45–52, use a graphing utility to graph the curve represented by the parametric equations. x 4 sin , x sin , y 41 cos 45. Cycloid: 46. Cycloid: 47. Prolate cycloid: 48. Prolate cycloid: y 1 cos x 3 2 sin , x 2 4 sin , y 1 3 2 cos y 2 4 cos 49. Hypocycloid: x 3 cos3 , y 3 sin3 50. Curtate cycloid: 51. Witch of Agnesi: 52. Folium of Descartes: y 8 4 cos x 8 4 sin , x 2 cot , x 3t 1 t 3, y 2 sin2 y 3t 2 1 t 3 In Exercises 53–56, match the parametric equations with the correct graph and describe the domain and range. [The graphs are labeled (a), (b), (c), and (d).] (a) (c) y 2 1 −2 −1 −1 x 1 2 y 5 (b) (d) y 2 1 1 −1 −1 −2 y 4 x 5 −5 −5 −4 2 −4 x x 57. (a) (b) (c) (d) 58. (a) (b) (c) (d) 60, 60, 45, 45, 15, 15, 30, 30, v0 v0 v0 v0 v0 v0 v0 v0 88 132 88 132 60 100 60 100 feet per second feet per second feet per second feet per second feet per second feet per second feet per second feet per second Model It 59. Sports The center field fence in Yankee Stadium is 7 feet high and 408 feet from home plate. A baseball is hit at a point 3 feet above the ground. It leaves the bat at an angle of degrees with the horizontal at a speed of 100 miles per hour (see figure). 7 ft θ 3 ft 408 ft Not drawn to scale (a) Write a set of parametric equations that model the path of the baseball. (b) Use a graphing utility to graph the path of the baseball when 15. Is the hit a home run? (c) Use a graphing utility to graph the path of the baseball when 23. Is the hit a home run? (d) Find the minimum angle required for the hit to be a home run. 333202_1006.qxd 12/8/05 9:05 AM Page 778 778 Chapter 10 Topics in Analytic Geometry 60. Sports An archer releases an arrow from a bow at a point 5 feet above the ground. The arrow leaves the bow at an 10 angle of with the horizontal and at an initial speed of 240 feet per second. (a) Write a set of parametric equations that model the path of the arrow. (b) Assuming the ground is level, find the distance the arrow travels before it hits the ground. (Ignore air resistance.) (c) Use a graphing utility to graph the path of the arrow and approximate its maximum height. (d) Find the total time the arrow is in the air. 61. Projectile Motion Eliminate the parameter t from the and y h v0 sin t 16t2 parametric equations x v0 cos t for the motion of a projectile to show that the rectangular equation is y 16 sec 2 x 2 tan x h. 2 v0 62. Path of a Projectile The path of a projectile is given by the rectangular equation y 7 x 0.02x 2. (a) Use the result of Exercise 61 to find the parametric equations of the path. h, v0, . and Find (b) Use a graphing utility to graph the rectangular equation for the path of the projectile. Confirm your answer in part (a) by sketching the curve represented by the parametric equations. (c) Use a graphing utility to approximate the maximum height of the projectile and its range. 63. Curtate Cycloid A wheel of radius units rolls along a straight line without slipping. The curve traced by a point P is called a curtate cycloid (see figure). Use the angle shown in the figure to find a set of parametric equations for the curve. that is units from the center b < a b a π ( a, a + b) y 2a P b θ a (0, a − b) π a π 2 a x 64. Epicycloid A circle of radius one unit rolls around the outside of a circle of radius two units without slipping. The curve traced by a point on the circumference of the smaller circle is called an epicycloid (see figure). Use the angle shown in the figure to find a set of parametric equations for the curve. y 4 3 1 θ 1 (x, y) 3 4 x Synthesis True or False? the statement is true or false. Justify your answer. In Exercises 65 and 66, determine whether 65. The two y t 2 1 rectangular equation. sets of parametric and y 9t 2 1 x 3t, equations x t, have the same 66. The graph of the parametric equations x t 2 and y t 2 is the line y x. 67. Writing Write a short paragraph explaining why parametric equations are useful. 68. Writing Explain the process of sketching a plane curve given by parametric equations. What is meant by the orientation of the curve? Skills Review In Exercises 69–72, solve the system of equations. 69. 71. 11 13 5x 7y 3x y 3a 2b c 2a b 3c a 3b 9c 8 3 16 70. 72. 9 14 3x 5y 4x 2y 5u 7v 9w 4 u 2v 3w 7 8u 2v w 20 In Exercises 73–76, find the reference angle in standard position. and , and sketch 73. 75. 105 2 3 230 74. 76. 5 6 333202_1007.qxd 12/8/05 9:06 AM Page 779 10.7 Polar Coordinates Section 10.7 Polar Coordinates 779 What you should learn • Plot points on the polar coordinate system. • Convert points from rectangular to polar form and vice versa. • Convert equations from rectangular to polar form and vice versa. Why you should learn it Polar coordinates offer a different mathematical perspective on graphing. For instance, in Exercises 1–8 on page 7
83, you are asked to find multiple representations of polar coordinates. Introduction on the rectangular coordinate system, where and So far, you have been representing graphs of equations as collections of points x, y y represent the directed In this section, you will distances from the coordinate axes to the point study a different system called the polar coordinate system. x x, y. To form the polar coordinate system in the plane, fix a point called the an initial ray called the polar axis, as P in the plane can be assigned polar O, O pole (or origin), and construct from shown in Figure 10.57. Then each point coordinates r as follows. PO to r, 1. 2. directed distance from directed angle, counterclockwise from polar axis to segment OP P r= ( , )θ r = directed distance θ O = directed angle Polar axis FIGURE 10.57 Example 1 Plotting Points on the Polar Coordinate System a. The point r, 2, 3 lies two units from the pole on the terminal side of the angle 3, as shown in Figure 10.58. b. The point side of the angle r, 3, 6 6, r, 3, 116 lies three units from the pole on the terminal as shown in Figure 10.59. coincides with the point 3, 6, as shown c. The point in Figure 10.60. =θ π 3 2,( θ π 6 3, −( π ) 6 π 3 2 2 3 =θ 0 π 11 6 3,( π 11 6 ) π 3 2 FIGURE 10.58 FIGURE 10.59 FIGURE 10.60 Now try Exercise 1. 333202_1007.qxd 12/8/05 9:06 AM Page 780 780 Chapter 10 Topics in Analytic Geometry Exploration Most graphing calculators have a polar graphing mode. If yours does, graph the equation r 3. (Use a setting in which 6 ≤ x ≤ 6 You should obtain a circle of radius 3. and 4 ≤ y ≤ 4.) a. Use the trace feature to cursor around the circle. Can you locate the point 3, 54? b. Can you find other polar representations of the point 3, 54? how you did it. If so, explain π 2 π 3 2 π 3, −( π ) 3 4 = −, = −3, − 3, − 4 FIGURE 10.61 ( y r θ Pole (Origin) x FIGURE 10.62 3, = ... 4 θ (r, ) (x, y) y x Polar axis (x-axis) x, y In rectangular coordinates, each point has a unique representation. and represent the same point, as illustrated in Example 1. Another way r. represent This is not true for polar coordinates. For instance, the coordinates r, 2 to obtain multiple representations of a point is to use negative values for r Because the same point. In general, the point is a directed distance, the coordinates r, r, r, and r, can be represented as r, r, ± 2n 1 r, r, ± 2n or is any integer. Moreover, the pole is represented by 0, , where is any n where angle. Example 2 Multiple Representations of Points Plot the point point, using 3, 34 2 < < 2. and find three additional polar representations of this Solution The point is shown in Figure 10.61. Three other representations are as follows. 3, 3 4 3, 3 4 3, 3 4 2 3, 5 4 Add 2 . to 3, 7 4 3, 4 Now try Exercise 3. Replace r by –r; subtract from . Replace r by –r; add . to Coordinate Conversion x x, y Moreover, for To establish the relationship between polar and rectangular coordinates, let the polar axis coincide with the positive -axis and the pole with the origin, as shown it follows that in Figure 10.62. Because r 2 x 2 y 2. the definitions of the trigonometric functions imply that tan y x lies on a circle of radius cos x r sin y r r > 0, and r, , , . If r < 0, you can show that the same relationships hold. Coordinate Conversion The polar coordinates as follows. r, are related to the rectangular coordinates x, y Polar-to-Rectangular x r cos y r sin Rectangular-to-Polar tan y x r 2 x 2 y 2 333202_1007.qxd 12/8/05 9:06 AM Page 781 2, 0) 1 −1 −2 FIGURE 10.63 π ) 6 ) 3 2 2 Section 10.7 Polar Coordinates 781 Example 3 Polar-to-Rectangular Conversion Convert each point to rectangular coordinates. 