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17 using the natural log 360. 3(1.04)3t = 8 using the common log 361. 34x − 5 = 38 using the common log 362. 50e−0.12t = 10 using the natural log For the following exercises, use a calculator to solve the equation. Unless indicated otherwise, round all answers to the nearest ten-thousandth. 349. log11 ⎝−2x2 − 7x⎞ ⎛ ⎠ = log11 (x − 2) 350. ln(2x + 9) = ln(−5x) 351. log9 (3 − x) = log9 (4x − 8) 352. ⎞ ⎛ ⎝x2 + 13 ⎠ = log(7x + 3) log 363. 7e3x − 5 + 7.9 = 47 364. ln(3) + ln(4.4x + 6.8) = 2 365. log(−0.7x − 9) = 1 + 5log(5) 366. 752 Chapter 6 Exponential and Logarithmic Functions Atmospheric pressure P in pounds per square inch is represented by the formula P = 14.7e−0.21x, where x is the number of miles above sea level. To the nearest foot, how high is the peak of a mountain with an atmospheric pressure of 8.369 pounds per square inch? (Hint: there are 5280 feet in a mile) 367. The magnitude M of an earthquake is represented by ⎛ log ⎝ the equation M = 2 3 ⎞ where E is the amount of ⎠ earthquake E E0 energy released by the and E0 = 104.4 is the assigned minimal measure released by an earthquake. To the nearest hundredth, what would the magnitude be of an earthquake releasing 1.4 ⋅ 1013 joules of energy? in joules Extensions 368. Use the definition of a logarithm along with the one- to-one property of logarithms to prove that b logb x = x. 369. the formula for continually compounding Recall interest, y = Aekt. Use the definition of a logarithm along with properties of logarithms to solve the formula for time t such that t is equal to a single logarithm. kt the interest compound Recall ⎝1
+ r k 370. A = a⎛ with properties of logarithms to solve the formula for time t.. Use the definition of a logarithm along formula ⎞ ⎠ ⎞ ⎝T0 − Ts 371. Newton’s Law of Cooling states that the temperature T of an object at any time t can be described by the ⎠e−kt equation T = Ts + ⎛ the temperature of the surrounding environment, T0 is the initial temperature of the object, and k is the cooling rate. Use the definition of a logarithm along with properties of logarithms to solve the formula for time t such that t is equal to a single logarithm. where Ts is, This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 6 Exponential and Logarithmic Functions 753 6.7 \| Exponential and Logarithmic Models Learning Objectives In this section, you will: 6.7.1 Model exponential growth and decay. 6.7.2 Use Newton’s Law of Cooling. 6.7.3 Use logistic-growth models. 6.7.4 Choose an appropriate model for data. 6.7.5 Express an exponential model in base e. Figure 6.46 A nuclear research reactor inside the Neely Nuclear Research Center on the Georgia Institute of Technology campus (credit: Georgia Tech Research Institute) We have already explored some basic applications of exponential and logarithmic functions. In this section, we explore some important applications in more depth, including radioactive isotopes and Newton’s Law of Cooling. Modeling Exponential Growth and Decay In real-world applications, we need to model the behavior of a function. In mathematical modeling, we choose a familiar general function with properties that suggest that it will model the real-world phenomenon we wish to analyze. In the case of rapid growth, we may choose the exponential growth function: y = A0 ekt is equal to the value at time zero, e is Euler’s constant, and k is a positive constant that determines the rate where A0 (percentage) of growth. We may use the exponential growth function in applications involving doubling time, the time it takes for a quantity to double. Such phenomena as wildlife populations, financial investments, biological samples, and natural resources may exhibit growth based on a doubling time. In
some applications, however, as we will see when we discuss the logistic equation, the logistic model sometimes fits the data better than the exponential model. On the other hand, if a quantity is falling rapidly toward zero, without ever reaching zero, then we should probably choose the exponential decay model. Again, we have the form y = A0 ekt where A0 is the starting value, and e is Euler’s constant. Now k is a negative constant that determines the rate of decay. We may use the exponential decay model when we are calculating half-life, or the time it takes for a substance to exponentially decay to half of its original quantity. We use half-life in applications involving radioactive isotopes. 754 Chapter 6 Exponential and Logarithmic Functions In our choice of a function to serve as a mathematical model, we often use data points gathered by careful observation and measurement to construct points on a graph and hope we can recognize the shape of the graph. Exponential growth and decay graphs have a distinctive shape, as we can see in Figure 6.47 and Figure 6.48. It is important to remember that, although parts of each of the two graphs seem to lie on the x-axis, they are really a tiny distance above the x-axis. Figure 6.47 A graph showing exponential growth. The equation is y = 2e3x. Figure 6.48 A graph showing exponential decay. The equation is y = 3e−2x. Exponential growth and decay often involve very large or very small numbers. To describe these numbers, we often use orders of magnitude. The order of magnitude is the power of ten, when the number is expressed in scientific notation, with one digit to the left of the decimal. For example, the distance to the nearest star, Proxima Centauri, measured in kilometers, is 40,113,497,200,000 kilometers. Expressed in scientific notation, this is 4.01134972 × 1013. So, we could describe this number as having order of magnitude 1013. Characteristics of the Exponential Function, y = A0ekt An exponential function with the form y = A0 ekt has the following characteristics: • one-to-one function • horizontal asymptote: y = 0 This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 6 Exponential and Logarithmic Functions 755 • domain
: ( – ∞, ∞) • range: (0, ∞) • x intercept: none • y-intercept: ⎛ ⎝0, A0 ⎞ ⎠ • increasing if k > 0 (see Figure 6.49) • decreasing if k < 0 (see Figure 6.49) Figure 6.49 An exponential function models exponential growth when k > 0 and exponential decay when k < 0. Example 6.65 Graphing Exponential Growth A population of bacteria doubles every hour. If the culture started with 10 bacteria, graph the population as a function of time. Solution When an amount grows at a fixed percent per unit time, the growth is exponential. To find A0 A0 doubles from 10 to 20. The formula is derived as follows we use the fact that is the amount at time zero, so A0 = 10. To find k, use the fact that after one hour (t = 1) the population 20 = 10ek ⋅ 1 2 = ek ln2 = k Divide by 10 Take the natural logarithm so k = ln(2). Thus the equation we want to graph is y = 10e(ln2)t Figure 6.50. = 10(eln2) t = 10 · 2 t. The graph is shown in 756 Chapter 6 Exponential and Logarithmic Functions Figure 6.50 The graph of y = 10e(ln2)t Analysis The population of bacteria after ten hours is 10,240. We could describe this amount is being of the order of magnitude 104. The population of bacteria after twenty hours is 10,485,760 which is of the order of magnitude 107, so we could say that the population has increased by three orders of magnitude in ten hours. Half-Life We now turn to exponential decay. One of the common terms associated with exponential decay, as stated above, is halflife, the length of time it takes an exponentially decaying quantity to decrease to half its original amount. Every radioactive isotope has a half-life, and the process describing the exponential decay of an isotope is called radioactive decay. To find the half-life of a function describing exponential decay, solve the following equation: We find that the half-life depends only on the constant k and not on the starting quantity A0. The formula is derived as follows A0 = Ao ekt 1 2 1 2 A0 = Ao ekt = ekt 1 2
⎛ ⎞ 1 ln ⎠ = kt ⎝ 2 −ln(2) = kt ln(2) k = t − Divide by A0. Take the natural log. Apply laws of logarithms. Divide by k. Since t, the time, is positive, k must, as expected, be negative. This gives us the half-life formula t = − ln(2) k (6.16) This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 6 Exponential and Logarithmic Functions 757 Given the half-life, find the decay rate. 1. Write A = Ao ekt. 2. Replace A by 1 2 A0 and replace t by the given half-life. 3. Solve to find k. Express k as an exact value (do not round). Note: It is also possible to find the decay rate using k = − ln(2) t. Example 6.66 Finding the Function that Describes Radioactive Decay The half-life of carbon-14 is 5,730 years. Express the amount of carbon-14 remaining as a function of time, t. Solution This formula is derived as follows. () The continuous growth formula. A = A0 ekt 0.5A0 = A0 ek ⋅ 5730 Substitute the half-life for t and 0.5A0 for f (t). 0.5 = e5730k ln(0.5) = 5730k ln(0.5) 5730 ⎛ ⎝ Divide by A0. Take the natural log of both sides. Divide by the coefficient o k. k = ⎞ ⎠t ln(0.5) 5730 Substitute for r in the continuous growth formula. A = A0 e The function that describes this continuous decay is f (t) = A0 e ln(0.5) 5730 ≈ − 1.2097 is negative, as expected in the case of exponential decay. ⎛ ⎝ ln(0.5) 5730 ⎞ ⎠t. We observe that the coefficient of t, The half-life of plutonium-244 is 80,000,000 years. Find function gives the amount of carbon-14 6.65 remaining as a function of time, measured in years.
Radiocarbon Dating The formula for radioactive decay is important in radiocarbon dating, which is used to calculate the approximate date a plant or animal died. Radiocarbon dating was discovered in 1949 by Willard Libby, who won a Nobel Prize for his discovery. It compares the difference between the ratio of two isotopes of carbon in an organic artifact or fossil to the ratio of those two isotopes in the air. It is believed to be accurate to within about 1% error for plants or animals that died within the last 60,000 years. Carbon-14 is a radioactive isotope of carbon that has a half-life of 5,730 years. It occurs in small quantities in the carbon dioxide in the air we breathe. Most of the carbon on Earth is carbon-12, which has an atomic weight of 12 and is not radioactive. Scientists have determined the ratio of carbon-14 to carbon-12 in the air for the last 60,000 years, using tree rings and other organic samples of known dates—although the ratio has changed slightly over the centuries. 758 Chapter 6 Exponential and Logarithmic Functions As long as a plant or animal is alive, the ratio of the two isotopes of carbon in its body is close to the ratio in the atmosphere. When it dies, the carbon-14 in its body decays and is not replaced. By comparing the ratio of carbon-14 to carbon-12 in a decaying sample to the known ratio in the atmosphere, the date the plant or animal died can be approximated. Since the half-life of carbon-14 is 5,730 years, the formula for the amount of carbon-14 remaining after t years is ⎛ ⎝ ln(0.5) 5730 ⎞ ⎠t A ≈ A0 e where • A is the amount of carbon-14 remaining • A0 is the amount of carbon-14 when the plant or animal began decaying. This formula is derived as follows: A = A0 ekt The continuous growth formula. 0.5A0 = A0 ek ⋅ 5730 Substitute the half-life for t and 0.5A0 for f (t). 0.5 = e5730k ln(0.5) = 5730k ln(0.5) 5730 ⎛ ⎝ Divide by A0. Take the natural log of both sides. Divide by the coefficient o k. k = �
� ⎠t ln(0.5) 5730 Substitute for r in the continuous growth formula. A = A0 e To find the age of an object, we solve this equation for t : ⎞ ⎠ ⎛ A ln ⎝ A0 −0.000121 t = (6.17) Out of necessity, we neglect here the many details that a scientist takes into consideration when doing carbon-14 dating, and we only look at the basic formula. The ratio of carbon-14 to carbon-12 in the atmosphere is approximately 0.0000000001%. Let r be the ratio of carbon-14 to carbon-12 in the organic artifact or fossil to be dated, determined by a method called liquid scintillation. From the equation A ≈ A0 e−0.000121t we know the ratio of the percentage of carbon-14 in the object we are dating to the percentage of carbon-14 in the atmosphere is r = A to A0. We solve this equation for t, ≈ e−0.000121t get t = ln(r) −0.000121 Given the percentage of carbon-14 in an object, determine its age. 1. Express the given percentage of carbon-14 as an equivalent decimal, k. 2. Substitute for k in the equation t = ln(r) −0.000121 and solve for the age, t. Example 6.67 Finding the Age of a Bone A bone fragment is found that contains 20% of its original carbon-14. To the nearest year, how old is the bone? Solution This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 6 Exponential and Logarithmic Functions 759 We substitute 20% = 0.20 for k in the equation and solve for t : t = ln(r) −0.000121 ln(0.20) −0.000121 = ≈ 13301 Use the general form of the equation. Substitute for r. Round to the nearest year. The bone fragment is about 13,301 years old. Analysis The instruments that measure the percentage of carbon-14 are extremely sensitive and, as we mention above, a scientist will need to do much more work than we did in order to be satisfied. Even so, carbon dating is only accurate to about 1%, so this age should be given as 13,301 years ±
1% or 13,301 years ± 133 years. Cesium-137 has a half-life of about 30 years. If we begin with 200 mg of cesium-137, will it take more or 6.66 less than 230 years until only 1 milligram remains? Calculating Doubling Time For decaying quantities, we determined how long it took for half of a substance to decay. For growing quantities, we might want to find out how long it takes for a quantity to double. As we mentioned above, the time it takes for a quantity to double is called the doubling time. Given the basic exponential growth equation A = A0 ekt quantity has doubled, that is, by solving 2A0 = A0 ekt The formula is derived as follows:, doubling time can be found by solving for when the original. 2A0 = A0 ekt 2 = ekt ln2 = kt t = ln2 k Divide by A0. Take the natural logarithm. Divide by the coefficient o t. Thus the doubling time is Example 6.68 t = ln2 k (6.18) Finding a Function That Describes Exponential Growth According to Moore’s Law, the doubling time for the number of transistors that can be put on a computer chip is approximately two years. Give a function that describes this behavior. Solution The formula is derived as follows: 760 Chapter 6 Exponential and Logarithmic Functions t = ln2 k 2 = ln2 k k = ln2 2 t ln2 2 A = A0 e The doubling time formula. Use a doubling time of two years. Multiply by k and divide by 2. Substitute k into the continuous growth formula. The function is A = A0 e t ln2 2. 6.67 Recent data suggests that, as of 2013, the rate of growth predicted by Moore’s Law no longer holds. Growth has slowed to a doubling time of approximately three years. Find the new function that takes that longer doubling time into account. Using Newton’s Law of Cooling Exponential decay can also be applied to temperature. When a hot object is left in surrounding air that is at a lower temperature, the object’s temperature will decrease exponentially, leveling off as it approaches the surrounding air temperature. On a graph of the temperature function, the leveling off will correspond to a horizontal asymptote at the temperature of the surrounding air. Unless