3, 2, b. a. 6 x Solution a. For the point r, 2, , you have the following. x r cos 2 cos 2 y r sin 2 sin 0 x, y 2, 0. (See Figure 10.63.) you have the following. b. For the point The rectangular coordinates are , r, 3, 6 33 2 31 2 x 3 cos y 3 sin 6 6 3 2 3 2 The rectangular coordinates are x, y 3 2 . , 3 2 Now try Exercise 13. Example 4 Rectangular-to-Polar Conversion Convert each point to polar coordinates. a. 1, 1 b. 0, 2 Solution a. For the second-quadrant point x, y 1, 1, you have 1 2 0 1 tan y x 3 . 4 Because lies in the same quadrant as 1 2 1 2 r x 2 y 2 use positive x, y, 2 r. π 2 2 (x, y) = (−1, 1) θ (r, ) = 2, ( −2 −1 FIGURE 10.64 1 π 4 )3 −1 π 2 (x, y) = (0, 2) 1 −1 −2 −1 FIGURE 10.65 θ (r, ) = 2, ( π 2 ) So, one set of polar coordinates is 10.64. r, 2, 34, as shown in Figure b. Because the point x, y 0, 2 lies on the positive -axis, choose y 1 2 0 2 and r 2. This implies that one set of polar coordinates is in Figure 10.65. r, 2, 2, as shown Now try Exercise 19. 333202_1007.qxd 12/8/05 9:06 AM Page 782 782 Chapter 10 Topics in Analytic Geometry Equation Conversion By comparing Examples 3 and 4, you can see that point conversion from the polar to the rectangular system is straightforward, whereas point conversion from the rectangular to the polar system is more involved. For equations, the opposite is x true. To convert a rectangular equation to polar form, you simply replace by r cos can be by written in polar form as follows. For instance, the rectangular equation r sin . y x 2 and y y x 2 r sin r cos 2 r sec tan Rectangular equation Polar equation Simplest form On the other hand, converting a polar equation to rectangular form requires considerable ingenuity. Example 5 demonstrates several polar-to-rectangular conversions that enable you to sketch the graphs of some polar equations. Example 5 Converting Polar Equations to Rectangular Form Describe the graph of each polar equation and find the corresponding rectangular equation. a. r 2 b. 3 c. r sec Solution a. The graph of the polar equation r 2 consists of all points that are two units from the pole. In other words, this graph is a circle centered at the origin with a radius of 2, as shown in Figure 10.66. You can confirm this by converting to rectangular form, using the relationship r 2 x 2 y 2. r 2 r 2 22 x 2 y 2 22 Polar equation Rectangular equation b. The graph of the polar equation 3 consists of all points on the line that with the positive polar axis, as shown in Figure 10.67. 3 makes an angle of To convert to rectangular form, make use of the relationship tan yx. 3 Polar equation tan 3 y 3x Rectangular equation c. The graph of the polar equation r sec is not evident by simple inspection, so convert to rectangular form by using the relationship r sec Polar equation r cos 1 r cos x. x 1 Rectangular equation Now you see that the graph is a vertical line, as shown in Figure 10.68. π FIGURE 10.66 π FIGURE 10.67 FIGURE 10.68 Now try Exercise 65. 333202_1007.qxd 12/8/05 11:09 AM Page 783 Section 10.7 Polar Coordinates 783 10.7 Exercises VOCABULARY CHECK: Fill in the blanks. 1. The origin of the polar coordinate system is called the ________. 2. For the point r, , r is the ________ ________ from and is the ________ ________ counterclockwise from the polar axis to the line segment PO to OP. 3. To plot the point r, , 4. The polar coordinates use the ________ coordinate system. r, are related to the rectangular coordinates x, y as follows: x ________ y ________ tan ________ r2 ________ PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. In Exercises 1–8, plot the point given in polar coordinates and find two additional polar representations of the point, using 2 < < 2. 4, 3 0, 7 6 2, 2.36 22, 4.71 1. 3. 5. 7. 1, 3 4 5 16, 2 3, 1.57 5, 2.36 2. 4. 6. 8. In Exercises 9–16, a point in polar coordinates is given. Convert the point to rectangular coordinates. 9. 3, 2 π 2 10. 3, ( )3 π 2 11. 1, 5 4 π 2 12. 0, π 2 0 2 4 ( (r, ) = −1, θ )5 π 4 0 2 4 (r, ) = (0, − ) π θ 2, 3 4 2.5, 1.1 13. 15. 2, 7 6 8.25, 3.5 14. 16. 17. 19. In Exercises 17–26, a point in rectangular coordinates is given. Convert the point to polar coordinates. 3, 3 0, 5 3, 1 3, 1 5, 12 1, 1 6, 0 3, 4 3, 3 6, 9 22. 23. 18. 26. 25. 21. 24. 20. In Exercises 27–32, use a graphing utility to find one set of polar coordinates for the point given in rectangular coordinates. 3, 2 3, 2 5 2, 4 5, 2 3, 2, 32 7 4, 3 28. 32. 31. 27. 30. 29. 3 2 In Exercises 33– 48, convert the rectangular equation to polar form. Assume a > 0. 33. 35. 37. 39. 41. 43. 45. 47. x2 y2 9 y 4 x 10 3x y 2 0 xy 16 y2 8x 16 0 x2 y2 a2 x2 y2 2ax 0 40. 36. 38. 34. x2 y2 16 y x x 4a 3x 5y 2 0 2xy 1 x2 y22 9x2 y2 x2 y2 9a2 46. 48. x2 y2 2ay 0 42. 44. 333202_1007.qxd 12/8/05 9:06 AM Page 784 784 Chapter 10 Topics in Analytic Geometry In Exercises 49–64,convert the polar equation to rectangular form. 49. 51. 53. 55. 57. 59. r 4 sin 2 3 r 4 r 4 csc r2 cos r 2 sin 3 61. r 63. r 2 1 sin 6 2 3 sin 50. 52. 54. 56. 58. 60. r 2 cos 5 3 r 10 r 3 sec r 2 sin 2 r 3 cos 2 62. r 64. r 1 1 cos 6 2 cos 3 sin In Exercises 65–70, describe the graph of the polar equation and find the corresponding rectangular equation. Sketch its graph. r 6 65. 66. r 8 3 4 r 2 csc 68. 70. 67. 6 r 3 sec 69. Synthesis In Exercises 71 and 72, determine whether True or False? the statement is true or false. Justify your answer. r, 1 for some integer represent the same point on the polar coordinate system. 2n r, 2 2 71. If then and 1 n, 72. If r1 r2, r2, then point on the polar coordinate system. r1, and represent the same 73. Convert the polar equation to rectangular form and verify that it is the equation of a circle. Find the radius of the circle and the rectangular coordinates of the center of the circle. r 2h cos k sin 74. Convert the polar equation r cos 3 sin to rectangular form and identify the graph. 75. Think About It r1, 1 and is 2 r1 2 r2 r2, (a) Show that the distance between the points . 2 2 2r1r2 cos (b) Describe the positions of the points relative to each other for Simplify the Distance Formula for this case. Is the simplification what you expected? Explain. 2. 1 1 (c) Simplify the Distance Formula for 1 2 90. Is the simplification what you expected? Explain. (d) Choose two points on the polar coord
inate system and find the distance between them. Then choose different polar representations of the same two points and apply the Distance Formula again. Discuss the result. 76. Exploration (a) Set the window format of your graphing utility on rectangular coordinates and locate the cursor at any position off the coordinate axes. Move the cursor horizontally and observe any changes in the displayed coordinates of the points. Explain the changes in the coordinates. Now repeat the process moving the cursor vertically. (b) Set the window format of your graphing utility on polar coordinates and locate the cursor at any position off the coordinate axes. Move the cursor horizontally and observe any changes in the displayed coordinates of the points. Explain the changes in the coordinates. Now repeat the process moving the cursor vertically. (c) Explain why the results of parts (a) and (b) are not the same. Skills Review In Exercises 77–80, use the properties of logarithms to expand the expression as a sum, difference, and/or constant multiple of logarithms. (Assume all variables are positive.) log6 x2z 3y ln xx 42 77. 79. log4 2x y ln 5x2x2 1 78. 80. In Exercises 81–84, condense the expression to the logarithm of a single quantity. log7 x log7 3y 1 2 ln 6 ln y lnx 3 ln x lnx 2 log5 a 8 log5 x 1 81. 84. 83. 82. In Exercises 85–90, use Cramer’s Rule to solve the system of equations. 85. 87. 89. 7y y 5x 3x 3a x 2a a 2x 5x 2b b 3b y 3y 4y 11 3 c 0 3c 0 9c 8 2z z 2z 1 2 4 86. 88. 90. 10 5 9w 3w w 2x3 3x 5y 4x 2y 5u 7v 2v 2v 2x1 x2 2x2 x2 u 8u 2x1 2x1 6x3 15 7 0 4 5 2 91. In Exercises 91–94, use a determinant to determine whether the points are collinear. 4, 3, 6, 7, 2, 1 2, 4, 0, 1, 4, 5 6, 4, 1, 3, 1.5, 2.5 92. 93. 94. 2.3, 5, 0.5, 0, 1.5, 3 333202_1008.qxd 12/8/05 9:08 AM Page 785 10.8 Graphs of Polar Equations Section 10.8 Graphs of Polar Equations 785 What you should learn • Graph polar equations by point plotting. • Use symmetry to sketch graphs of polar equations. • Use zeros and maximum r-values to sketch graphs of polar equations. • Recognize special polar graphs. Why you should learn it Equations of several common figures are simpler in polar form than in rectangular form. For instance, Exercise 6 on page 791 shows the graph of a circle and its polar equation. Introduction In previous chapters, you spent a lot of time learning how to sketch graphs on rectangular coordinate systems. You began with the basic point-plotting method, which was then enhanced by sketching aids such as symmetry, intercepts, asymptotes, periods, and shifts. This section approaches curve sketching on the polar coordinate system similarly, beginning with a demonstration of point plotting. Example 1 Graphing a Polar Equation by Point Plotting Sketch the graph of the polar equation r 4 sin . Solution The sine function is periodic, so you can get a full range of -values by consid0 ≤ ≤ 2, ering values of r as shown in the following table. in the interval r 0 0 6 2 3 23 11 6 2 23 2 0 2 4 2 0 If you plot these points as shown in Figure 10.69, it appears that the graph is a circle of radius 2 whose center is at the point x, y 0, 2. π 2 Circle: r = 4 sin FIGURE 10.69 Now try Exercise 21. You can confirm the graph in Figure 10.69 by converting the polar equation to rectangular form and then sketching the graph of the rectangular equation. You can also use a graphing utility set to polar mode and graph the polar equation or set the graphing utility to parametric mode and graph a parametric representation. 333202_1008.qxd 12/8/05 9:08 AM Page 786 786 Chapter 10 Topics in Analytic Geometry Symmetry the graph is traced out twice. In Figure 10.69, note that as Moreover, note that the graph is symmetric with respect to the line Had you known about this symmetry and retracing ahead of time, you could have used fewer points. increases from 0 to 2. Symmetry with respect to the line 2 is one of three important types 2 of symmetry to consider in polar curve sketching. (See Figure 10.70.) π 2 − θ (−r, − ) θ θ π )− (r, π π θ (r, ) θ 0 π π 3 2 Symmetry with Respect to the Line 2 FIGURE 10.70 π 2 π 3 2 θ (rr, ) θ 0 θ (r, − ) π )− θ (−r, θπ(r, )+ (−r, θ ) π 3 2 Symmetry with Respect to the Polar Axis Symmetry with Respect to the Pole This is Note in Example 2 that cos cos . because the cosine function is even. Recall from Section 4.2 that the cosine function is even and the sine function is odd. That is, sin sin . Tests for Symmetry in Polar Coordinates The graph of a polar equation is symmetric with respect to the following if the given substitution yields an equivalent equation. 1. The line 2: 2. The polar axis: 3. The pole: Replace Replace Replace r, r, r, by by by r, r, or r, r, . or r, . or r, . Example 2 Using Symmetry to Sketch a Polar Graph π 2 r = 3 + 2 cos θ Use symmetry to sketch the graph of r 3 2 cos . r, Solution Replacing So, you can conclude that the curve is symmetric with respect to the polar axis. Plotting the points in the table and using polar axis symmetry, you obtain the graph shown in Figure 10.71. This graph is called a limaçon. r 3 2 cos 3 2 cos . produces r, by FIGURE 10.71 Now try Exercise 27. 333202_1008.qxd 12/8/05 9:08 AM Page 787 Spiral of Archimedes Section 10.8 Graphs of Polar Equations 787 π 4 The three tests for symmetry in polar coordinates listed on page 786 are sufficient to guarantee symmetry, but they are not necessary. For instance, Figure 10.72 shows the graph of to be symmetric with respect to the line 2, and yet the tests on page 786 fail to indicate symmetry because neither of the following replacements yields an equivalent equation Original Equation r 2 r 2 Replacement r, r, by by r, r, New Equation r 2 r 3 The equations discussed in Examples 1 and 2 are of the form r 4 sin f sin r 3 2 cos gcos . and and The graph of the first equation is symmetric with respect to the line the graph of the second equation is symmetric with respect to the polar axis. This observation can be generalized to yield the following tests. 2, FIGURE 10.72 Quick Tests for Symmetry in Polar Coordinates 1. The graph of r f sin 2. The graph of r gcos is symmetric with respect to the line 2 is symmetric with respect to the polar axis. . Zeros and Maximum r-Values r Two additional aids to graphing of polar equations involve knowing the -values For is maximum and knowing the for which instance, in Example 1, the maximum value of and 2, this occurs when when 0. -values for which r r 4 sin is for as shown in Figure 10.69. Moreover, r 0. r 4, r 0 Example 3 Sketching a Polar Graph Sketch the graph of r 1 2 cos . Solution From the equation r 1 2 cos , you can obtain the following Symmetry: Maximum value of r : Zero of : r when With respect to the polar axis r 3 r 0 3 -values in the interval when 11 6 π5 3 π 3 2 θ π 4 3 Limaçon: r = 1 − 2 cos The table shows several corresponding points, you can sketch the graph shown in Figure 10.73. By plotting the 0, . 0 6 1 0.73 .73 3 Note how the negative -values determine the inner loop of the graph in Figure 10.73. This graph, like the one in Figure 10.71, is a limaçon. r FIGURE 10.73 Now try Exercise 29. 333202_1008.qxd 12/8/05 9:08 AM Page 788 788 Chapter 10 Topics in Analytic Geometry Some curves reach their zeros and maximum -values at more than one r point, as shown in Example 4. Example 4 Sketching a Polar Graph Sketch the graph of r 2 cos 3. Solution Symmetry: Maximum value of r : Zeros of : r 3 0, , 2, 3 or With respect to the polar axis r 2 when 0, 3, 23, r 0 when 6, 2, 56 3 2, 32, 52 r 0 2 12 2 6 0 4 3 5 12 2 2 2 or 2 0 By plotting these points and using the specified symmetry, zeros, and maximum values, you can obtain the graph shown in Figure 10.74. This graph is called a rose curve, and each of the loops on the graph is called a petal of the rose curve. Note how the entire curve is generated as increases from 0 to . Exploration Notice that the rose curve in Example 4 has three petals. How many petals do the rose r 2 cos 4 curves given by r 2 sin 3 and have? Determine the numbers of petals for the curves given by r 2 sin n, r 2 cos n and n where is a positive integer. Te Use a graphing utility in polar mode to verify the graph of r 2 cos 3 shown in Figure 10.74. 0 ≤ ≤ 2 3 FIGURE 10.74 ≤ ≤ Now try Exercise 33 333202_1008.qxd 12/8/05 9:08 AM Page 789 Section 10.8 Graphs of Polar Equations 789 Special Polar Graphs Several important types of graphs have equations that are simpler in polar form than in rectangular form. For example, the circle r 4 sin in Example 1 has the more complicated rectangular equation x 2 y 2 2 4. Several other types of graphs that have simple polar equations are shown below. Limaçons r a ± b cos r a ± b sin a > 0, b > 0 Rose Curves n n petals if 2n petals if n ≥ 2 is odd, n is even Limaçon with inner loop Cardioid (heart-shaped Dimpled limaçon Convex limaçon cos n Rose curve r a cos n Rose curve r a sin n Rose curve r a sin n Rose curve Circles and Lemniscates cos Circle r a sin Circle r 2 a 2 sin 2 Lemniscate r 2 a 2 cos 2 Lemniscate 333202_1008.qxd 12/8/05 9:08 AM Page 790 790 Chapter 10 Topics in Analytic Geometry π 2 ( −3 Example 5 Sketching a Rose Curve Sketch the graph of r 3 cos 2. π (3, ) π (3, 0) 0 3 1 2 Solution Type of curve: Symmetry: r = 3 cos 2 θ π 3 2 ( −3, π 2 ) FIGURE 10.75 π 2 ( 33, π 4 ) r2 = 9 sin 2 θ π 3 2 FIGURE 10.76 Rose curve with 2n 4 petals With respect to polar axis, the line and the pole r 3 r 0 0, 2, , 32 4, 34 when when 2, Maximum value of r : Zeros of r: Using this information together with the additional points shown in the following table, you obtain the graph shown in Figure 10.75 Now try Exercise 35. Example 6 Sketching a Lemniscate Sketch the graph of r 2 9 sin 2. Solution Type of curve: Symmetry: Maximum value of r : Lemniscate With respect to the pole r 3 when 4 Zeros of : r r 0 when 0, 2 this equation has no solution points. So, you restrict the values of If sin 2 < 0, to those for which sin 2 ≥ 0. 0 ≤ ≤ 2 or ≤ ≤ 3 2 Moreover, using symmetry, y