the room temperature is zero, this will correspond to a vertical shift of the generic exponential decay function. This translation leads to Newton’s Law of Cooling, the scientific formula for temperature as a function of time as an object’s temperature is equalized with the ambient temperature This formula is derived as follows: T(t) = aekt + Ts ) T(t) = Abct + Ts T(t) = Aeln(bct T(t) = Aectlnb T(t) = Aekt + Ts + Ts + Ts Laws of logarithms. Laws of logarithms. Rename the constant c ln b, calling it k. Newton’s Law of Cooling The temperature of an object, T, in surrounding air with temperature Ts will behave according to the formula T(t) = Aekt + Ts (6.19) where • t is time • A is the difference between the initial temperature of the object and the surroundings • k is a constant, the continuous rate of cooling of the object This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 6 Exponential and Logarithmic Functions 761 Given a set of conditions, apply Newton’s Law of Cooling. 1. Set Ts equal to the y-coordinate of the horizontal asymptote (usually the ambient temperature). 2. Substitute the given values into the continuous growth formula T(t) = Aek t + Ts to find the parameters A and k. 3. Substitute in the desired time to find the temperature or the desired temperature to find the time. Example 6.69 Using Newton’s Law of Cooling A cheesecake is taken out of the oven with an ideal internal temperature of 165°F, and is placed into a 35°F refrigerator. After 10 minutes, the cheesecake has cooled to 150°F. If we must wait until the cheesecake has cooled to 70°F before we eat it, how long will we have to wait? Solution Because the surrounding air temperature in the refrigerator is 35 degrees, the cheesecake’s temperature will decay exponentially toward 35, following the equation We know the initial temperature was 165, so T(0) = 165. T(t) = Aekt + 35 165 = Aek0 + 35 Substitute (0, 165). A = 130 Solve for A. We were given another
data point, T(10) = 150, which we can use to solve for k. = e10k Substitute (10, 150). 150 = 130ek10 + 35 115 = 130ek10 115 130 ⎞ 115 ⎠ = 10k 130 ⎛ 115 ln ⎝ 130 10 This gives us the equation for the cooling of the cheesecake: T(t) = 130e – 0.0123t k = Divide by 130. ⎛ ln ⎝ Subtract 35. ⎞ ⎠ = − 0.0123 Divide by the coefficient o k. + 35. Take the natural log of both sides. Now we can solve for the time it will take for the temperature to cool to 70 degrees. 70 = 130e−0.0123t 35 = 130e−0.0123t 35 130 ln( 35 130 ) = − 0.0123t = e−0.0123t + 35 Substitute in 70 for T(t). Subtract 35. Divide by 130. Take the natural log of both sides t = ln( 35 130) −0.0123 ≈ 106.68 Divide by the coefficient o t. It will take about 107 minutes, or one hour and 47 minutes, for the cheesecake to cool to 70°F. 762 Chapter 6 Exponential and Logarithmic Functions 6.68 A pitcher of water at 40 degrees Fahrenheit is placed into a 70 degree room. One hour later, the temperature has risen to 45 degrees. How long will it take for the temperature to rise to 60 degrees? Using Logistic Growth Models Exponential growth cannot continue forever. Exponential models, while they may be useful in the short term, tend to fall apart the longer they continue. Consider an aspiring writer who writes a single line on day one and plans to double the number of lines she writes each day for a month. By the end of the month, she must write over 17 billion lines, or one-halfbillion pages. It is impractical, if not impossible, for anyone to write that much in such a short period of time. Eventually, an exponential model must begin to approach some limiting value, and then the growth is forced to slow. For this reason, it is often better to use a model with an upper bound instead of an exponential growth model, though the exponential growth model is still useful over a short term, before approaching the limiting value. The logistic
growth model is approximately exponential at first, but it has a reduced rate of growth as the output approaches the model’s upper bound, called the carrying capacity. For constants a, b, and c, the logistic growth of a population over time x is represented by the model f (x) = c 1 + ae−bx The graph in Figure 6.51 shows how the growth rate changes over time. The graph increases from left to right, but the growth rate only increases until it reaches its point of maximum growth rate, at which point the rate of increase decreases. Figure 6.51 Logistic Growth The logistic growth model is f (x) = c 1 + ae−bx where • • • c 1 + a is the initial value c is the carrying capacity, or limiting value b is a constant determined by the rate of growth. This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 6 Exponential and Logarithmic Functions 763 Example 6.70 Using the Logistic-Growth Model An influenza epidemic spreads through a population rapidly, at a rate that depends on two factors: The more people who have the flu, the more rapidly it spreads, and also the more uninfected people there are, the more rapidly it spreads. These two factors make the logistic model a good one to study the spread of communicable diseases. And, clearly, there is a maximum value for the number of people infected: the entire population. For example, at time t = 0 there is one person in a community of 1,000 people who has the flu. So, in that community, at most 1,000 people can have the flu. Researchers find that for this particular strain of the flu, the logistic growth constant is b = 0.6030. Estimate the number of people in this community who will have had this flu after ten days. Predict how many people in this community will have had this flu after a long period of time has passed. Solution We substitute the given data into the logistic growth model f (x) = c 1 + ae−bx Because at most 1,000 people, the entire population of the community, can get the flu, we know the limiting value is c = 1000. To find a, we use the formula that the number of cases at time t = 0 is c from which 1 + a = 1, it follows that a = 999. This model
predicts that, after ten days, the number of people who have had the flu is 1 + 999e−0.6030x ≈ 293.8. Because the actual number must be a whole number (a person has either f (x) = 1000 had the flu or not) we round to 294. In the long term, the number of people who will contract the flu is the limiting value, c = 1000. Analysis Remember that, because we are dealing with a virus, we cannot predict with certainty the number of people infected. The model only approximates the number of people infected and will not give us exact or actual values. The graph in Figure 6.52 gives a good picture of how this model fits the data. 764 Chapter 6 Exponential and Logarithmic Functions Figure 6.52 The graph of f (x) = 1000 1 + 999e−0.6030x 6.69 Using the model in Example 6.70, estimate the number of cases of flu on day 15. Choosing an Appropriate Model for Data Now that we have discussed various mathematical models, we need to learn how to choose the appropriate model for the raw data we have. Many factors influence the choice of a mathematical model, among which are experience, scientific laws, and patterns in the data itself. Not all data can be described by elementary functions. Sometimes, a function is chosen that approximates the data over a given interval. For instance, suppose data were gathered on the number of homes bought in the United States from the years 1960 to 2013. After plotting these data in a scatter plot, we notice that the shape of the data from the years 2000 to 2013 follow a logarithmic curve. We could restrict the interval from 2000 to 2010, apply regression analysis using a logarithmic model, and use it to predict the number of home buyers for the year 2015. Three kinds of functions that are often useful in mathematical models are linear functions, exponential functions, and logarithmic functions. If the data lies on a straight line, or seems to lie approximately along a straight line, a linear model may be best. If the data is non-linear, we often consider an exponential or logarithmic model, though other models, such as quadratic models, may also be considered. In choosing between an exponential model and a logarithmic model, we look at the way the data curves. This is called the concavity. If we draw a line between two data points,
and all (or most) of the data between those two points lies above that line, we say the curve is concave down. We can think of it as a bowl that bends downward and therefore cannot hold water. If all (or most) of the data between those two points lies below the line, we say the curve is concave up. In this case, we can think of a bowl that bends upward and can therefore hold water. An exponential curve, whether rising or falling, whether representing growth or decay, is always concave up away from its horizontal asymptote. A logarithmic curve is always concave away from its vertical asymptote. In the case of positive data, which is the most common case, an exponential curve is always concave up, and a logarithmic curve always concave down. A logistic curve changes concavity. It starts out concave up and then changes to concave down beyond a certain point, called a point of inflection. After using the graph to help us choose a type of function to use as a model, we substitute points, and solve to find the parameters. We reduce round-off error by choosing points as far apart as possible. This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 6 Exponential and Logarithmic Functions 765 Example 6.71 Choosing a Mathematical Model Does a linear, exponential, logarithmic, or logistic model best fit the values listed in Table 6.17? Find the model, and use a graph to check your choice.386 2.197 2.773 3.219 3.584 3.892 4.159 4.394 x y 1 0 Table 6.17 Solution First, plot the data on a graph as in Figure 6.53. For the purpose of graphing, round the data to two significant digits. Figure 6.53 Clearly, the points do not lie on a straight line, so we reject a linear model. If we draw a line between any two of the points, most or all of the points between those two points lie above the line, so the graph is concave down, suggesting a logarithmic model. We can try y = aln(bx). Plugging in the first point, (1,0), gives 0 = alnb. We reject the case that a = 0 (if it were, all outputs would be 0
), so we know ln(b) = 0. Thus b = 1 and y = aln(x). Next we can use the point (9,4.394) to solve for a : y = aln(x) 4.394 = aln(9) a = 4.394 ln(9) 766 Chapter 6 Exponential and Logarithmic Functions Because a = 4.394 ln(9) ≈ 2, an appropriate model for the data is y = 2ln(x). To check the accuracy of the model, we graph the function together with the given points as in Figure 6.54. Figure 6.54 The graph of y = 2lnx. We can conclude that the model is a good fit to the data. ⎛ ⎝x2⎞ Compare Figure 6.54 to the graph of y = ln ⎠ shown in Figure 6.55. ⎝x2⎞ ⎛ Figure 6.55 The graph of y = ln ⎠ This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 6 Exponential and Logarithmic Functions 767 ⎝x2⎞ ⎛ The graphs appear to be identical when x > 0. A quick check confirms this conclusion: y = ln x > 0. ⎠ = 2ln(x) for ⎝x2⎞ ⎛ the graph of y = ln However, if x < 0, ⎠ includes a “extra” branch, as shown in Figure 6.56. This occurs because, while y = 2ln(x) cannot have negative values in the domain (as such values would force the argument ⎛ ⎝x2⎞ to be negative), the function y = ln ⎠ can have negative domain values. Figure 6.56 6.70 Does a linear, exponential, or logarithmic model best fit the data in Table 6.18? Find the model.297 5.437 8.963 14.778 24.365 40.172 66.231 109.196 180.034 Table 6.18 Expressing an Exponential Model in Base e While powers and logarithms of any base can be used in modeling, the two most common bases are 10 and e. In science and mathematics, the base e is often
preferred. We can use laws of exponents and laws of logarithms to change any base to base e. Given a model with the form y = ab x, change it to the form y = A0 ekx. ⎝b x⎞. 1. Rewrite y = ab x as y = ae ln⎛ ⎠ 2. Use the power rule of logarithms to rewrite y as y = ae xln(b) = aeln(b)x. 3. Note that a = A0 and k = ln(b) in the equation y = A0 ekx. Example 6.72 768 Chapter 6 Exponential and Logarithmic Functions Changing to base e Change the function y = 2.5(3.1) x so that this same function is written in the form y = A0 ekx. Solution The formula is derived as follows x y = 2.5(3.1) ln⎛ ⎝3.1 x⎞ ⎠ = 2.5e = 2.5e xln3.1 = 2.5e(ln3.1) x Insert exponential and its inverse. Laws of logs. Commutative law of multiplication 6.71 Change the function y = 3(0.5) x to one having e as the base. Access these online resources for additional instruction and practice with exponential and logarithmic models. • Logarithm Application – pH (http://openstaxcollege.org/l/logph) • Exponential Model – Age Using Half-Life (http://openstaxcollege.org/l/expmodelhalf) • Newton’s Law of Cooling (http://openstaxcollege.org/l/newtoncooling) • Exponential Growth Given Doubling Time (http://openstaxcollege.org/l/expgrowthdbl) • Exponential Growth – Find Initial Amount Given Doubling Time (http://openstaxcollege.org/l/initialdouble) This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 6 Exponential and Logarithmic Functions 769 6.7 EXERCISES Verbal 372. With what kind of exponential model would half-life be associated? What role does half-life play in these models? What is carbon
dating? Why does it work? Give an 373. example in which carbon dating would be useful. 374. With what kind of exponential model would doubling time be associated? What role does doubling time play in these models? Define Newton’s Law of Cooling. Then name at least 375. three real-world situations where Newton’s Law of Cooling would be applied. What is an order of magnitude? Why are orders of 376. magnitude useful? Give an example to explain. Numeric The temperature of an object in degrees Fahrenheit is equation represented by + 72. To the nearest degree, what is 377. after t minutes T(t) = 68e−0.0174t the temperature of the object after one and a half hours? the x f(x) –2 0.694 –1 0.833 0 1 2 3 4 5 1 1.2 1.44 1.728 2.074 2.488 For the following exercises, use the logistic growth model f (x) = 150 1 + 8e−2x. 384. Rewrite f (x) = 1.68(0.65) x as an exponential equation with base e to five significant digits. 378. Find and interpret f (0). Round to the nearest tenth. Technology 379. Find and interpret f (4). Round to the nearest tenth. 380. Find the carrying capacity. 381. Graph the model. For the following exercises, enter the data from each table into a graphing calculator and graph the resulting scatter plots. Determine whether the data from the table could represent a function that linear, exponential, or logarithmic. is 382. Determine whether the data from the table could best be represented as a function that is linear, exponential, or logarithmic. Then write a formula for a model that represents the data. 385. 383. 770 Chapter 6 Exponential and Logarithmic Functions x f(x) x f(x.079 5.296 6.159 6.828 7.375 7.838 8.238 8.592.4 2.88 3.456 4.147 4.977 5.972 7.166 8.6 10.32 10 8.908 10 12.383 386. 387. This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 6 Exponential and Logarithmic Functions 771 x f
(x) 4 5 6 7 8 9 9.429 9.972 10.415 10.79 11.115 11.401 10 11.657 11 11.889 12 12.101 13 12.295 388. x f(x) 1.25 5.75 2.25 8.75 3.56 12.68 4.2 14.6 5.65 18.95 6.75 22.25 7.25 23.75 8.6 27.8 9.25 29.75 10.5 33.5 For the following exercises, use a graphing calculator and this scenario: the population of a fish farm in t years is modeled by the equation P(t) = 1000 1 + 9e−0.6t. 389. Graph the function. 390. What is the initial population of fish? To the nearest tenth, what is the doubling time for the 391. fish population? To the nearest whole number, what will the fish 392. population be after 2 years? To the nearest tenth, how long will it take for the 393. population to reach 900? What is the carrying capacity for the fish population? 394. Justify your answer using the graph of P. Extensions 395. A substance has a half-life of 2.045 minutes. If the initial amount of the substance was 132.8 grams, how many 772 Chapter 6 Exponential and Logarithmic Functions half-lives will have passed before the substance decays to 8.3 grams? What is the total time of decay? 396. The formula for an increasing population is given by P(t) = P0 ert where P0 is the initial population and r > 0. Derive a general formula for the time t it takes for the population to increase by a factor of M. 397. Recall the formula for calculating the magnitude of an The half-life of Radium-226 is 1590 years. What is the to four annual decay rate? Express the decimal result significant digits and the percentage to two significant digits. The half-life of Erbium-165 is 10.4 hours. What is 407. the hourly decay rate? Express the decimal result to four significant digits and the percentage to two significant digits. earthquake, M = 2 3 S S0 this equation algebraically for the seismic moment S. ⎞ ⎠. Show each step for solving ⎛ log ⎝ 398. y = What is the y-inter
cept of the logistic growth model 1 + ae−rx? Show the steps for calculation. What c does this point tell us about the population? 399. Prove that b x = e xln(b) for positive b ≠ 1. Real-World Applications A wooden artifact from an archeological dig contains 408. 60 percent of the carbon-14 that is present in living trees. To the nearest year, about how many years old is the artifact? (The half-life of carbon-14 is 5730 years.) A research student is working with a culture of 409. bacteria that doubles in size every twenty minutes. The initial population count was 1350 bacteria. Rounding to five significant digits, write an exponential equation representing this situation. To the nearest whole number, what is the population size after 3 hours? For the following exercises, use this scenario: A doctor prescribes 125 milligrams of a therapeutic drug that decays by about 30% each hour. For the following exercises, use this scenario: A biologist recorded a count of 360 bacteria present in a culture after 5 minutes and 1000 bacteria present after 20 minutes. 400. To the nearest hour, what is the half-life of the drug? Write an exponential model representing the amount 401. of the drug remaining in the patient’s system after t hours. Then use the formula to find the amount of the drug that would remain in the patient’s system after 3 hours. Round to the nearest milligram. Using the model found in the previous exercise, find the result. Round to the nearest 402. f (10) and interpret hundredth. For the following exercises, use this scenario: A tumor is injected with 0.5 grams of Iodine-125, which has a decay rate of 1.15% per day. To the nearest day, how long will it take for half of the 403. Iodine-125 to decay? Write an exponential model representing the amount 404. of Iodine-125 remaining in the tumor after t days. Then use the formula to find the amount of Iodine-125 that would remain in the tumor after 60 days. Round to the nearest tenth of a gram. A scientist begins with 250 grams of a radioactive 405. substance. After 250 minutes, the sample has decayed to 32 grams. Rounding to five significant digits, write an exponential equation representing this situation. To the nearest minute, what is the half-life of this substance? 406. This content
is available for free at https://cnx.org/content/col11758/1.5 To the nearest whole number, what was the initial 410. population in the culture? an Rounding to six significant digits, write 411. exponential equation representing this situation. To the nearest minute, how long did it take the population to double? For the following exercises, use this scenario: A pot of temperature of 100° boiling soup with an internal Fahrenheit was taken off the stove to cool in a 69° F room. After fifteen minutes, the internal temperature of the soup was 95° F. Use Newton’s Law of Cooling to write a formula that 412. models this situation. To the nearest minute, how long will it take the soup 413. to cool to 80° F? To the nearest degree, what will the temperature be 414. after 2 and a half hours? For the following exercises, use this scenario: A turkey is taken out of the oven with an internal temperature of 165°F and is allowed to cool in a 75°F room. After half an hour, the internal temperature of the turkey is 145°F. 415. Write a formula that models this situation. To the nearest degree, what will the temperature be 416. after 50 minutes? Chapter 6 Exponential and Logarithmic Functions 773 To the nearest minute, how long will it take the turkey 417. to cool to 110° F? D. f (t) = 4.75 1 + 13e−0.83925t For the following exercises, find the value of the number shown on each logarithmic scale. Round all answers to the nearest thousandth. 418. 419. 420. sounds on a logarithmic scale: Whisper: 10−10 Plot each set of approximate values of intensity of W m2, Vacuum: 10−4 W m2, Jet: 102 W m2 421. S S0 ⎛ log ⎝ earthquake, M = 2 3 Recall the formula for calculating the magnitude of an ⎞ ⎠. One magnitude 3.9 on the MMS scale. If a second earthquake has 750 times as much energy as the first, find the magnitude of the second quake. Round to the nearest hundredth. earthquake has For the following exercises, use this scenario: The equation N(t) = models the number of people in a 500 1 + 49e−0.7t town who have heard a rumor after t days
. 422. How many people started the rumor? To the nearest whole number, how many people will 423. have heard the rumor after 3 days? 424. As t increases without bound, what value does N(t) approach? Interpret your answer. For the following exercise, choose the correct answer choice. A doctor and injects a patient with 13 milligrams of 425. radioactive dye that decays exponentially. After 12 minutes, there are 4.75 milligrams of dye remaining in the patient’s system. Which is an appropriate model for this situation? A. B. C. f (t) = 13(0.0805) t f (t) = 13e0.9195t f (t) = 13e( − 0.0839t) 774 Chapter 6 Exponential and Logarithmic Functions 6.8 \| Fitting Exponential Models to Data Learning Objectives In this section, you will: 6.8.1 Build an exponential model from data. 6.8.2 Build a logarithmic model from data. 6.8.3 Build a logistic model from data. In previous sections of this chapter, we were either given a function explicitly to graph or evaluate, or we were given a set of points that were guaranteed to lie on the curve. Then we used algebra to find the equation that fit the points exactly. In this section, we use a modeling technique called regression analysis to find a curve that models data collected from realworld observations. With regression analysis, we don’t expect all the points to lie perfectly on the curve. The idea is to find a model that best fits the data. Then we use the model to make predictions about future events. Do not be confused by the word model. In mathematics, we often use the terms function, equation, and model interchangeably, even though they each have their own formal definition. The term model is typically used to indicate that the equation or function approximates a real-world situation. We will concentrate on three types of regression models in this section: exponential, logarithmic, and logistic. Having already worked with each of these functions gives us an advantage. Knowing their formal definitions, the behavior of their graphs, and some of their real-world applications gives us the opportunity to deepen our understanding. As each regression model is presented, key features and definitions of its associated function are included for review. Take a moment to rethink each of these functions, reflect on the work
we’ve done so far, and then explore the ways regression is used to model realworld phenomena. Building an Exponential Model from Data As we’ve learned, there are a multitude of situations that can be modeled by exponential functions, such as investment growth, radioactive decay, atmospheric pressure changes, and temperatures of a cooling object. What do these phenomena have in common? For one thing, all the models either increase or decrease as time moves forward. But that’s not the whole story. It’s the way data increase or decrease that helps us determine whether it is best modeled by an exponential equation. Knowing the behavior of exponential functions in general allows us to recognize when to use exponential regression, so let’s review exponential growth and decay. Recall that exponential functions have the form y = ab x or y = A0 ekx form most commonly used on graphing utilities, y = ab x learned about the exponential function y = ab x (assume a > 0) :. When performing regression analysis, we use the. Take a moment to reflect on the characteristics we’ve already • b must be greater than zero and not equal to one. • The initial value of the model is y = a. ◦ ◦ If b > 1, slowly at first, but then increase more and more rapidly, without bound. the function models exponential growth. As x increases, the outputs of the model increase If 0 < b < 1, the function models exponential decay. As x increases, the outputs for the model decrease rapidly at first and then level off to become asymptotic to the x-axis. In other words, the outputs never become equal to or less than zero. As part of the results, your calculator will display a number known as the correlation coefficient, labeled by the variable r, or r 2. (You may have to change the calculator’s settings for these to be shown.) The values are an indication of the “goodness of fit” of the regression equation to the data. We more commonly use the value of r 2 instead of r, but the closer either value is to 1, the better the regression equation approximates the data. This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 6 Exponential and Logarithmic Functions 775 Exponential Regression Exponential regression is used to model situations in which growth begins slowly and then accelerates rapidly without bound,
or where decay begins rapidly and then slows down to get closer and closer to zero. We use the command “ExpReg” on a graphing utility to fit an exponential function to a set of data points. This returns an equation of the form, y = ab x Note that: • b must be non-negative. • when b > 1, we have an exponential growth model. • when 0 < b < 1, we have an exponential decay model. Given a set of data, perform exponential regression using a graphing utility. 1. Use the STAT then EDIT menu to enter given data. a. Clear any existing data from the lists. b. List the input values in the L1 column. c. List the output values in the L2 column. 2. Graph and observe a scatter plot of the data using the STATPLOT feature. a. Use ZOOM [9] to adjust axes to fit the data. b. Verify the data follow an exponential pattern. 3. Find the equation that models the data. a. Select “ExpReg” from the STAT then CALC menu. b. Use the values returned for a and b to record the model, y = ab x. 4. Graph the model in the same window as the scatterplot to verify it is a good fit for the data. Example 6.73 Using Exponential Regression to Fit a Model to Data In 2007, a university study was published investigating the crash risk of alcohol impaired driving. Data from 2,871 crashes were used to measure the association of a person’s blood alcohol level (BAC) with the risk of being in an accident. Table 6.19 shows results from the study [9]. The relative risk is a measure of how many times more likely a person is to crash. So, for example, a person with a BAC of 0.09 is 3.54 times as likely to crash as a person who has not been drinking alcohol. 9. Source: Indiana University Center for Studies of Law in Action, 2007 776 Chapter 6 Exponential and Logarithmic Functions BAC Relative Risk of Crashing 0 1 0.01 0.03 0.05 0.07 0.09 1.03 1.06 1.38 2.09 3.54 BAC 0.11 0.13 0.15 0.17 0.19 0.21 Relative Risk of Crashing 6.41 12.6 22.1 39.05 65.32 99
.78 Table 6.19 a. Let x represent the BAC level, and let y represent the corresponding relative risk. Use exponential regression to fit a model to these data. b. After 6 drinks, a person weighing 160 pounds will have a BAC of about 0.16. How many times more likely is a person with this weight to crash if they drive after having a 6-pack of beer? Round to the nearest hundredth. Solution a. Using the STAT then EDIT menu on a graphing utility, list the BAC values in L1 and the relative risk values in L2. Then use the STATPLOT feature to verify that the scatterplot follows the exponential pattern shown in Figure 6.57: Figure 6.57 Use the “ExpReg” command from the STAT then CALC menu to obtain the exponential model, Converting from scientific notation, we have: y = 0.58304829(2.20720213E10) y = 0.58304829(22,072,021,300) x x This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 6 Exponential and Logarithmic Functions 777 Notice that r 2 ≈ 0.97 which indicates the model is a good fit to the data. To see this, graph the model in the same window as the scatterplot to verify it is a good fit as shown in Figure 6.58: Figure 6.58 b. Use the model to estimate the risk associated with a BAC of 0.16. Substitute 0.16 for x in the model and solve for y. y = 0.58304829(22,072,021,300) x Use the regression model found in part (a). = 0.58304829(22,072,021,300)0.16 Substitute 0.16 for x. ≈ 26.35 Round to the nearest hundredth. If a 160-pound person drives after having 6 drinks, he or she is about 26.35 times more likely to crash than if driving while sober. 6.72 Table 6.20 shows a recent graduate’s credit card balance each month after graduation. Month 1 2 3 4 5 6 7 8 620.00 761.88 899.80 1039.93 1270.63 1589.04 1851.31 2154.92 Debt ($) Table 6.20