ou need to consider only the first of these two intervals. By finding a few additional points (see table below), you can obtain the graph shown in Figure 10.76. r ±3sin 2 0 0 12 ±3 2 4 ±3 5 12 ±3 2 2 0 Now try Exercise 39. 333202_1008.qxd 12/8/05 9:08 AM Page 791 10.8 Exercises Section 10.8 Graphs of Polar Equations 791 2. The graph of VOCABULARY CHECK: Fill in the blanks. r f sin 1. The graph of r gcos r 2 cos r 2 cos r 2 4 sin 2 r 1 sin 3. The equation 4. The equation 5. The equation 6. The equation represents a ________. represents a ________. represents a ________. is symmetric with respect to the line ________. is symmetric with respect to the ________ ________. represents a ________ ________. PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. In Exercises 1– 6, identify the type of polar graph. 1. π 2 r = 3 cos 2θ 2 sin θ 3. r = 3(1 − θ 2 cos ) π 2 4. π 2 r = 16 cos 2θ 2 0 2 0 5 5. π 2 r = 6 sin 2θ 6. π 2 r = 3 cosθ 0 21 65 21 4 5 0 In Exercises 7–12, test for symmetry with respect to /2, 7. the polar axis, and the pole. r 16 cos 3 8. r r 5 4 cos 2 1 sin r 2 16 cos 2 9. 11. r 3 2 cos r 2 36 sin 2 10. 12. In Exercises 13–16, find the maximum value of zeros of r. r and any 13. 15. r 101 sin r 4 cos 3 14. 16. r 6 12 cos r 3 sin 2 In Exercises 17–40, sketch the graph of the polar equation using symmetry, zeros, maximum -values, and any other additional points. r 17. 19. 21. 23. 25. 27. 29. 31. 33. 35. 37. r 5 r 6 r 3 sin r 31 cos r 41 sin r 3 6 sin r 1 2 sin r 3 4 cos r 5 sin 2 r 2 sec 3 sin 2 cos r 18. 20. 22. 24. 26. 28. 30. 32. 34. 36. r 2 r 3 4 r 4 cos r 41 sin r 21 cos r 4 3 sin r 1 2 cos r 4 3 cos r 3 cos 2 r 5 csc 38. r 6 2 sin 3 cos 39. r 2 9 cos 2 40. r 2 4 sin In Exercises 41– 46, use a graphing utility to graph the polar equation. Describe your viewing window. 41. 43. 45. r 8 cos r 32 sin r 8 sin cos 2 42. 44. 46. r cos 2 r 2 cos3 2 r 2 csc 5 In Exercises 47–52, use a graphing utility to graph the polar equation. Find an interval for for which the graph is traced only once. 47. r 3 4 cos 48. r 5 4 cos 333202_1008.qxd 12/8/05 9:08 AM Page 792 792 Chapter 10 Topics in Analytic Geometry 49. r 2 cos3 2 51. r 2 9 sin 2 50. 52. r 3 sin5 2 r 2 1 In Exercises 53–56, use a graphing utility to graph the polar equation and show that the indicated line is an asymptote of the graph. Name of Graph 53. Conchoid 54. Conchoid 55. Hyperbolic spiral 56. Strophoid Synthesis Polar Equation r 2 sec r 2 csc r 3 r 2 cos 2 sec Asymptote x 1 y 1 y 3 x 2 True or False? In Exercises 57 and 58, determine whether the statement is true or false. Justify your answer. 57. In the polar coordinate system, if a graph that has symmetry with respect to the polar axis were folded on the line 0, the portion of the graph above the polar axis would coincide with the portion of the graph below the polar axis. 58. In the polar coordinate system, if a graph that has symmetry with respect to the pole were folded on the line 34, the portion of the graph on one side of the fold would coincide with the portion of the graph on the other side of the fold. 59. Exploration Sketch the graph of over each interval. Describe the part of the graph obtained in each case. r 6 cos (a) 0 ≤ ≤ (c) 2 ≤ ≤ 2 (b) (d 60. Graphical Reasoning Use a graphing utility to graph the 0, for (a) polar equation 4, Use the graphs to describe (b) the effect of the angle Write the equation as a function of r 61 cos 2. . for part (c). and (c) r f is rotated about the pole through an Show that the equation of the rotated graph is sin 61. The graph of . angle r f . 62. Consider the graph of r f sin . (a) Show that if the graph is rotated counterclockwise 2 radians about the pole, the equation of the rotated graph is r f cos . (b) Show that if the graph is rotated counterclockwise radians about the pole, the equation of the rotated graph is r f sin . (c) Show that if the graph is rotated counterclockwise radians about the pole, the equation of the rotated 32 graph is r f cos . In Exercises 63– 66, use the results of Exercises 61 and 62. 63. Write an equation for the limaçon r 2 sin after it has been rotated through the given angle. 3 2 r 2 sin 2 64. Write an equation for the rose curve (b) (d) (c) (a) 2 4 after it has been rotated through the given angle. (a) 6 (b) 2 (c) 2 3 (d) 65. Sketch the graph of each equation. (a) r 1 sin r 1 sin (b) 4 66. Sketch the graph of each equation. (a) r 3 sec r 3 sec (c) 3 r 3 sec (b) r 3 sec (d) 4 2 67. Exploration Use a graphing utility to graph and identify r 2 k sin for k 0, 1, 2, and 3. 68. Exploration Consider the equation r 3 sin k. (a) Use a graphing utility to graph the equation for Find the interval for over which the graph is k 1.5. traced only once. (b) Use a graphing utility to graph the equation for Find the interval for over which the graph is k 2.5. traced only once. (c) Is it possible to find an interval for over which the k? graph is traced only once for any rational number Explain. Skills Review In Exercises 69–72, find the zeros (if any) of the rational function. 69. 71. f x x2 9 x 1 f x 5 3 x 2 70. 72. f x 6 4 x2 4 f x x 3 27 x2 4 In Exercises 73 and 74, find the standard form of the equation of the ellipse with the given characteristics. Then sketch the ellipse. 73. Vertices: 4, 2, 2, 2; minor axis of length 4 74. Foci: 3, 2, 3, 4; major axis of length 8 333202_1009.qxd 12/8/05 9:09 AM Page 793 10.9 Polar Equations of Conics Section 10.9 Polar Equations of Conics 793 What you should learn • Define conics in terms of eccentricity. • Write and graph equations of conics in polar form. • Use equations of conics in polar form to model real-life problems. Why you should learn it The orbits of planets and satellites can be modeled with polar equations. For instance, in Exercise 58 on page 798, a polar equation is used to model the orbit of a satellite. Alternative Definition of Conic In Sections 10.3 and 10.4, you learned that the rectangular equations of ellipses and hyperbolas take simple forms when the origin lies at their centers. As it happens, there are many important applications of conics in which it is more convenient to use one of the foci as the origin. In this section, you will learn that polar equations of conics take simple forms if one of the foci lies at the pole. To begin, consider the following alternative definition of conic that uses the concept of eccentricity. Alternative Definition of Conic The locus of a point in the plane that moves so that its distance from a fixed point (focus) is in a constant ratio to its distance from a fixed line (directrix) is a conic. The constant ratio is the eccentricity of the conic and is denoted by Moreover, the conic is an ellipse if and a hyperbola if (See Figure 10.77.) a parabola if e 1, e < 1, e > 1. e. In Figure 10.77, note that for each type of conic, the focus is at the pole. π 2 π 2 Directrix π 2 Directrix Q P P Q 0 F = (0, 0) 0 F = (0, 0) 0 < e < 1 Ellipse: PF PQ FIGURE 10.77 < 1 e 1 Parabola: PF PQ 1 Directrix Q P 0 F = (0, 0) P′ Q′ e > 1 Hyperbola PF PF PQ PQ > 1 Digital Image © 1996 Corbis; Original image courtesy of NASA/Corbis Polar Equations of Conics The benefit of locating a focus of a conic at the pole is that the equation of the conic takes on a simpler form. For a proof of the polar equations of conics, see Proofs in Mathematics on page 808. Polar Equations of Conics The graph of a polar equation of the form 1. r ep 1 ± e cos or 2. r ep 1 ± e sin p is a conic, where the focus (pole) and the directrix. e > 0 is the eccentricity and is the distance between 333202_1009.qxd 12/8/05 9:09 AM Page 794 794 