a. Use exponential regression to fit a model to these data. b. If spending continues at this rate, what will the graduate’s credit card debt be one year after graduating? 778 Chapter 6 Exponential and Logarithmic Functions Is it reasonable to assume that an exponential regression model will represent a situation indefinitely? No. Remember that models are formed by real-world data gathered for regression. It is usually reasonable to make estimates within the interval of original observation (interpolation). However, when a model is used to make predictions, it is important to use reasoning skills to determine whether the model makes sense for inputs far beyond the original observation interval (extrapolation). Building a Logarithmic Model from Data Just as with exponential functions, there are many real-world applications for logarithmic functions: intensity of sound, pH levels of solutions, yields of chemical reactions, production of goods, and growth of infants. As with exponential models, data modeled by logarithmic functions are either always increasing or always decreasing as time moves forward. Again, it is the way they increase or decrease that helps us determine whether a logarithmic model is best. Recall that logarithmic functions increase or decrease rapidly at first, but then steadily slow as time moves on. By reflecting on the characteristics we’ve already learned about this function, we can better analyze real world situations that reflect this type of growth or decay. When performing logarithmic regression analysis, we use the form of the logarithmic function most commonly used on graphing utilities, y = a + bln(x). For this function • All input values, x, must be greater than zero. • The point (1, a) is on the graph of the model. • • If b > 0, If b < 0, the model is increasing. Growth increases rapidly at first and then steadily slows over time. the model is decreasing. Decay occurs rapidly at first and then steadily slows over time. Logarithmic Regression Logarithmic regression is used to model situations where growth or decay accelerates rapidly at first and then slows over time. We use the command “LnReg” on a graphing utility to fit a logarithmic function to a set of data points. This returns an equation of the form, y = a + bln(x) Note that • all input values, x, must be non-negative. • when b > 0, the model is
increasing. • when b < 0, the model is decreasing. Given a set of data, perform logarithmic regression using a graphing utility. 1. Use the STAT then EDIT menu to enter given data. a. Clear any existing data from the lists. b. List the input values in the L1 column. c. List the output values in the L2 column. 2. Graph and observe a scatter plot of the data using the STATPLOT feature. a. Use ZOOM [9] to adjust axes to fit the data. b. Verify the data follow a logarithmic pattern. 3. Find the equation that models the data. a. Select “LnReg” from the STAT then CALC menu. b. Use the values returned for a and b to record the model, y = a + bln(x). 4. Graph the model in the same window as the scatterplot to verify it is a good fit for the data. This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 6 Exponential and Logarithmic Functions 779 Example 6.74 Using Logarithmic Regression to Fit a Model to Data Due to advances in medicine and higher standards of living, life expectancy has been increasing in most developed countries since the beginning of the 20th century. Table 6.21 shows the average life expectancies, in years, of Americans from 1900–2010[10]. Year 1900 1910 1920 1930 1940 1950 Life Expectancy(Years) 47.3 50.0 54.1 59.7 62.9 68.2 Year 1960 1970 1980 1990 2000 2010 Life Expectancy(Years) 69.7 70.8 73.7 75.4 76.8 78.7 Table 6.21 a. Let x represent time in decades starting with x = 1 for the year 1900, x = 2 for the year 1910, and so on. Let y represent the corresponding life expectancy. Use logarithmic regression to fit a model to these data. b. Use the model to predict the average American life expectancy for the year 2030. Solution a. Using the STAT then EDIT menu on a graphing utility, list the years using values 1–12 in L1 and the corresponding life expectancy in L2. Then use the STATPLOT feature to verify that the scatterplot follows a logarithmic pattern as shown in Figure 6.59: Figure 6.59 10. Source
: Center for Disease Control and Prevention, 2013 780 Chapter 6 Exponential and Logarithmic Functions Use the “LnReg” command from the STAT then CALC menu to obtain the logarithmic model, y = 42.52722583 + 13.85752327ln(x) Next, graph the model in the same window as the scatterplot to verify it is a good fit as shown in Figure 6.60: Figure 6.60 b. To predict the life expectancy of an American in the year 2030, substitute x = 14 for the in the model and solve for y : y = 42.52722583 + 13.85752327ln(x) Use the regression model found in part (a). = 42.52722583 + 13.85752327ln(14) Substitute 14 for x. ≈ 79.1 Round to the nearest tenth. If life expectancy continues to increase at this pace, the average life expectancy of an American will be 79.1 by the year 2030. This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 6 Exponential and Logarithmic Functions 781 6.73 Sales of a video game released in the year 2000 took off at first, but then steadily slowed as time moved on. Table 6.22 shows the number of games sold, in thousands, from the years 2000–2010. Year 2000 2001 2002 2003 2004 2005 Number Sold (thousands) 142 149 154 155 159 161 Year 2006 2007 2008 2009 2010 Number Sold (thousands) 163 164 164 166 167 - - Table 6.22 a. Let x represent time in years starting with x = 1 for the year 2000. Let y represent the number of games sold in thousands. Use logarithmic regression to fit a model to these data. b. If games continue to sell at this rate, how many games will sell in 2015? Round to the nearest thousand. Building a Logistic Model from Data Like exponential and logarithmic growth, logistic growth increases over time. One of the most notable differences with logistic growth models is that, at a certain point, growth steadily slows and the function approaches an upper bound, or limiting value. Because of this, logistic regression is best for modeling phenomena where there are limits in expansion, such as availability of living space or nutrients. It is worth pointing out that logistic functions actually model resource-limited exponential growth
. There are many examples of this type of growth in real-world situations, including population growth and spread of disease, rumors, and even stains in fabric. When performing logistic regression analysis, we use the form most commonly used on graphing utilities: y = c 1 + ae−bx Recall that: c 1 + a • is the initial value of the model. • when b > 0, the model increases rapidly at first until it reaches its point of maximum growth rate, ⎛ ⎝ ln(a) b, ⎞ ⎠. At c 2 that point, growth steadily slows and the function becomes asymptotic to the upper bound y = c. • c is the limiting value, sometimes called the carrying capacity, of the model. Logistic Regression Logistic regression is used to model situations where growth accelerates rapidly at first and then steadily slows to an upper limit. We use the command “Logistic” on a graphing utility to fit a logistic function to a set of data points. This returns an equation of the form y = c 1 + ae−bx Note that • The initial value of the model is c 1 + a. 782 Chapter 6 Exponential and Logarithmic Functions • Output values for the model grow closer and closer to y = c as time increases. Given a set of data, perform logistic regression using a graphing utility. 1. Use the STAT then EDIT menu to enter given data. a. Clear any existing data from the lists. b. List the input values in the L1 column. c. List the output values in the L2 column. 2. Graph and observe a scatter plot of the data using the STATPLOT feature. a. Use ZOOM [9] to adjust axes to fit the data. b. Verify the data follow a logistic pattern. 3. Find the equation that models the data. a. Select “Logistic” from the STAT then CALC menu. b. Use the values returned for a, b, and c to record the model, y = c 1 + ae−bx. 4. Graph the model in the same window as the scatterplot to verify it is a good fit for the data. Example 6.75 Using Logistic Regression to Fit a Model to Data Mobile telephone service has increased rapidly in America since the mid 1990s. Today, almost all residents have cellular service. Table 6.23 shows
the percentage of Americans with cellular service between the years 1995 and 2012 [11]. 11. Source: The World Bank, 2013 This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 6 Exponential and Logarithmic Functions 783 Year Americans with Cellular Service (%) Year Americans with Cellular Service (%) 1995 12.69 1996 16.35 1997 20.29 1998 25.08 1999 30.81 2000 38.75 2001 45.00 2002 49.16 2003 55.15 Table 6.23 2004 62.852 2005 68.63 2006 76.64 2007 82.47 2008 85.68 2009 89.14 2010 91.86 2011 95.28 2012 98.17 a. Let x represent time in years starting with x = 0 for the year 1995. Let y represent the corresponding percentage of residents with cellular service. Use logistic regression to fit a model to these data. b. Use the model to calculate the percentage of Americans with cell service in the year 2013. Round to the nearest tenth of a percent. c. Discuss the value returned for the upper limit, c. What does this tell you about the model? What would the limiting value be if the model were exact? Solution a. Using the STAT then EDIT menu on a graphing utility, list the years using values 0–15 in L1 and the corresponding percentage in L2. Then use the STATPLOT feature to verify that the scatterplot follows a logistic pattern as shown in Figure 6.61: 784 Chapter 6 Exponential and Logarithmic Functions Figure 6.61 Use the “Logistic” command from the STAT then CALC menu to obtain the logistic model, y = 105.7379526 1 + 6.88328979e−0.2595440013x Next, graph the model in the same window as shown in Figure 6.62 the scatterplot to verify it is a good fit: Figure 6.62 b. To approximate the percentage of Americans with cellular service in the year 2013, substitute x = 18 for the in the model and solve for y : This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 6 Exponential and Logarithmic Functions 785 y = = 105.7379526 1 + 6.88328979e−0.2595440013x 105.7379526 1 + 6.
88328979e−0.2595440013(18) Use the regression model found in part (a). Substitute 18 for x. ≈ 99.3 Round to the nearest tenth According to the model, about 98.8% of Americans had cellular service in 2013. c. The model gives a limiting value of about 105. This means that the maximum possible percentage of Americans with cellular service would be 105%, which is impossible. (How could over 100% of a population have cellular service?) If the model were exact, the limiting value would be c = 100 and the model’s outputs would get very close to, but never actually reach 100%. After all, there will always be someone out there without cellular service! Table 6.24 shows the population, in thousands, of harbor seals in the Wadden Sea over the years 1997 to 6.74 2012. Year Seal Population (Thousands) Year Seal Population (Thousands) 1997 3.493 2005 19.590 1998 5.282 2006 21.955 1999 6.357 2007 22.862 2000 9.201 2008 23.869 2001 11.224 2009 24.243 2002 12.964 2010 24.344 2003 16.226 2011 24.919 2004 18.137 2012 25.108 Table 6.24 a. Let x represent time in years starting with x = 0 for the year 1997. Let y represent the number of seals in thousands. Use logistic regression to fit a model to these data. b. Use the model to predict the seal population for the year 2020. c. To the nearest whole number, what is the limiting value of this model? Access this online resource for additional instruction and practice with exponential function models. • Exponential Regression on a Calculator (http://openstaxcollege.org/l/pregresscalc) 786 Chapter 6 Exponential and Logarithmic Functions this website (http://openstaxcollege.org/l/PreCalcLPC04) Visit Learningpod. for additional practice questions from This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 6 Exponential and Logarithmic Functions 787 6.8 EXERCISES Verbal What situations are best modeled by a logistic 426. equation? Give an example, and state a case for why the example is a good fit. What is a carrying capacity? What kind of model has 427. a carrying capacity built into
its formula? Why does this make sense? What is regression analysis? Describe the process of 428. performing regression analysis on a graphing utility. What might a scatterplot of data points look like if it 429. were best described by a logarithmic model? What does the y-intercept on the graph of a logistic 430. equation correspond to for a population modeled by that equation? Graphical For the following exercises, match the given function of best fit with the appropriate scatterplot in Figure 6.63 through Figure 6.67. Answer using the letter beneath the matching graph. Figure 6.64 Figure 6.63 Figure 6.65 788 Chapter 6 Exponential and Logarithmic Functions x as Rewrite the exponential model A(t) = 1550(1.085) an equivalent model with base e. Express the exponent to four significant digits. A logarithmic model is given by the equation 438. h(p) = 67.682 − 5.792ln(p). To the nearest hundredth, for what value of p does h(p) = 62? 439. P(t) = A logistic model 90 1 + 5e−0.42t. To the nearest hundredth, for what is given by the equation Figure 6.66 Figure 6.67 431. y = 10.209e−0.294x 432. y = 5.598 − 1.912ln(x) 433. y = 2.104(1.479) x 434. y = 4.607 + 2.733ln(x) 435. y = 14.005 1 + 2.79e−0.812x Numeric To the nearest whole number, what is the initial value 436. of a population modeled by the logistic equation P(t) = 1 + 6.995e−0.68t? What is the carrying capacity? 175 437. This content is available for free at https://cnx.org/content/col11758/1.5 value of t does P(t) = 45? What is the y-intercept on the graph of the logistic 440. model given in the previous exercise? Technology the following exercises, use this scenario: The For population P of a koi pond over x months is modeled by 68 1 + 16e−0.28x. the function P(x) = Graph the population model to show the population 441
. over a span of 3 years. 442. What was the initial population of koi? How many koi will the pond have after one and a half 443. years? How many months will it take before there are 20 koi 444. in the pond? Use the intersect feature to approximate the number 445. of months it will take before the population of the pond reaches half its carrying capacity. For the following exercises, use this scenario: The population P of an endangered species habitat for wolves 558 1 + 54.8e−0.462x, is modeled by the function P(x) = where x is given in years. Graph the population model to show the population 446. over a span of 10 years. What was the initial population of wolves transported 447. to the habitat? How many wolves will 448. years? the habitat have after 3 How many years will it take before there are 100 449. wolves in the habitat? 450. Chapter 6 Exponential and Logarithmic Functions 789 Use the intersect feature to approximate the number of years it will take before the population of the habitat reaches half its carrying capacity. For the following exercises, refer to Table 6.25. x 1 2 3 4 5 6 f(x) 1125 1495 2310 3294 4650 6361 Table 6.25 Use a graphing calculator to create a scatter diagram 451. of the data. Use the regression feature to find an exponential 452. function that best fits the data in the table. Write the exponential function as an exponential 453. equation with base e. x 1 2 3 4 5 6 f(x) 555 383 307 210 158 122 Table 6.26 Use a graphing calculator to create a scatter diagram 456. of the data. Use the regression feature to find an exponential 457. function that best fits the data in the table. Write the exponential function as an exponential 458. equation with base e. Graph the exponential equation on the scatter 459. diagram. Use the intersect feature to find the value of x for 460. which f (x) = 250. Graph the exponential equation on the scatter 454. diagram. For the following exercises, refer to Table 6.27. Use the intersect feature to find the value of x for 455. which f (x) = 4000. For the following exercises, refer to Table 6.26. 790 Chapter 6 Exponential and Logarithmic Functions x 1 2 3 4 5 6 f(x)
5.1 6.3 7.3 7.7 8.1 8.6 Table 6.27 Use a graphing calculator to create a scatter diagram 461. of the data. Use the LOGarithm option of the REGression feature 462. to find a logarithmic function of the form y = a + bln(x) that best fits the data in the table. Use the logarithmic function to find the value of the 463. function when x = 10. Graph the logarithmic equation on the scatter 464. diagram(x) 7.5 6 5.2 4.3 3.9 3.4 3.1 2.9 Table 6.28 Use a graphing calculator to create a scatter diagram 466. of the data. 467. Use the LOGarithm option of the REGression feature to find a logarithmic function of the form y = a + bln(x) that best fits the data in the table. Use the intersect feature to find the value of x for 465. which f (x) = 7. Use the logarithmic function to find the value of the 468. function when x = 10. For the following exercises, refer to Table 6.28. Graph the logarithmic equation on the scatter 469. diagram. Use the intersect feature to find the value of x for 470. which f (x) = 8. For the following exercises, refer to Table 6.29. This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 6 Exponential and Logarithmic Functions 791 (x) 8.7 12.3 15.4 18.5 20.7 22.5 23.3 24 24.6 10 24.8 Table 6.29 x f (x) 0 2 4 5 7 8 12 28.6 52.8 70.3 99.9 112.5 10 125.8 11 127.9 15 135.1 17 135.9 Table 6.30 Use a graphing calculator to create a scatter diagram 471. of the data. Use a graphing calculator to create a scatter diagram 476. of the data. 472. logistic growth model of the form y = Use the LOGISTIC regression option to find a that best c 1 + ae−bx 477. logistic growth model of the form y = Use the LOG
ISTIC regression option to find a that best c 1 + ae−bx fits the data in the table. fits the data in the table. 473. Graph the logistic equation on the scatter diagram. 478. Graph the logistic equation on the scatter diagram. To the nearest whole number, what is the predicted 474. carrying capacity of the model? To the nearest whole number, what is the predicted 479. carrying capacity of the model? Use the intersect feature to find the value of x for 475. which the model reaches half its carrying capacity. Use the intersect feature to find the value of x for 480. which the model reaches half its carrying capacity. For the following exercises, refer to Table 6.30. Extensions 481. a population is given by P(t) = Recall that the general form of a logistic equation for such that the c 1 + ae−bt, initial population at time t = 0 is P(0) = P0. Show algebraically that c − P(t) P(t) = c − P0 P0 e−bt. 792 Chapter 6 Exponential and Logarithmic Functions 482. Use a graphing utility to find an exponential regression formula f (x) and a logarithmic regression formula g(x) for the points (1.5, 1.5) and (8.5, 8.5). Round all numbers to 6 decimal places. Graph the points and both formulas along with the line y = x on the same the relationship of the axis. Make a conjecture about regression formulas. Verify the conjecture made in the previous exercise. 483. Round all numbers to six decimal places when necessary. 484. Find the inverse function f −1 (x) for the logistic 1 + ae−bx. Show all steps. c function f (x) = Use the result from the previous exercise to graph the 485. logistic model P(t) = 20 1 + 4e−0.5t on the same axis. What are the intercepts and asymptotes of each function? along with its inverse This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 6 Exponential and Logarithmic Functions 793 CHAPTER 6 REVIEW KEY TERMS annual percentage rate (APR) the yearly interest rate earned by an investment account, also called nominal rate carrying capacity in a logistic
model, the limiting value of the output change-of-base formula a formula for converting a logarithm with any base to a quotient of logarithms with any other base. common logarithm the exponent to which 10 must be raised to get x; log10 (x) is written simply as log(x). compound interest interest earned on the total balance, not just the principal doubling time the time it takes for a quantity to double exponential growth a model that grows by a rate proportional to the amount present extraneous solution a solution introduced while solving an equation that does not satisfy the conditions of the original equation half-life the length of time it takes for a substance to exponentially decay to half of its original quantity logarithm the exponent to which b must be raised to get x; written y = logb (x) logistic growth model a function of the form f (x) = c 1 + ae−bx where c 1 + a is the initial value, c is the carrying capacity, or limiting value, and b is a constant determined by the rate of growth natural logarithm the exponent to which the number e must be raised to get x; loge (x) is written as ln(x). Newton’s Law of Cooling the scientific formula for temperature as a function of time as an object’s temperature is equalized with the ambient temperature nominal rate the yearly interest rate earned by an investment account, also called annual percentage rate order of magnitude the power of ten, when a number is expressed in scientific notation, with one non-zero digit to the left of the decimal power rule for logarithms a rule of logarithms that states that the log of a power is equal to the product of the exponent and the log of its base product rule for logarithms a rule of logarithms that states that the log of a product is equal to a sum of logarithms quotient rule for logarithms a rule of logarithms that states that the log of a quotient is equal to a difference of logarithms KEY EQUATIONS 794 Chapter 6 Exponential and Logarithmic Functions definition of the exponential function f (x) = b x, where b > 0, b ≠ 1 definition of exponential growth f (x) = ab x, where a > 0, b > 0, b ≠ 1 compound interest formula nt ⎝1 + r
n, where ⎞ A(t) = P⎛ ⎠ A(t) is the account value at time t t is the number of years P is the initial investment, often called the principal r is the annual percentage rate (APR), or nominal rate n is the number of compounding periods in one year continuous growth formula, where A(t) = aert t is the number of unit time periods of growth a is the starting amount (in the continuous compounding formula a is replaced with P, the principal) e is the mathematical constant, e ≈ 2.718282 General Form for the Translation of the Parent Function f (x) = b x f (x) = ab x + c + d Definition of the logarithmic function For x > 0, b > 0, b ≠ 1, y = logb (x) if and only if b y = x. Definition of the common logarithm For x > 0, y = log(x) if and only if 10 y = x. Definition of the natural logarithm For x > 0, y = ln(x) if and only if e y = x. General Form for the Translation of the Parent Logarithmic Function f (x) = logb (x) f (x) = alogb (x + c) + d The Product Rule for Logarithms logb(MN) = logb (M) + logb (N) The Quotient Rule for Logarithms logb ⎛ ⎝ M N ⎞ ⎠ = logb M − logb N The Power Rule for Logarithms logb (M n ) = nlogb M The Change-of-Base Formula logb M= logn M logn b n > 0, n ≠ 1, b ≠ 1 This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 6 Exponential and Logarithmic Functions 795 One-to-one property for exponential functions Definition of a logarithm For any algebraic expressions S and T and any positive real number b, where bS if and only if S = T. = bT For any algebraic expression S and positive real numbers b and c, where b ≠ 1, logb(S) = c if and only if bc = S. One-to
-one property for logarithmic functions For any algebraic expressions S and T and any positive real number b, where b ≠ 1, logb S = logb T if and only if S = T. Half-life formula If A = A0 ekt, k < 0, the half-life is t = − ln(2) k. Carbon-14 dating ⎞ ⎠ ⎛ A ln ⎝ A0 −0.000121. t = A0 A is the amount of carbon-14 when the plant or animal died t is the amount of carbon-14 remaining today is the age of the fossil in years Doubling time formula If A = A0 ekt, k > 0, the doubling time is t = ln2 k Newton’s Law of Cooling T(t) = Aekt the continuous rate of cooling. + Ts, where Ts is the ambient temperature, A = T(0) − Ts, and k is KEY CONCEPTS 6.1 Exponential Functions • An exponential function is defined as a function with a positive constant other than 1 raised to a variable exponent. See Example 6.1. • A function is evaluated by solving at a specific value. See Example 6.2 and Example 6.3. • An exponential model can be found when the growth rate and initial value are known. See Example 6.4. • An exponential model can be found when the two data points from the model are known. See Example 6.5. • An exponential model can be found using two data points from the graph of the model. See Example 6.6. • An exponential model can be found using two data points from the graph and a calculator. See Example 6.7. • The value of an account at any time t can be calculated using the compound interest formula when the principal, annual interest rate, and compounding periods are known. See Example 6.8. • The initial investment of an account can be found using the compound interest formula when the value of the account, annual interest rate, compounding periods, and life span of the account are known. See Example 6.9. 796 Chapter 6 Exponential and Logarithmic Functions • The number e is a mathematical constant often used as the base of real world exponential growth and decay models. Its decimal approximation is e ≈ 2.718282. • Scientific and graphing calculators have the key [e x ] or
⎡ ⎣exp(x)⎤ ⎦ for calculating powers of e. See Example 6.10. • Continuous growth or decay models are exponential models that use e as the base. Continuous growth and decay models can be found when the initial value and growth or decay rate are known. See Example 6.11 and Example 6.12. 6.2 Graphs of Exponential Functions • The graph of the function f (x) = b x has a y-intercept at (0, 1), domain (−∞, ∞), range (0, ∞), and horizontal asymptote y = 0. See Example 6.13. • • If b > 1, will increase without bound. the function is increasing. The left tail of the graph will approach the asymptote y = 0, and the right tail If 0 < b < 1, will approach the asymptote y = 0. the function is decreasing. The left tail of the graph will increase without bound, and the right tail • The equation f (x) = b x + d represents a vertical shift of the parent function f (x) = b x. • The equation f (x) = b x + c represents a horizontal shift of the parent function f (x) = b x. See Example 6.14. • Approximate solutions of the equation f (x) = b x + c + d can be found using a graphing calculator. See Example 6.15. • The equation f (x) = ab x, where a > 0, represents a vertical stretch if \|a\| > 1 or compression if 0 < \|a\| < 1 of the parent function f (x) = b x. See Example 6.16. • When the parent function f (x) = b x is multiplied by − 1, the result, f (x) = − b x, is a reflection about the x- axis. When the input is multiplied by − 1, 6.17. the result, f (x) = b−x, is a reflection about the y-axis. See Example • All translations of the exponential function can be summarized by the general equation f (x) = ab x + c + d. See Table 6.9. • Using the general equation f (x) = ab x + c + d, we can write the equation of a function given its description. See Example 6.18. 6.3
Logarithmic Functions • The inverse of an exponential function is a logarithmic function, and the inverse of a logarithmic function is an exponential function. • Logarithmic equations can be written in an equivalent exponential form, using the definition of a logarithm. See Example 6.19. • Exponential equations can be written in their equivalent logarithmic form using the definition of a logarithm See Example 6.20. • Logarithmic functions with base b can be evaluated mentally using previous knowledge of powers of b. See Example 6.21 and Example 6.22. • Common logarithms can be evaluated mentally using previous knowledge of powers of 10. See Example 6.23. • When common logarithms cannot be evaluated mentally, a calculator can be used. See Example 6.24. • Real-world exponential problems with base 10 can be rewritten as a common logarithm and then evaluated using a calculator. See Example 6.25. This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 6 Exponential and Logarithmic Functions 797 • Natural logarithms can be evaluated using a calculator Example 6.26. 6.4 Graphs of Logarithmic Functions • To find the domain of a logarithmic function, set up an inequality showing the argument greater than zero, and solve for x. See Example 6.27 and Example 6.28 • The graph of the parent function f (x) = logb (x) has an x-intercept at (1, 0), domain (0, ∞), range (−∞, ∞), vertical asymptote x = 0, and ◦ ◦ if b > 1, the function is increasing. if 0 < b < 1, the function is decreasing. See Example 6.29. • The equation f (x) = logb (x + c) shifts the parent function y = logb (x) horizontally ◦ ◦ left c units if c > 0. right c units if c < 0. See Example 6.30. • The equation f (x) = logb (x) + d shifts the parent function y = logb (x) vertically ◦ up d units if d > 0. ◦ down d units if d < 0. See Example 6.31. • For any constant a > 0,
the equation f (x) = alogb (x) ◦ stretches the parent function y = logb (x) vertically by a factor of a if \|a\| > 1. ◦ compresses the parent function y = logb (x) vertically by a factor of a if \|a\| < 1. See Example 6.32 and Example 6.33. • When the parent function y = logb (x) is multiplied by − 1, the result is a reflection about the x-axis. When the input is multiplied by − 1, the result is a reflection about the y-axis. ◦ The equation f (x) = − logb (x) represents a reflection of the parent function about the x-axis. ◦ The equation f (x) = logb (−x) represents a reflection of the parent function about the y-axis. See Example 6.34. ◦ A graphing calculator may be used to approximate solutions to some logarithmic equations See Example 6.35. • All translations of the logarithmic function can be summarized by the general equation f (x) = alogb (x + c) + d. See Table 6.15. • Given an equation with the general form f (x) = alogb (x + c) + d, we can identify the vertical asymptote x = − c for the transformation. See Example 6.36. • Using the general equation f (x) = alogb (x + c) + d, we can write the equation of a logarithmic function given its graph. See Example 6.37. 798 Chapter 6 Exponential and Logarithmic Functions 6.5 Logarithmic Properties • We can use the product rule of logarithms to rewrite the log of a product as a sum of logarithms. See Example 6.38. • We can use the quotient rule of logarithms to rewrite the log of a quotient as a difference of logarithms. See Example 6.39. • We can use the power rule for logarithms to rewrite the log of a power as the product of the exponent and the log of its base. See Example 6.40, Example 6.41, and Example 6.42. • We can use the product rule, the quotient rule, and the power rule together to combine or expand a logarithm with a complex input