Chapter 10 Topics in Analytic Geometry Equations of the form r ep 1 ± e cos gcos Vertical directrix correspond to conics with a vertical directrix and symmetry with respect to the polar axis. Equations of the form r ep 1 ± e sin gsin Horizontal directrix 2. correspond to conics with a horizontal directrix and symmetry with respect to the line Moreover, the converse is also true—that is, any conic with a focus at the pole and having a horizontal or vertical directrix can be represented by one of the given equations. Example 1 Identifying a Conic from Its Equation Identify the type of conic represented by the equation r Algebraic Solution To identify the type of conic, rewrite the equation in the form r ep1 ± e cos . r 15 3 2 cos 5 1 23 cos e 2 3 < 1, Because is an ellipse. Write original equation. Divide numerator and denominator by 3. you can conclude that the graph 15 3 2 cos . Graphical Solution 0 You can start sketching the graph by plotting points from . the Because the equation is of the form to r graph of is symmetric with respect to the polar axis. So, you can complete the sketch, as shown in Figure 10.78. From this, you can conclude that the graph is an ellipse. r gcos , π 2 r = 15 2 cos θ − 3 π(3, ) (15, 0) 0 3 6 9 12 18 21 Now try Exercise 11. FIGURE 10.78 For the ellipse in Figure 10.78, the major axis is horizontal and the vertices 15, 0 lie at To find the b2 a 2 c 2 length of the to conclude that major axis, you can use the equations 2a 18. and So, the length of the axis is e ca and minor 3, . b2 a 2 c 2 a2 ea2 a21 e 2. e 2 3, Because b 45 35. analysis for hyperbolas yields Ellipse b2 921 2 2 45, you have So, the length of the minor axis is 3 which 2b 65. implies that A similar b2 c 2 a 2 ea2 a2 a2e 2 1. Hyperbola 333202_1009.qxd 12/8/05 9:09 AM Page 795 Section 10.9 Polar Equations of Conics 795 Example 2 Sketching a Conic from Its Polar Equation Identify the conic r 32 3 5 sin and sketch its graph. Solution Dividing the numerator and denominator by 3, you have r 323 1 53 sin e 5 . 3 > 1, Because lies on the line and Because the length of the transverse axis is 12, you can see that b, the graph is a hyperbola. The transverse axis of the hyperbola 16, 32. a 6. To find and the vertices occur at 2, 4, 2 write 4 8 0 b 2 a 2e 2 1 625 3 2 1 64. b 8. So, hyperbola are Finally, you can use and a b to d
etermine that the asymptotes of the y 10 ± 3 4 x. The graph is shown in Figure 10.79. π 2 ( −16, )3 π 2 π ( ) 4, 2 r = 32 3 + 5 sin θ FIGURE 10.79 Te c h n o l o g y Use a graphing utility set in polar mode to verify the four orientations shown at the right. Remember that e must be positive, but p can be positive or negative. π 2 Directrix: y = 3 (0, 0 + sin θ FIGURE 10.80 Now try Exercise 19. In the next example, you are asked to find a polar equation of a specified conic. To do this, let be the distance between the pole and the directrix. p 1. Horizontal directrix above the pole: 2. Horizontal directrix below the pole: 3. Vertical directrix to the right of the pole: 4. Vertical directrix to the left of the pole: r r r r ep 1 e sin ep 1 e sin ep 1 e cos ep 1 e cos Example 3 Finding the Polar Equation of a Conic Find the polar equation of the parabola whose focus is the pole and whose directrix is the line y 3. Solution From Figure 10.80, you can see that the directrix is horizontal and above the pole, so you can choose an equation of the form r ep 1 e sin . Moreover, because the eccentricity of a parabola is between the pole and the directrix is p 3, you have the equation e 1 and the distance r 3 1 sin . Now try Exercise 33. 333202_1009.qxd 12/8/05 9:09 AM Page 796 796 Chapter 10 Topics in Analytic Geometry Applications Kepler’s Laws (listed below), named after the German astronomer Johannes Kepler (1571–1630), can be used to describe the orbits of the planets about the sun. 1. Each planet moves in an elliptical orbit with the sun at one focus. 2. A ray from the sun to the planet sweeps out equal areas of the ellipse in equal times. 3. The square of the period (the time it takes for a planet to orbit the sun) is proportional to the cube of the mean distance between the planet and the sun. Although Kepler simply stated these laws on the basis of observation, they were later validated by Isaac Newton (1642–1727). In fact, Newton was able to show that each law can be deduced from a set of universal laws of motion and gravitation that govern the movement of all heavenly bodies, including comets and satellites. This is illustrated in the next example, which involves the comet named after the English mathematician and physicist Edmund Halley (1656–1742). astronomical If you use Earth as a reference with a period of 1 year and a distance of 1 astronomical unit (an is defined as the mean distance between Earth and the sun, or about 93 million miles), the proportionality constant in Kepler’s third law is 1. For example, because Mars has a mean distance to the sun of So, the period of Mars is astronomical units, its period d 1.524 P 1.88 is given by d 3 P2. years. unit P π 2 Sun π 0 Earth Halley’s comet Example 4 Halley’s Comet Halley’s comet has an elliptical orbit with an eccentricity of The length of the major axis of the orbit is approximately 35.88 astronomical units. Find a polar equation for the orbit. How close does Halley’s comet come to the sun? e 0.967. Solution Using a vertical axis, as shown in Figure 10.81, choose an equation of the form r ep1 e sin . 2 Because the vertices of the ellipse occur when 32, and you can determine the length of the major axis to be the sum of r the -values of the vertices. That is, 2a 0.967p 1 0.967 0.967p 1 0.967 29.79p 35.88. ep 0.9671.204 1.164. p 1.204 So, and equation, you have 1.164 1 0.967 sin r Using this value of ep in the r where (the focus), substitute is measured in astronomical units. To find the closest point to the sun in this equation to obtain 2 π 3 2 r 1.164 1 0.967 sin2 0.59 astronomical unit 55,000,000 miles. FIGURE 10.81 Now try Exercise 57. 333202_1009.qxd 12/8/05 9:09 AM Page 797 Section 10.9 Polar Equations of Conics 797 10.9 Exercises VOCABULARY CHECK: In Exercises 1–3, fill in the blanks. 1. The locus of a point in the plane that moves so that its distance from a fixed point (focus) is in a constant ratio to its distance from a fixed line (directrix) is a ________. 2. The constant ratio is the ________ of the conic and is denoted by ________. 3. An equation of the form r ep 1 e cos 4. Match the conic with its eccentricity. has a ________ directrix to the ________ of the pole. (a) e < 1 (i) parabola (b) e 1 (ii) hyperbola (c) e > 1 (iii) ellipse PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. In Exercises 1–4, write the polar equation of the conic for e 1, e 1.5. Identify the conic for each equation. Verify your answers with a graphing utility. e 0.5, and 1. r 3. r 4e 1 e cos 4e 1 e sin 2. r 4. r 4e 1 e cos 4e 1 e sin In Exercises 5–10, match the polar equation with its graph. [The graphs are labeled (a), (b), (c), (d), (e), and (f).] (a) (c) (eb) (d) π 2 π 2 (f. r 7. r 9. r 2 1 cos 3 1 2 sin 4 2 cos 6. r 8. r 10. r 3 2 cos 2 1 sin 4 1 3 sin In Exercises 11–24, identify the conic and sketch its graph. 11. r 13. r 15. r 17. r 19. r 21. r 23. r 2 1 cos 5 1 sin 2 2 cos 6 2 sin 3 2 4 sin 3 2 6 cos 4 2 cos 12. r 14. r 16. r 18. r 20. r 3 1 sin 6 1 cos 3 3 sin 9 3 2 cos 5 1 2 cos 22. r 24. r 3 2 6 sin 2 2 3 sin In Exercises 25–28, use a graphing utility to graph the polar equation. Identify the graph. 25. r 1 1 sin 27. r 3 4 2 cos 26. r 28. r 5 2 4 sin 4 1 2 cos 333202_1009.qxd 12/8/05 9:09 AM Page 798 798 Chapter 10 Topics in Analytic Geometry In Exercises 29–32, use a graphing utility to graph the rotated conic. 29. r 30. r 31. r 2 1 cos 4 3 3 sin 3 6 2 sin 6 (See Exercise 11.) (See Exercise 16.) (See Exercise 17.) 