. See Example 6.43, Example 6.44, and Example 6.45. • The rules of logarithms can also be used to condense sums, differences, and products with the same base as a single logarithm. See Example 6.46, Example 6.47, Example 6.48, and Example 6.49. • We can convert a logarithm with any base to a quotient of logarithms with any other base using the change-of-base formula. See Example 6.50. • The change-of-base formula is often used to rewrite a logarithm with a base other than 10 and e as the quotient of natural or common logs. That way a calculator can be used to evaluate. See Example 6.51. 6.6 Exponential and Logarithmic Equations • We can solve many exponential equations by using the rules of exponents to rewrite each side as a power with the same base. Then we use the fact that exponential functions are one-to-one to set the exponents equal to one another and solve for the unknown. • When we are given an exponential equation where the bases are explicitly shown as being equal, set the exponents equal to one another and solve for the unknown. See Example 6.52. • When we are given an exponential equation where the bases are not explicitly shown as being equal, rewrite each side of the equation as powers of the same base, then set the exponents equal to one another and solve for the unknown. See Example 6.53, Example 6.54, and Example 6.55. • When an exponential equation cannot be rewritten with a common base, solve by taking the logarithm of each side. See Example 6.56. • We can solve exponential equations with base e, by applying the natural logarithm of both sides because exponential and logarithmic functions are inverses of each other. See Example 6.57 and Example 6.58. • After solving an exponential equation, check each solution in the original equation to find and eliminate any extraneous solutions. See Example 6.59. • When given an equation of the form logb(S) = c, where S is an algebraic expression, we can use the definition of = S, and solve for the unknown. See a logarithm to rewrite the equation as the equivalent exponential equation bc Example 6.60 and Example 6.61. • We can also
use graphing to solve equations with the form logb(S) = c. We graph both equations y = logb(S) and y = c on the same coordinate plane and identify the solution as the x-value of the intersecting point. See Example 6.62. • When given an equation of the form logb S = logb T, where S and T are algebraic expressions, we can use the one-to-one property of logarithms to solve the equation S = T for the unknown. See Example 6.63. • Combining the skills learned in this and previous sections, we can solve equations that model real world situations, whether the unknown is in an exponent or in the argument of a logarithm. See Example 6.64. This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 6 Exponential and Logarithmic Functions 799 6.7 Exponential and Logarithmic Models • The basic exponential function is f (x) = ab x. If b > 1, we have exponential growth; if 0 < b < 1, we have exponential decay. • We can also write this formula in terms of continuous growth as A = A0 ekx, where A0 is the starting value. If A0 is positive, then we have exponential growth when k > 0 and exponential decay when k < 0. See Example 6.65. • In general, we solve problems involving exponential growth or decay in two steps. First, we set up a model and use the model to find the parameters. Then we use the formula with these parameters to predict growth and decay. See Example 6.66. • We can find the age, t, of an organic artifact by measuring the amount, k, of carbon-14 remaining in the artifact and using the formula t = ln(k) −0.000121 to solve for t. See Example 6.67. • Given a substance’s doubling time or half-time, we can find a function that represents its exponential growth or decay. See Example 6.68. • We can use Newton’s Law of Cooling to find how long it will take for a cooling object to reach a desired temperature, or to find what temperature an object will be after a given time. See Example 6.69. • We can use logistic growth functions to model real-world situations where the rate of growth changes over
time, such as population growth, spread of disease, and spread of rumors. See Example 6.70. • We can use real-world data gathered over time to observe trends. Knowledge of linear, exponential, logarithmic, and logistic graphs help us to develop models that best fit our data. See Example 6.71. • Any exponential function with the form y = ab x can be rewritten as an equivalent exponential function with the form y = A0 ekx where k = lnb. See Example 6.72. 6.8 Fitting Exponential Models to Data • Exponential regression is used to model situations where growth begins slowly and then accelerates rapidly without bound, or where decay begins rapidly and then slows down to get closer and closer to zero. • We use the command “ExpReg” on a graphing utility to fit function of the form y = ab x to a set of data points. See Example 6.73. • Logarithmic regression is used to model situations where growth or decay accelerates rapidly at first and then slows over time. • We use the command “LnReg” on a graphing utility to fit a function of the form y = a + bln(x) to a set of data points. See Example 6.74. • Logistic regression is used to model situations where growth accelerates rapidly at first and then steadily slows as the function approaches an upper limit. • We use the command “Logistic” on a graphing utility to fit a function of the form y = c 1 + ae−bx to a set of data points. See Example 6.75. CHAPTER 6 REVIEW EXERCISES Exponential Functions t 486. Determine whether the function y = 156(0.825) represents exponential growth, exponential decay, or neither. Explain 487. The population of a herd of deer is represented by the, where t is given in years. To function A(t) = 205(1.13) the nearest whole number, what will the herd population be after 6 years? t 800 Chapter 6 Exponential and Logarithmic Functions 488. Find an exponential equation that passes through the points (2, 2.25) and (5, 60.75). 489. Determine whether Table 6.31 could represent a function that is linear, exponential, or neither. If it appears to be exponential, find a function that passes through the points.
2 3 4 0.9 0.27 0.081 x f(x) 1 3 Table 6.31 490. A retirement account is opened with an initial deposit of $8,500 and earns 8.12% interest compounded monthly. What will the account be worth in 20 years? 491. Hsu-Mei wants to save $5,000 for a down payment on a car. To the nearest dollar, how much will she need to invest in an account now with 7.5% APR, compounded daily, in order to reach her goal in 3 years? 492. Does equation y = 2.294e−0.654t represent continuous growth, continuous decay, or neither? Explain. the Logarithmic Functions 498. Rewrite log17 (4913) = x as exponential equation. an equivalent 499. Rewrite ln(s) = t as an equivalent exponential equation. 500. Rewrite a equation. − 2 5 = b as an equivalent logarithmic deposit 493. Suppose an investment account is opened with an of $10,500 earning 6.25% interest, initial compounded continuously. How much will the account be worth after 25 years? 501. Rewrite e−3.5 = h as an equivalent logarithmic equation. 502. Solve for x log64(x) = 1 3 to exponential form. Graphs of Exponential Functions the 494. Graph function f (x) = 3.5(2) domain and range and give the y-intercept. x. State the 503. Evaluate log5 ⎛ ⎝ 1 125 ⎞ ⎠ without using a calculator. 495. Graph the function f (x and its reflection about the y-axis on the same axes, and give the y-intercept. x is reflected about the y496. The graph of f (x) = 6.5 axis and stretched vertically by a factor of 7. What is the equation of the new function, g(x)? State its y-intercept, domain, and range. 497. The graph below shows transformations of the graph. What is the equation for the transformation? of f (x) = 2 x 504. Evaluate log(0.000001) without using a calculator. 505. Evaluate log(4.005) using a calculator. Round to the nearest thousandth. ⎝e−0.8648⎞ �
�� 506. Evaluate ln ⎠ without using a calculator. ⎛ 3 507. Evaluate ln ⎝ 18 ⎞ using a calculator. Round to the ⎠ nearest thousandth. Graphs of Logarithmic Functions 508. Graph the function g(x) = log(7x + 21) − 4. This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 6 Exponential and Logarithmic Functions 801 509. Graph the function h(x) = 2ln(9 − 3x) + 1. 524. Solve State the domain, vertical asymptote, and end 510. behavior of the function g(x) = ln(4x + 20) − 17. Logarithmic Properties 511. Rewrite ln(7r ⋅ 11st) in expanded form. 512. Rewrite log8 (x) + log8 (5) + log8 (y) + log8 (13) in compact form. 513. Rewrite logm ⎛ ⎝ 67 83 ⎞ ⎠ in expanded form. 514. Rewrite ln(z) − ln(x) − ln(y) in compact form. ⎛ 515. Rewrite ln ⎝ ⎞ as a product. ⎠ 1 x5 516. Rewrite − log y ⎛ ⎝ 1 12 ⎞ ⎠ as a single logarithm. ⎛ r 2 s11 517. Use properties of logarithms to expand log ⎝ t 14 ⎞ ⎠. 518. Use ⎛ ⎝2b b + 1 ln b − 1 properties of logarithms to expand ⎞ ⎠. 519. Condense the expression 5ln(b) + ln(c) + ln(4 − a) 2 to a single logarithm. 520. Condense the expression 3log7 v + 6log7 w − log7 u 3 to a single logarithm. 521. Rewrite log3 (12.75) to base e. 522. Rewrite 512x − 17 = 125 as a logarithm. Then apply the change of base formula to solve for
x using the common log. Round to the nearest thousandth. Exponential and Logarithmic Equations 523. Solve 2163x ⋅ 216 side with a common base. x = 363x + 2 by rewriting each 125 −x − 3 = 53 by rewriting each side with ⎞ ⎠ 1 625 a common base. ⎛ ⎝ 525. Use logarithms to find the exact solution for 7 ⋅ 17−9x − 7 = 49. If there is no solution, write no solution. 526. Use logarithms to find the exact solution for 3e6n − 2 + 1 = − 60. If there is no solution, write no solution. 527. Find the exact solution for 5e3x no solution, write no solution. − 4 = 6. If there is 528. Find the exact solution for 2e5x − 2 − 9 = − 56. If there is no solution, write no solution. 529. Find the exact solution for 52x − 3 = 7 is no solution, write no solution. x + 1. If there 530. Find the exact solution for e2x there is no solution, write no solution. − e x − 110 = 0. If 531. Use the definition of a logarithm to solve. − 5log7 (10n) = 5. 532. 47. Use the definition of a logarithm to find the exact solution for 9 + 6ln(a + 3) = 33. 533. Use the one-to-one property of logarithms to find an exact solution for log8 (7) + log8 (−4x) = log8 (5). If there is no solution, write no solution. 534. Use the one-to-one property of logarithms to find an ⎞ ⎛ ⎝5x2 − 5 ⎠ = ln(56). If there is exact solution for ln(5) + ln no solution, write no solution. 535. The formula for measuring sound intensity in ⎛ decibels D is defined by the equation D = 10log ⎝ ⎞ ⎠, where I is the intensity of the sound in watts per square I I0 meter and I0 = 10−12 is the lowest level of sound that the average person can hear. How many decibels
are emitted from a large orchestra with a sound intensity of 6.3 ⋅ 10−3 watts per square meter? 802 Chapter 6 Exponential and Logarithmic Functions 536. The population of a city is modeled by the equation P(t) = 256, 114e0.25t where t is measured in years. If the city continues to grow at this rate, how many years will it take for the population to reach one million? 546. 537. Find the inverse function f −1 for the exponential function f (x. 538. Find the inverse function f −1 for the logarithmic function f (x) = 0.25 ⋅ log2 ⎞ ⎛ ⎝x3 + 1 ⎠. Exponential and Logarithmic Models For the following exercises, use this scenario: A doctor prescribes 300 milligrams of a therapeutic drug that decays by about 17% each hour. 539. To the nearest minute, what is the half-life of the drug? 540. Write an exponential model representing the amount of the drug remaining in the patient’s system after t hours. Then use the formula to find the amount of the drug that would remain in the patient’s system after 24 hours. Round to the nearest hundredth of a gram. For the following exercises, use this scenario: A soup with an internal temperature of 350° Fahrenheit was taken off the stove to cool in a 71°F room. After fifteen minutes, the internal temperature of the soup was 175°F. 541. Use Newton’s Law of Cooling to write a formula that models this situation. 547. 542. How many minutes will it take the soup to cool to 85°F(x) 3.05 4.42 6.4 9.28 13.46 19.52 28.3 41.04 59.5 10 86.28 For the following exercises, use this scenario: The equation N(t) = models the number of people in 1200 1 + 199e−0.625t a school who have heard a rumor after t days. 543. How many people started the rumor? 544. To the nearest tenth, how many days will it be before the rumor spreads to half the carrying capacity? 545. What is the carrying capacity? For the following exercises, enter the data from each table into a graphing calculator and graph the resulting scatter plots. Determine
whether the data from the table would likely represent a function that is linear, exponential, or logarithmic. This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 6 Exponential and Logarithmic Functions 803 x f(x) 0.5 18.05 model. Then use the appropriate regression feature to find an equation that models the data. When necessary, round values to five decimal places. 551. 1 3 5 7 10 12 13 15 17 20 17 15.33 14.55 14.04 13.5 13.22 13.1 12.88 12.69 12.45 (x) 409.4 260.7 170.4 110.6 74 44.7 32.4 19.5 12.7 10 8.1 548. Find a formula for an exponential equation that goes through the points (−2, 100) and (0, 4). Then express the formula as an equivalent equation with base e. 552. by 250, 000 Fitting Exponential Models to Data 549. What modeled is the carrying capacity for a population equation logistic the P(t) = 1 + 499e−0.45t? What is the initial population for the model? 550. The population of a culture of bacteria is modeled by the logistic equation P(t) = 14, 250 1 + 29e−0.62t, where t is in days. To the nearest tenth, how many days will it take the culture to reach 75% of its carrying capacity? For the following exercises, use a graphing utility to create a scatter diagram of the data given in the table. Observe the shape of the scatter diagram to determine whether the data is best described by an exponential, logarithmic, or logistic 804 Chapter 6 Exponential and Logarithmic Functions x f(x) 0.15 36.21 0.25 28.88 0.5 24.39 0.75 18.28 1 16.5 1.5 12.99 2 9.91 2.25 8.57 2.75 7.23 3 5.99 3.5 4.81 x 0 2 4 5 7 8 f(x) 9 22.6 44.2 62.1 96.9 113.4 10 133.4 11 137.6 15 148.4 17 149.3 553. CHAPTER 6 PRACTICE TEST 554. The population of a
pod of bottlenose dolphins is modeled by the function A(t) = 8(1.17), where t is given in years. To the nearest whole number, what will the pod population be after 3 years? t 558. Graph the function f (x) = 5(0.5)−x and its reflection across the y-axis on the same axes, and give the y-intercept. 555. Find an exponential equation that passes through the points (0, 4) and (2, 9). f (x) = x ⎞ ⎠ ⎛ ⎝ 1 2. What is the equation for the transformation? 559. The graph shows transformations of the graph of 556. Drew wants to save $2,500 to go to the next World Cup. To the nearest dollar, how much will he need to invest in an account now with 6.25% APR, compounding daily, in order to reach his goal in 4 years? 557. An investment account was opened with an initial deposit of $9,600 and earns 7.4% interest, compounded continuously. How much will the account be worth after 15 years? This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 6 Exponential and Logarithmic Functions 805 571. 4ln(c) + ln(d) + Condense ln(a) 3 + the expression ln(b + 3) 3 to a single logarithm. 572. Rewrite 163x − 5 = 1000 as a logarithm. Then apply the change of base formula to solve for x using the natural log. Round to the nearest thousandth. x 573. Solve ⎛ ⎝ ⋅ 1 243 side with a common base. 1 81 ⎞ ⎠ −3x − 1 = ⎞ ⎠ ⎛ ⎝ 1 9 by rewriting each 574. Use logarithms to find the exact solution for − 9e10a − 8 − 5 = − 41. If there is no solution, write no solution. 560. Rewrite log8.5 (614.125) = a as an equivalent exponential equation. 575. Find the exact solution for 10e4x + 2 + 5 = 56. If there is no solution, write no solution. Rewrite e 561. equation.