32. r 5 1 2 cos 23 (See Exercise 20.) In Exercises 33–48, find a polar equation of the conic with its focus at the pole. Directrix Conic Eccentricity 33. Parabola 34. Parabola 35. Ellipse 36. Ellipse 37. Hyperbola 38. Hyperbola Conic 39. Parabola 40. Parabola 41. Parabola 42. Parabola 43. Ellipse 44. Ellipse 45. Ellipse 46. Hyperbola 47. Hyperbola 48. Hyperbola Vertex or Vertices 1, 2 6, 0 5, 10, 2 2, 0, 10, 2, 2, 4, 32 20, 0, 4, 2, 0, 8, 0 1, 32, 9, 32 4, 2, 1, 2 49. Planetary Motion The planets travel in elliptical orbits with the sun at one focus. Assume that the focus is at the pole, the major axis lies on the polar axis, and the length 2a (see figure). Show that the polar of the major axis is r a1 e21 e cos equation of the orbit is where e is the eccentricity. π 2 Planet r θ Sun a 50. Planetary Motion Use the result of Exercise 49 to show ) from the and the maximum dis- that the minimum distance ( sun to the planet is aphelion tance ( r a1 e ) is r a1 e. perihelion distance distance Planetary Motion In Exercises 51–56, use the results of Exercises 49 and 50 to find the polar equation of the planet’s orbit and the perihelion and aphelion distances. 51. Earth 52. Saturn 53. Venus 54. Mercury 55. Mars 56. Jupiter e 0.0167 a 95.956 106 miles, a 1.427 109 kilometers, a 108.209 106 kilometers, a 35.98 106 miles, a 141.63 106 miles, a 778.41 106 kilometers, e 0.2056 e 0.0934 e 0.0542 e 0.0068 e 0.0484 57. Astronomy The comet Encke has an elliptical orbit with The length of the major axis an eccentricity of of the orbit is approximately 4.42 astronomical units. Find a polar equation for the orbit. How close does the comet come to the sun? e 0.847. Model It 58. Satellite Tracking A satellite in a 100-mile-high circular orbit around Earth has a velocity of approximately 17,500 miles per hour. If this velocity is multiplied by the satellite will have the minimum velocity necessary to escape Earth’s gravity and it will follow a parabolic path with the center of Earth as the focus (see figure). 2, π 2 Circular orbit 4100 miles Parabolic path 0 Not drawn to scale (a) Find a polar equation of the parabolic path of the satellite (assume the radius of Earth is 4000 miles). (b) Use a graphing utility to graph the equation you found in part (a). 0 and the satellite when 30. (c) Find the distance between the surface of the Earth (d) Find the distance between the surface of Earth and the satellite when 60. 333202_1009.qxd 12/8/05 9:09 AM Page 799 Synthesis True or False? statement is true or false. Justify your answer. In Exercises 59–61, determine whether the 59. For a given value of 2, the graph of e > 1 over the interval 0 to r ex 1 e cos is the same as the graph of r ex 1 e cos . 60. The graph of r 4 3 3 sin has a horizontal directrix above the pole. 61. The conic represented by the following equation is an ellipse. r 2 16 9 4 cos 4 62. Writing In your own words, define the term eccentricity and explain how it can be used to classify conics. 63. Show that the polar equation of the ellipse x 2 a 2 y 2 b2 1 is r 2 b2 1 e 2 cos2 . 64. Show that the polar equation of the hyperbola x 2 a 2 y 2 b2 1 is r 2 b2 1 e 2 cos2 . In Exercises 65–70, use the results of Exercises 63 and 64 to write the polar form of the equation of the conic. 65. 67. x 2 169 x 2 9 y 2 144 1 y 2 16 1 69. Hyperbola One focus: 70. Ellipse Vertices: One focus: Vertices: 1 66. 68. 1 y 2 16 y 2 4 x 2 25 x 2 36 5, 2 4, 2, 4, 2 4, 0 5, 0, 5, 71. Exploration Consider the polar equation r 4 1 0.4 cos . Section 10.9 Polar Equations of Conics 799 (a) Identify the conic without graphing the equation. (b) Without graphing the following polar equations, describe how each differs from the given polar equation. r1 4 1 0.4 cos , r2 4 1 0.4 sin (c) Use a graphing utility to verify your results in part (b). 72. Exploration The equation r ep 1 ± e sin is the equation of an ellipse with What happens to the lengths of both the major axis and the minor axis when changes? the value of Use an example to explain your reasoning. remains fixed and the value of e < 1. p e Skills Review In Exercises 73–78, solve the trigonometric equation. 73. 75. 43 tan 3 1 12 sin2 9 77. 2 cot x 5 cos 2 74. 76. 6 cos x 2 1 9 csc2 x 10 2 78. 2 sec 2 csc 4 In Exercises 79–82, find the exact value of the trigonometric are in Quadrant IV and function given that sin u 3 5 cosu v cosu v u and cos v 1/2. sinu v sinu v and 79. 80. 82. 81. v In Exercises 83 and 84, find the exact value
s of cos 2u, using the double-angle formulas. tan 2u and sin 2u , 83. sin u 4 5 , 2 < u < 84. tan u 3, 3 2 < u < 2 In Exercises 85–88, find a formula for sequence. an for the arithmetic 85. 87. a1 a3 0, d 1 4 27, a8 72 86. 88. a1 a1 13, d 3 5, a4 9.5 In Exercises 89–92, evaluate the expression. Do not use a calculator. 89. 91. 12C9 10P3 18C16 90. 92. 29 P2 333202_100R.qxd 12/8/05 9:11 AM Page 800 800 Chapter 10 Topics in Analytic Geometry 10 Chapter Summary What did you learn? Section 10.1 Find the inclination of a line (p. 728). Find the angle between two lines (p. 729). Find the distance between a point and a line (p. 730). Section 10.2 Recognize a conic as the intersection of a plane and a double-napped cone (p. 735). Write equations of parabolas in standard form and graph parabolas (p. 736). Use the reflective property of parabolas to solve real-life problems (p. 738). Section 10.3 Write equations of ellipses in standard form and graph ellipses (p. 744). Use properties of ellipses to model and solve real-life problems (p. 748). Find the eccentricities of ellipses (p. 748). Section 10.4 Write equations of hyperbolas in standard form (p. 753). Find asymptotes of and graph hyperbolas (p. 755). Use properties of hyperbolas to solve real-life problems (p. 758). Classify conics from their general equations (p. 759). Section 10.5 Rotate the coordinate axes to eliminate the xy-term in equations of conics (p. 763). Use the discriminant to classify conics (p. 767). Section 10.6 Evaluate sets of parametric equations for given values of the parameter (p. 771). Sketch curves that are represented by sets of parametric equations (p. 772). and rewrite the equations as single rectangular equations (p. 773). Find sets of parametric equations for graphs (p. 774). Section 10.7 Plot points on the polar coordinate system (p. 779). Convert points from rectangular to polar form and vice versa (p. 780). Convert equations from rectangular to polar form and vice versa (p. 782). Section 10.8 Graph polar equations by point plotting (p. 785). Use symmetry (p. 786), zeros, and maximum r-values (p. 787) to sketch graphs of polar equations. Recognize special polar graphs (p. 789). Section 10.9 Define conics in terms of eccentricity and write and graph equations of conics in polar form (p. 793). Use equations of conics in polar form to model real-life problems (p. 796). Review Exercises 1–4 5–8 9, 10 11, 12 13–16 17–20 21–24 25, 26 27–30 31–34 35–38 39, 40 41–44 45–48 49–52 53, 54 55–60 61–64 65–68 69–76 77–88 89–98 89–98 99–102 103–110 111, 112 333202_100R.qxd 12/8/05 9:11 AM Page 801 10 Review Exercises 10.1 In Exercises 1–4, find the inclination degrees) of the line with the given characteristics. (in radians and 1. Passes through the points 2. Passes through the points 1, 2 3, 4 and 2, 5 2, 7 and 3. Equation: 4. Equation: y 2x 4 6x 7y 5 0 In Exercises 5–8, find the angle between the lines. 5. 4x y 2 5x y 1 62 −1 −1 (in radians and degrees) 5x 3y 3 2x 3y 1 y 3 2 1 −1 θ x 1 2 7. 2x 7y 8 0.4x y 0 8. 0.02x 0.07y 0.18 0.09x 0.04y 0.17 In Exercises 9 and 10, find the distance between the point and the line. Point 1, 2 0, 4 9. 10. Line x y 3 0 x 2y 2 0 10.2 In Exercises 11 and 12, state what type of conic is formed by the intersection of the plane and the double-napped cone. 11. 12. Review Exercises 801 In Exercises 17 and 18, find an equation of the tangent line to the parabola at the given point, and find the -intercept of the line. x 17. 18. x2 2y, x2 2y, 2, 2 4, 8 19. Architecture A parabolic archway is 12 meters high at the vertex. At a height of 10 meters, the width of the archway is 8 meters (see figure). How wide is the archway at ground level? y y − , 10) ( 4 (0, 12) (4, 10) x 1.5 cm x FIGURE FOR 19 FIGURE FOR 20 20. Flashlight The light bulb in a flashlight is at the focus of its parabolic reflector, 1.5 centimeters from the vertex of the reflector (see figure). Write an equation of a cross section of the flashlight’s reflector with its focus on the positive -axis and its vertex at the origin. x In Exercises 21–24, find the standard form of the 10.3 equation of the ellipse with the given characteristics. Then graph the ellipse. 21. Vertices: 22. Vertices: 23. Vertices: 2, 0, 2, 2 3, 0, 7, 0; 2, 0, 2, 4; 0, 1, 4, 1; foci: 0, 0, 4, 0 foci: 2, 1, 2, 3 endpoints of the minor axis: 24. Vertices: axis: 6, 5, 2, 5 4, 1, 4, 11; endpoints of the minor 25. Architecture A semielliptical archway is to be formed over the entrance to an estate. The arch is to be set on pillars that are 10 feet apart and is to have a height (atop the pillars) of 4 feet. Where should the foci be placed in order to sketch the arch? In Exercises 13–16, find the standard form of the equation of the parabola with the given characteristics. Then graph the parabola. 26. Wading Pool You are building a wading pool that is in the shape of an ellipse. Your plans give an equation for the elliptical shape of the pool measured in feet as 13. Vertex: Focus: 15. Vertex: 0, 0 4, 0 0, 2 14. Vertex: Focus: 16. Vertex: 2, 0 0, 0 2, 2 Directrix: x 3 Directrix: y 0 x2 324 y2 196 1. Find the longest distance across the pool, the shortest distance, and the distance between the foci. 333202_100R.qxd 12/8/05 9:11 AM Page 802 802 Chapter 10 Topics in Analytic Geometry In Exercises 27–30, find the center, vertices, foci, and eccentricity of the ellipse. x 22 81 x 52 1 y 12 100 y 32 36 1 1 16x2 9y2 32x 72y 16 0 4x2 25y2 16x 150y 141 0 27. 28. 29. 30. In Exercises 31–34, find the standard form of the 10.4 equation of the hyperbola with the given characteristics. 31. Vertices: 32. Vertices: 0, ±1; foci: 2, 2, 2, 2; 0, ±3 foci: 33. Foci: 34. Foci: 0, 0, 8, 0; 3, ±2; 4, 2, 4, 2 y ±2x 4 asymptotes: asymptotes: y ±2x 3 In Exercises 35 –38, find the center, vertices, foci, and the equations of the asymptotes of the hyperbola, and sketch its graph using the asymptotes as an aid. y 52 4 1 35. x 32 16 y 12 4 x2 1 9x2 16y2 18x 32y 151 0 4x2 25y2 8x 150y 121 0 36. 37. 38. 39. LORAN Radio transmitting station A located 200 miles east of transmitting station B. A ship is in an area to the north and 40 miles west of station A. Synchronized radio pulses transmitted at 186,000 miles per second by the two stations are received 0.0005 second sooner from station A than from station B. How far north is the ship? is 40. Locating an Explosion Two of your friends live 4 miles apart and on the same “east-west” street, and you live halfway between them. You are having a three-way phone conversation when you hear an explosion. Six seconds later, your friend to the east hears the explosion, and your friend to the west hears it 8 seconds after you do. Find equations of two hyperbolas that would locate the explosion. (Assume that the coordinate system is measured in feet and that sound travels at 1100 feet per second.) In Exercises 41–44, classify the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola. 41. 42. 43. 44. 5x2 2y2 10x 4y 17 0 4y2 5x 3y 7 0 3x 2 2y 2 12x 12y 29 0 4x 2 4y 2 4x 8y 11 0 10.5 In Exercises 45 –48, rotate the axes to eliminate the xy -term in the equation. Then write the equation in standard form. Sketch the graph of the resulting equation, showing both sets of axes. 45. 46. 47. 48. xy 4 0 x2 10xy y2 1 0 5x2 2xy 5y2 12 0 4x2 8xy 4y2 72 x 92 y 0 In Exercises 49–52, (a) use the discriminant to classify the graph, (b) use the Quadratic Formula to solve for and (c) use a graphing utility to graph the equation. y, 49. 50. 51. 52. 16x2 24xy 9y2 30x 40y 0 13x2 8xy 7y2 45 0 x2 y2 2xy 22 x 22 y 2 0 x2 10xy y2 1 0 In Exercises 53 and 54, complete the table for each and 10.6 set of parametric equations. Plot the points sketch a graph of the parametric equations. x, y 53. x 3t 2 and y 7 4t 54. x 1 5 t and In Exercises 55–60, (a) sketch the curve represented by the parametric equations (indicate the orientation of the curve) and (b) eliminate the parameter and write the corresponding rectangular equation whose graph represents the curve. Adjust the domain of the resulting rectangular equation, if necessary. (c) Verify your result with a graphing utility. 55. 57. 59. x 2t y 4t x t2 y t x 6 cos y 6 sin 56. 58. 60. x 1 4t y 2 3t x t 4 y t2 x 3 3 cos y 2 5 sin 333202_100R.qxd 12/8/05 9:11 AM Page 803 61. Find a parametric representation of the circle with center 5, 4 and radius 6. 62. Find a parametric representation of the ellipse with center major axis horizontal and eight units in length, 3, 4, and minor axis six units in length. 63. Find a parametric representation of the hyperbola with vertices 0, ±4 and foci 0, ±5. P 64. Involute of a Circle The involute of a circle is described of a string that is held taut as it is by the endpoint unwound from a spool (see figure). The spool does not rotate. Show that a parametric representation of the involute of a circle is x r cos sin y r sin cos . y P θr x 65. 2 < < 2. In Exercises 65–68, plot the point given in polar 10.7 coordinates and find two additional polar representations of the point, using 2, 4 5, 3 7, 4.19 3, 2.62 66. 67. 68. In Exercises 69–72, a point in polar coordinates is given. Convert the point to rectangular coordinates. 5 4 2, 0, 1, 3 3, 3 4 69. 70. 71. 72. 2 Review Exercises 803 In Exercises 77–82, convert the rectangular equation to polar form. 77. 79. 81. x 2 y 2 49 x2 y2 6y 0 xy 5 78. 80. 82. x 2 y 2 20 x 2 y 2 4x 0 xy 2 In Exercises 83–88, convert the polar equation to rectangular form. 83. 85. 87. r 5 r 3 cos r2 sin 84. 86. 88. r 12 r 8 sin r 2 cos 2 In Exercises 89–98, determine the symmetry of 10.8 the r. maximum value of , and any zeros of Then sketch the graph of the polar equation (plot additional points if necessary). r r, 89. 91. 93. 95. 97. r 4 r 4 sin 2 r 21 cos r 2 6 sin r 3 cos 2 90. 92. 94. 96. 98. r 11 r cos 5 r 3 4 cos r 5 5 cos r cos 2 In Exercises 99 –102, identify the type of polar graph and use a graphing utility to graph the equation. 99. 100. 101. 102. r 32 cos r 31 2 cos r 4 cos 3 r 2 9 cos 2 In Exerci
ses 103–106, identify the conic and sketch 10.9 its graph. 103. r 104. r 105. r 106. r 1 1 2 sin 2 1 sin 4 5 3 cos 16 4 5 cos In Exercises 73–76, a point in rectangular coordinates is given. Convert the point to polar coordinates. In Exercises 107–110, find a polar equation of the conic with its focus at the pole. 73. 74. 75. 76. 0, 2 5, 5 4, 6 3, 4 107. Parabola 108. Parabola 109. Ellipse 110. Hyperbola Vertex: Vertex: 2, 2, 2 5, 0, 1, Vertices: Vertices: 1, 0, 7, 0 333202_100R.qxd 12/8/05 9:12 AM Page 804 804 Chapter 10 Topics in Analytic Geometry 111. Explorer 18 On November 26, 1963, the United States launched Explorer 18. Its low and high points above the surface of Earth were 119 miles and 122,800 miles, respectively (see figure). The center of Earth was at one focus of the orbit. Find the polar equation of the orbit and find the distance between the surface of Earth (assume Earth has a radius of 4000 miles) and the satellite when 3. π 2 Explorer 18 r π 3 Earth 0 a 112. Asteroid An asteroid takes a parabolic path with Earth as its focus. It is about 6,000,000 miles from Earth at its closest approach. Write the polar equation of the path of the asteroid with its vertex at Find the distance between the asteroid and Earth when 3. 2. Synthesis True or False? the statement is true or false. Justify your answer. In Exercises 113–116, determine whether 113. When B 0 in an equation of the form Ax2 Bxy Cy2 Dx Ey F 0 the graph of the equation can be a parabola only if also. C 0 114. The graph of 1 4 x 2 y 4 1 is a hyperbola. 115. Only one set of parametric equations can represent the line y 3 2x. 116. There is a unique polar coordinate representation of each point in the plane. 