1 2 = m as an equivalent logarithmic 576. Find the exact solution for − 5e−4x − 1 − 4 = 64. If there is no solution, write no solution. 562. Solve for x by converting the logarithmic equation log 1 7 (x) = 2 to exponential form. 563. Evaluate log(10,000,000) without using a calculator. 564. Evaluate ln(0.716) using a calculator. Round to the nearest thousandth. 565. Graph the function g(x) = log(12 − 6x) + 3. State the domain, vertical asymptote, and end 566. behavior of the function f (x) = log5 (39 − 13x) + 7. 577. Find the exact solution for 2 is no solution, write no solution. x − 3 = 62x − 1. If there 578. Find the exact solution for e2x there is no solution, write no solution. − e x − 72 = 0. If 579. Use the definition of a logarithm to find the exact solution for 4log(2n) − 7 = − 11 580. Use the one-to-one property of logarithms to find ⎛ ⎞ ⎝4x2 − 10 an exact solution for log ⎠ + log(3) = log(51) If there is no solution, write no solution. 581. The formula for measuring sound intensity in 567. Rewrite log(17a ⋅ 2b) as a sum. 568. Rewrite logt (96) − logt (8) in compact form. 569. Rewrite log8 ⎛ ⎜a ⎝ 1 b ⎞ ⎟ as a product. ⎠ properties 570. Use ⎞ ⎛ 3 ⎝y3 z2 ⋅ x − 4 ln ⎠. of logarithm to expand I I0 ⎛ decibels D is defined by the equation D = 10log ⎝ ⎞ ⎠, where I is the intensity of the sound in watts per square meter and I0 = 10−12 is the lowest level of sound that the average person can hear. How many decibels are emitted from a rock concert with a sound intensity of 4.7
⋅ 10−1 watts per square meter? 582. A radiation safety officer is working with 112 grams of a radioactive substance. After 17 days, the sample has decayed to 80 grams. Rounding to five significant digits, write an exponential equation representing this situation. To the nearest day, what is the half-life of this substance? 806 Chapter 6 Exponential and Logarithmic Functions 587. The population of a lake of fish is modeled by the logistic equation P(t) = 16, 120 1 + 25e−0.75t, where t is time in years. To the nearest hundredth, how many years will it take the lake to reach 80% of its carrying capacity? For the following exercises, use a graphing utility to create a scatter diagram of the data given in the table. Observe the shape of the scatter diagram to determine whether the data is best described by an exponential, logarithmic, or logistic model. Then use the appropriate regression feature to find an equation that models the data. When necessary, round values to five decimal places. 588(x) 20 21.6 29.2 36.4 46.6 55.7 72.6 87.1 107.2 10 138.1 589. 583. Write the formula found in the previous exercise as an equivalent equation with base e. Express the exponent to five significant digits. 584. A bottle of soda with a temperature of 71° Fahrenheit was taken off a shelf and placed in a refrigerator with an internal temperature of 35° F. After ten minutes, the soda was 63° F. Use the internal Newton’s Law of Cooling to write a formula that models the this situation. To the nearest degree, what will temperature of the soda be after one hour? temperature of 585. The population of a wildlife habitat is modeled by 360 the equation P(t) = 1 + 6.2e−0.35t, where t is given in years. How many animals were originally transported to the habitat? How many years will it take before the habitat reaches half its capacity? 586. Enter the data from Table 6.32 into a graphing calculator and graph the resulting scatter plot. Determine whether the data from the table would likely represent a function that is linear, exponential, or logarithmic(x) 3 8.55 11.79 14.09 15.88 17.33 18.57 19.64 20.58
10 21.42 Table 6.32 This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 6 Exponential and Logarithmic Functions 807 x 3 4 5 6 7 8 9 f(x) 13.98 17.84 20.01 22.7 24.1 26.15 27.37 10 28.38 11 29.97 12 31.07 13 31.43 590. x 0 f(x) 2.2 0.5 2.9 1 3.9 1.5 4.8 2 3 4 5 6 7 8 6.4 9.3 12.3 15 16.2 17.3 17.9 808 Chapter 6 Exponential and Logarithmic Functions This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 7 The Unit Circle: Sine and Cosine Functions 809 7 \| THE UNIT CIRCLE: SINE AND COSINE FUNCTIONS Figure 7.1 The tide rises and falls at regular, predictable intervals. (credit: Andrea Schaffer, Flickr) Chapter Outline 7.1 Angles 7.2 Right Triangle Trigonometry 7.3 Unit Circle 7.4 The Other Trigonometric Functions Introduction Life is dense with phenomena that repeat in regular intervals. Each day, for example, the tides rise and fall in response to the gravitational pull of the moon. Similarly, the progression from day to night occurs as a result of Earth’s rotation, and the pattern of the seasons repeats in response to Earth’s revolution around the sun. Outside of nature, many stocks that mirror a company’s profits are influenced by changes in the economic business cycle. In mathematics, a function that repeats its values in regular intervals is known as a periodic function. The graphs of such functions show a general shape reflective of a pattern that keeps repeating. This means the graph of the function has the same output at exactly the same place in every cycle. And this translates to all the cycles of the function having exactly the same length. So, if we know all the details of one full cycle of a true periodic function, then we know the state of the function’s outputs at all times, future and past. In this chapter, we will investigate various examples of periodic functions. 810 Chapter 7 The Unit Circle: Sine and Cosine Functions 7.1 \| Angles In this section you will: Learning
Objectives 7.1.1 Draw angles in standard position. 7.1.2 Convert between degrees and radians. 7.1.3 Find coterminal angles. 7.1.4 Find the length of a circular arc. 7.1.5 Use linear and angular speed to describe motion on a circular path. A golfer swings to hit a ball over a sand trap and onto the green. An airline pilot maneuvers a plane toward a narrow runway. A dress designer creates the latest fashion. What do they all have in common? They all work with angles, and so do all of us at one time or another. Sometimes we need to measure angles exactly with instruments. Other times we estimate them or judge them by eye. Either way, the proper angle can make the difference between success and failure in many undertakings. In this section, we will examine properties of angles. Drawing Angles in Standard Position Properly defining an angle first requires that we define a ray. A ray is a directed line segment. It consists of one point on a line and all points extending in one direction from that point. The first point is called the endpoint of the ray. We can refer to a specific ray by stating its endpoint and any other point on it. The ray in Figure 7.2 can be named as ray EF, or in symbol ⟶ form EF. Figure 7.2 An angle is the union of two rays having a common endpoint. The endpoint is called the vertex of the angle, and the two ⟶. Angles can be named using a point ⟶ and EF rays are the sides of the angle. The angle in Figure 7.3 is formed from ED on each ray and the vertex, such as angle DEF, or in symbol form ∠ DEF. Figure 7.3 Greek letters are often used as variables for the measure of an angle. Table 7.1 is a list of Greek letters commonly used to represent angles, and a sample angle is shown in Figure 7.4. This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 7 The Unit Circle: Sine and Cosine Functions 811 θ φ or ϕ α β γ theta phi alpha beta gamma Table 7.1 Figure 7.4 Angle theta, shown as ∠ θ Angle creation is a dynamic process. We start with two rays lying on top of one another. We leave one fixed in place, and
rotate the other. The fixed ray is the initial side, and the rotated ray is the terminal side. In order to identify the different sides, we indicate the rotation with a small arrow close to the vertex as in Figure 7.5. Figure 7.5 As we discussed at the beginning of the section, there are many applications for angles, but in order to use them correctly, we must be able to measure them. The measure of an angle is the amount of rotation from the initial side to the terminal side. Probably the most familiar unit of angle measurement is the degree. One degree is 1 360 of a circular rotation, so a complete circular rotation contains 360 degrees. An angle measured in degrees should always include the unit “degrees” after the number, or include the degree symbol °. For example, 90 degrees = 90°. To formalize our work, we will begin by drawing angles on an x-y coordinate plane. Angles can occur in any position on the coordinate plane, but for the purpose of comparison, the convention is to illustrate them in the same position whenever possible. An angle is in standard position if its vertex is located at the origin, and its initial side extends along the positive x-axis. See Figure 7.6. 812 Chapter 7 The Unit Circle: Sine and Cosine Functions Figure 7.6 If the angle is measured in a counterclockwise direction from the initial side to the terminal side, the angle is said to be a positive angle. If the angle is measured in a clockwise direction, the angle is said to be a negative angle. Drawing an angle in standard position always starts the same way—draw the initial side along the positive x-axis. To place the terminal side of the angle, we must calculate the fraction of a full rotation the angle represents. We do that by dividing the angle measure in degrees by 360°. For example, to draw a 90° angle, we calculate that 90°. So, the terminal side 360° = 1 4 will be one-fourth of the way around the circle, moving counterclockwise from the positive x-axis. To draw a 360° angle, we calculate that 360° = 1. So the terminal side will be 1 complete rotation around the circle, moving counterclockwise 360° from the positive x-axis. In this case, the initial side and the terminal side overlap. See Figure 7.7. Figure 7.7 Since we define an angle in standard position by its terminal side, we
have a special type of angle whose terminal side lies on an axis, a quadrantal angle. This type of angle can have a measure of 0°, 90°, 180°, 270°, or 360°. See Figure 7.8. This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 7 The Unit Circle: Sine and Cosine Functions 813 Figure 7.8 Quadrantal angles have a terminal side that lies along an axis. Examples are shown. Quadrantal Angles An angle is a quadrantal angle if its terminal side lies on an axis, including 0°, 90°, 180°, 270°, or 360°. Given an angle measure in degrees, draw the angle in standard position. 1. Express the angle measure as a fraction of 360°. 2. Reduce the fraction to simplest form. 3. Draw an angle that contains that same fraction of the circle, beginning on the positive x-axis and moving counterclockwise for positive angles and clockwise for negative angles. Example 7.1 Drawing an Angle in Standard Position Measured in Degrees a. Sketch an angle of 30° in standard position. b. Sketch an angle of −135° in standard position. Solution a. Divide the angle measure by 360°. To rewrite the fraction in a more familiar fraction, we can recognize that 30° 360° = 1 12 1 12 = 1 3 ⎞ ⎠ ⎛ ⎝ 1 4 One-twelfth equals one-third of a quarter, so by dividing a quarter rotation into thirds, we can sketch a line at 30°, as in Figure 7.9. 814 Chapter 7 The Unit Circle: Sine and Cosine Functions Figure 7.9 b. Divide the angle measure by 360°. In this case, we can recognize that −135° 360 Negative three-eighths is one and one-half times a quarter, so we place a line by moving clockwise one full quarter and one-half of another quarter, as in Figure 7.10. Figure 7.10 7.1 Show an angle of 240° on a circle in standard position. Converting Between Degrees and Radians Dividing a circle into 360 parts is an arbitrary choice, although it creates the familiar degree measurement. We may choose other ways to divide a circle. To find another unit, think of the process of drawing a circle. Imagine that you stop before
the This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 7 The Unit Circle: Sine and Cosine Functions 815 circle is completed. The portion that you drew is referred to as an arc. An arc may be a portion of a full circle, a full circle, or more than a full circle, represented by more than one full rotation. The length of the arc around an entire circle is called the circumference of that circle. The circumference of a circle is C = 2πr. If we divide both sides of this equation by r, we create the ratio of the circumference, which is always 2π, to the radius, regardless of the length of the radius. So the circumference of any circle is 2π ≈ 6.28 times the length of the radius. That means that if we took a string as long as the radius and used it to measure consecutive lengths around the circumference, there would be room for six full string-lengths and a little more than a quarter of a seventh, as shown in Figure 7.11. Figure 7.11 This brings us to our new angle measure. One radian is the measure of a central angle of a circle that intercepts an arc equal in length to the radius of that circle. A central angle is an angle formed at the center of a circle by two radii. Because the total circumference equals 2π times the radius, a full circular rotation is 2π radians. 2π radians = 360° π radians = 360° 1 radian = 180° = 180° 2 π ≈ 57.3° See Figure 7.12. Note that when an angle is described without a specific unit, it refers to radian measure. For example, an angle measure of 3 indicates 3 radians. In fact, radian measure is dimensionless, since it is the quotient of a length (circumference) divided by a length (radius) and the length units cancel. Figure 7.12 The angle t sweeps out a measure of one radian. Note that the length of the intercepted arc is the same as the length of the radius of the circle. 816 Chapter 7 The Unit Circle: Sine and Cosine Functions Relating Arc Lengths to Radius An arc length s is the length of the curve along the arc. Just as the full circumference of a circle always has a constant ratio to the radius, the arc length produced by any given angle also has