117. Consider an ellipse with the major axis horizontal and 10 units in length. The number in the standard form of the equation of the ellipse must be less than what real number? Explain the change in the shape of the ellipse as b approaches this number. b 118. The graph of the parametric equations and is shown in the figure. How would the graph and x 2 sect equations the x 2 sec t y 3 tan t change for y 3 tant? x = 2 sec t y = 3 tan t x 4 y 4 2 −2 −4 FIGURE FOR 118 119. A moving object is modeled by the parametric equations is time (see figure). x 4 cos t y 3 sin t, How would the path change for the following? where and t (a) (b) x 4 cos 2t, x 5 cos t, y 3 sin 2t y 3 sin t y 4 2 −2 −2 −4 x 2 120. Identify the type of symmetry each of the following polar points has with the point in the figure. (a) 4, 6 (b) 4, 6 (c) 4, 6 π 2 π( 4, 6 ) 0 2 121. What is the relationship between the graphs of the rectan- gular and polar equations? (a) x2 y2 25, r 5 (b) x y 0, 4 122. Geometry The area of the ellipse in the figure is twice the area of the circle. What is the length of the major axis? (Hint: The area of an ellipse is A ab. ) y (0, 10) −a ( , 0) x a ( , 0) (0, 10)− 333202_100R.qxd 12/8/05 9:12 AM Page 805 10 Chapter Test Chapter Test 805 Take this test as you would take a test in class. When you are finished, check your work against the answers given in the back of the book. 1. Find the inclination of the line 2. Find the angle between the lines 2x 7y 3 0. 3x 2y 4 0 3. Find the distance between the point 7, 5 and the line and 4x y 6 0. y 5 x. In Exercises 4–7, classify the conic and write the equation in standard form. Identify the center, vertices, foci, and asymptotes (if applicable).Then sketch the graph of the conic. 4. 5. 6. 7. y 2 4x 4 0 x 2 4y 2 4x 0 9x2 16y2 54x 32y 47 0 2x2 2y2 8x 4y 9 0 8. Find the standard form of the equation of the parabola with vertex vertical axis, and passing through the point 0, 4. 3, 2, with a 9. Find the standard form of the equation of the hyperbola with foci 0, 0 and 0, 4 and asymptotes y ± 1 2x 2. 10. (a) Determine the number of degrees the axis must be rotated to eliminate the xy -term of the conic x 2 6xy y 2 6 0. (b) Graph the conic from part (a) and use a graphing utility to confirm your result. 11. Sketch the curve represented by the parametric equations and Eliminate the parameter and write the corresponding rectangular equation. y 2 sin . x 2 3 cos 12. Find a set of parametric equations of the line passing through the points 2, 3 and 13. Convert the polar coordinate to rectangular form. 6, 4. (There are many correct answers.) 5 6 2, 2 2, 14. Convert the rectangular coordinate representations of this point. to polar form and find two additional polar 15. Convert the rectangular equation x 2 y 2 4y 0 to polar form. In Exercises 16–19, sketch the graph of the polar equation. Identify the type of graph. r 4 1 cos r 2 3 sin 16. 18. 17. r 4 2 cos 19. r 3 sin 2 20. Find a polar equation of the ellipse with focus at the pole, eccentricity directrix y 4. e 1 4, and 21. A straight road rises with an inclination of 0.15 radian from the horizontal. Find the slope of the road and the change in elevation over a one-mile stretch of the road. 22. A baseball is hit at a point 3 feet above the ground toward the left field fence. The fence is 10 feet high and 375 feet from home plate. The path of the baseball can be modeled and the y 3 115 sin t 16t 2. Will the baseball go over the fence if it is hit at an angle of Will the baseball go over the fence if 35? x 115 cos t parametric 30? equations by 333202_100R.qxd 12/8/05 9:12 AM Page 806 Proofs in Mathematics Inclination and Slope If a nonvertical line has inclination and slope (p. 728) m, then m tan . (x 2, y2) y2 x (x1, 0) θ x2 − x1 (x1, y1) d (x2, y2 Proof m 0, If lines because m 0 tan 0. the line is horizontal and 0. So, the result is true for horizontal If the line has a positive slope, it will intersect the -axis. Label this point is a second point on the line, the slope as shown in the figure. If x x2, y2 x1, 0, is m y2 x2 0 x1 y2 x2 x1 tan . The case in which the line has a negative slope can be proved in a similar manner. Distance Between a Point and a Line The distance between the point and the line x1, y1 (p. 730) Ax By C 0 is d Ax1 By1 C A2 B2 . Proof For simplicity’s sake, assume that the given line is neither horizontal nor vertical (see figure). By writing the equation in slope-intercept form x C B Ax By C 0 y A B you can see that the line has a slope of ing through BAx x1 y y1 is x1, y1 and perpendicular to the given line is . AC These two lines intersect at the point Ay1 A2 B2 ABx1 and y2 BA, So, the slope of the line passand its equation , x2, y2 where BC x BBx1 x2 Ay1 A2 B2 . m AB. x2, y2 is Finally, the distance between 2 y2 ABy1 A2 B2 By1 d x2 x1 B2x1 A2Ax1 Ax1 By1 C A2 B2 . and x1, y1 2 y1 x12 AC C2 B2Ax1 A2 B22 ABx1 A2y1 A2 B2 BC y12 By1 C2 y y 806 333202_100R.qxd 12/8/05 9:12 AM Page 807 Parabolic Paths There are many natural occurrences of parabolas in real life. For instance, the famous astronomer Galileo discovered in the 17th century that an object that is projected upward and obliquely to the pull of gravity travels in a parabolic path. Examples of this are the center of gravity of a jumping dolphin and the path of water molecules in a drinking fountain. Standard Equation of a Parabola The standard form of the equation of a parabola with vertex at follows. (p. 736) h, k is as x h2 4py k, p 0 y k2 4px h, p 0 Vertical axis, directrix: y k p Horizontal axis, directrix: x h p The focus lies on the axis units (directed distance) from the vertex. If the the equation takes one of the following forms. vertex is at the origin p 0, 0, x2 4py y2 4px Vertical axis Horizontal axis Axis: =x h Focusx, y) Vertex: )h k ( , Directrix: p− k =y Parabola with vertical axis Directrix: p− =x h p > 0 (x, y) Axis: y = k Focus: h p ( + , k ) Vertex: ( , )h k Parabola with horizontal axis Proof For the case in which the directrix is parallel to the -axis and the focus lies x, y is any point on the parabola, above the vertex, as shown in the top figure, if h, k p then, by definition, it is equidistant from the focus and the directrix y k p. So, you have x x h2 y k p2 y k p x h2 y k p2 y k p2 x h2 y2 2yk p k p2 y2 2yk p k p2 x h2 y2 2ky 2py k2 2pk p2 y2 2ky 2py k2 2pk p2 x h2 2py 2pk 2py 2pk x h2 4py k. For the case in which the directrix is parallel to the -axis and the focus lies to x, y is any point on the the right of the vertex, as shown in the bottom figure, if h p, k parabola, then, by definition, it is equidistant from the focus and the directrix x h p. So, you have y x h p2 y k2 x h p x h p2 y k2 x h p2 x2 2xh p h p2 y k2 x2 2xh p h p2 x2 2hx 2px h2 2ph p2 y k2 x2 2hx 2px h2 2ph p2 2px 2ph y k2 2px 2ph y k2 4px h. Note that if a parabola is centered at the origin, then the two equations above would simplify to respectively. y 2 4px, x 2 4py and 807 333202_100R.qxd 12/8/05 9:12 AM Page 808 Polar Equations of Conics The graph of a polar equation of the form (p. 793) 1. r ep 1 ± e cos or 2. r ep 1 ± e sin is a conic, where the focus (pole) and the directrix. e > 0 is the eccentricity and p is the distance between π 2 p P r= ( , )θ θ r F = (0, 0) x = r cos θ Directrix Q Proof A proof for other cases are similar. In the figure, consider a vertical directrix, right of the focus p is a point on the graph of r ep1 e cos F 0, 0. is shown here. The proofs of the units to the P r, p > 0 with If r ep 1 e cos 0 the distance between PQ p x P and the directrix is p r cos ep 1 e cos cos p 1 e cos p1 e cos 1 e cos e. r p P and the pole is simply PF r, the Moreover, because the distance between ratio of PF to PQ is PF PQ rr e e e and, by definition, the graph of the equation must be a conic. 808 333202_100R.qxd 12/8/05 9:12 AM Page 809 P.S. Problem Solving This collection of thought-provoking and challenging exercises further explores and expands upon concepts learned in this chapter. 1. Several mountain climbers are located in a mountain pass between two peaks. The angles of elevation to the two peaks are 0.84 radian and 1.10 radians. A range finder shows that the distances to the peaks are 3250 feet and 6700 feet, respectively (see figure). 5. A tour boat travels between two is

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