a constant relation to the radius, regardless of the length of the radius. This ratio, called the radian measure, is the same regardless of the radius of the circle—it depends only on the angle. This property allows us to define a measure of any angle as the ratio of the arc length s to the radius r. See Figure 7.13. If s = r, then θ = r r = 1 radian. s = rθ s θ = r Figure 7.13 (a) In an angle of 1 radian, the arc length s equals the radius r. (b) An angle of 2 radians has an arc length s = 2r. (c) A full revolution is 2π, or about 6.28 radians. To elaborate on this idea, consider two circles, one with radius 2 and the other with radius 3. Recall the circumference of a circle is C = 2πr, where r is the radius. The smaller circle then has circumference 2π(2) = 4π and the larger has circumference 2π(3) = 6π. Now we draw a 45° angle on the two circles, as in Figure 7.14. Figure 7.14 A 45° angle contains one-eighth of the circumference of a circle, regardless of the radius. Notice what happens if we find the ratio of the arc length divided by the radius of the circle. This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 7 The Unit Circle: Sine and Cosine Functions 817 Smaller circle: Larger circle the angle measures of both circles are the same, even though the arc length and radius differ. Since both ratios are 1 4 π, Radians One radian is the measure of the central angle of a circle such that the length of the arc between the initial side and the terminal side is equal to the radius of the circle. A full revolution (360°) equals 2π radians. A half revolution (180°) is equivalent to π radians. The radian measure of an angle is the ratio of the length of the arc subtended by the angle to the radius of the circle. In other words, if s is the length of an arc of a circle, and r is the radius of the circle, then the central angle containing that arc measures s r radians. In a circle of radius 1, the radian measure corresponds to the length of
the arc. A measure of 1 radian looks to be about 60°. Is that correct? Yes. It is approximately 57.3°. Because 2π radians equals 360°, 1 radian equals 360° 2π ≈ 57.3°. Using Radians Because radian measure is the ratio of two lengths, it is a unitless measure. For example, in Figure 7.13, suppose the radius were 2 inches and the distance along the arc were also 2 inches. When we calculate the radian measure of the angle, the “inches” cancel, and we have a result without units. Therefore, it is not necessary to write the label “radians” after a radian measure, and if we see an angle that is not labeled with “degrees” or the degree symbol, we can assume that it is a radian measure. Considering the most basic case, the unit circle (a circle with radius 1), we know that 1 rotation equals 360 degrees, 360°. We can also track one rotation around a circle by finding the circumference, C = 2πr, and for the unit circle C = 2π. These two different ways to rotate around a circle give us a way to convert from degrees to radians. 1 rotation = 360° = 2π radians 1 rotation = 180° = π radians 2 1 4 rotation = 90° = radians π 2 Identifying Special Angles Measured in Radians In addition to knowing the measurements in degrees and radians of a quarter revolution, a half revolution, and a full revolution, there are other frequently encountered angles in one revolution of a circle with which we should be familiar. It is common to encounter multiples of 30, 45, 60, and 90 degrees. These values are shown in Figure 7.15. Memorizing these angles will be very useful as we study the properties associated with angles. 818 Chapter 7 The Unit Circle: Sine and Cosine Functions Figure 7.15 Commonly encountered angles measured in degrees Now, we can list the corresponding radian values for the common measures of a circle corresponding to those listed in Figure 7.15, which are shown in Figure 7.16. Be sure you can verify each of these measures. Figure 7.16 Commonly encountered angles measured in radians Example 7.2 Finding a Radian Measure Find the radian measure of one-third of a full rotation. Solution This content is available for free at https://cn
x.org/content/col11758/1.5 Chapter 7 The Unit Circle: Sine and Cosine Functions 819 For any circle, the arc length along such a rotation would be one-third of the circumference. We know that So, 1 rotation = 2πr (2πr) s = 1 3 = 2πr 3 The radian measure would be the arc length divided by the radius. radian measure = 2πr 3 r = 2πr 3r = 2π 3 7.2 Find the radian measure of three-fourths of a full rotation. Converting Between Radians and Degrees Because degrees and radians both measure angles, we need to be able to convert between them. We can easily do so using a proportion where θ is the measure of the angle in degrees and θR is the measure of the angle in radians. This proportion shows that the measure of angle θ in degrees divided by 180 equals the measure of angle θ in radians divided by π. Or, phrased another way, degrees is to 180 as radians is to π. θ 180 θ R π = Degrees 180 = Radians π Converting between Radians and Degrees To convert between degrees and radians, use the proportion θR π = θ 180 Example 7.3 Converting Radians to Degrees Convert each radian measure to degrees. a. π 6 b. 3 820 Chapter 7 The Unit Circle: Sine and Cosine Functions Solution Because we are given radians and we want degrees, we should set up a proportion and solve it. a. We use the proportion, substituting the given information. θ 180 θ 180 θR π π 6 π = = θ = 180 6 θ = 30° b. We use the proportion, substituting the given information. θ 180 θ 180 = θ R π = 3 π 3(180) θ = π θ ≈ 172° 7.3 Convert − 3π 4 radians to degrees. Example 7.4 Converting Degrees to Radians Convert 15 degrees to radians. Solution In this example, we start with degrees and want radians, so we again set up a proportion, but we substitute the given information into a different part of the proportion 180 15 180 15π 180 π 12 Analysis Another way to think about this problem is by remembering that
30° = π 6. Because 15° = 1 2 (30°), we can find that 1 2 ⎛ ⎝ π 6 ⎞ ⎠ is π 12. This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 7 The Unit Circle: Sine and Cosine Functions 821 7.4 Convert 126° to radians. Finding Coterminal Angles Converting between degrees and radians can make working with angles easier in some applications. For other applications, we may need another type of conversion. Negative angles and angles greater than a full revolution are more awkward to work with than those in the range of 0° to 360°, or 0 to 2π. It would be convenient to replace those out-of-range angles with a corresponding angle within the range of a single revolution. It is possible for more than one angle to have the same terminal side. Look at Figure 7.17. The angle of 140° is a positive angle, measured counterclockwise. The angle of –220° is a negative angle, measured clockwise. But both angles have the same terminal side. If two angles in standard position have the same terminal side, they are coterminal angles. Every angle greater than 360° or less than 0° is coterminal with an angle between 0° and 360°, and it is often more convenient to find the coterminal angle within the range of 0° to 360° than to work with an angle that is outside that range. Figure 7.17 An angle of 140° and an angle of –220° are coterminal angles. Any angle has infinitely many coterminal angles because each time we add 360° to that angle—or subtract 360° from it—the resulting value has a terminal side in the same location. For example, 100° and 460° are coterminal for this reason, as is −260°. An angle’s reference angle is the measure of the smallest, positive, acute angle t formed by the terminal side of the angle t and the horizontal axis. Thus positive reference angles have terminal sides that lie in the first quadrant and can be used as models for angles in other quadrants. See Figure 7.18 for examples of reference angles for angles in different quadrants. 822 Chapter 7 The Unit Circle: Sine and Cosine Functions Figure 7.18 Cotermin
al and Reference Angles Coterminal angles are two angles in standard position that have the same terminal side. An angle’s reference angle is the size of the smallest acute angle, t′, formed by the terminal side of the angle t and the horizontal axis. Given an angle greater than 360°, find a coterminal angle between 0° and 360° 1. Subtract 360° from the given angle. 2. If the result is still greater than 360°, subtract 360° again till the result is between 0° and 360°. 3. The resulting angle is coterminal with the original angle. Example 7.5 Finding an Angle Coterminal with an Angle of Measure Greater Than 360° Find the least positive angle θ that is coterminal with an angle measuring 800°, where 0° ≤ θ < 360 °. Solution An angle with measure 800° is coterminal with an angle with measure 800 − 360 = 440°, but 440° is still greater than 360°, so we subtract 360° again to find another coterminal angle: 440 − 360 = 80°. The angle θ = 80° is coterminal with 800°. To put it another way, 800° equals 80° plus two full rotations, as shown in Figure 7.19. This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 7 The Unit Circle: Sine and Cosine Functions 823 Figure 7.19 7.5 Find an angle α that is coterminal with an angle measuring 870°, where 0° ≤ α < 360°. Given an angle with measure less than 0°, find a coterminal angle having a measure between 0° and 360°. 1. Add 360° to the given angle. 2. If the result is still less than 0°, add 360° again until the result is between 0° and 360°. 3. The resulting angle is coterminal with the original angle. Example 7.6 Finding an Angle Coterminal with an Angle Measuring Less Than 0° Show the angle with measure −45° on a circle and find a positive coterminal angle α such that 0° ≤ α < 360°. Solution Since 45° is half of 90°, we can start at the positive horizontal axis and measure clockwise half of a 90° angle. Because we can
find coterminal angles by adding or subtracting a full rotation of 360°, we can find a positive coterminal angle here by adding 360°. We can then show the angle on a circle, as in Figure 7.20. −45° + 360° = 315° 824 Chapter 7 The Unit Circle: Sine and Cosine Functions Figure 7.20 7.6 Find an angle β that is coterminal with an angle measuring −300° such that 0° ≤ β < 360°. Finding Coterminal Angles Measured in Radians We can find coterminal angles measured in radians in much the same way as we have found them using degrees. In both cases, we find coterminal angles by adding or subtracting one or more full rotations. Given an angle greater than 2π, find a coterminal angle between 0 and 2π. 1. Subtract 2π from the given angle. 2. If the result is still greater than 2π, subtract 2π again until the result is between 0 and 2π. 3. The resulting angle is coterminal with the original angle. Example 7.7 Finding Coterminal Angles Using Radians Find an angle β that is coterminal with 19π 4, where 0 ≤ β < 2π. Solution When working in degrees, we found coterminal angles by adding or subtracting 360 degrees, a full rotation. Likewise, in radians, we can find coterminal angles by adding or subtracting full rotations of 2π radians: 19π 4 − 2π = 19π 4 = 11π 4 − 8π 4 This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 7 The Unit Circle: Sine and Cosine Functions 825 The angle 11π 4 is coterminal, but not less than 2π, so we subtract another rotation. 11π 4 − 2π = 11π 4 = 3π 4 − 8π 4 The angle 3π 4 is coterminal with 19π 4, as shown in Figure 7.21. Figure 7.21 7.7 Find an angle of measure θ that is coterminal with an angle of measure − 17π 6 where 0 ≤ θ < 2π. Determining the Length of an Arc Recall that the radian measure θ of an angle was defined
as the ratio of the arc length s of a circular arc to the radius r of the circle, θ = s r. From this relationship, we can find arc length along a circle, given an angle. Arc Length on a Circle In a circle of radius r, the length of an arc s subtended by an angle with measure θ in radians, shown in Figure 7.22, is s = rθ (7.1) 826 Chapter 7 The Unit Circle: Sine and Cosine Functions Figure 7.22 Given a circle of radius r, calculate the length s of the arc subtended by a given angle of measure θ. 1. If necessary, convert θ to radians. 2. Multiply the radius r θ : s = rθ. Example 7.8 Finding the Length of an Arc Assume the orbit of Mercury around the sun is a perfect circle. Mercury is approximately 36 million miles from the sun. a. In one Earth day, Mercury completes 0.0114 of its total revolution. How many miles does it travel in one day? b. Use your answer from part (a) to determine the radian measure for Mercury’s movement in one Earth day. Solution a. Let’s begin by finding the circumference of Mercury’s orbit. C = 2πr = 2π(36 million miles) ≈ 226 million miles Since Mercury completes 0.0114 of its total revolution in one Earth day, we can now find the distance traveled. b. Now, we convert to radians. (0.0114)226 million miles = 2.58 million miles radian = arclength radius = 2.58 million miles 36 million miles = 0.0717 7.8 Find the arc length along a circle of radius 10 units subtended by an angle of 215°. This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 7 The Unit Circle: Sine and Cosine Functions 827 Finding the Area of a Sector of a Circle In addition to arc length, we can also use angles to find the area of a sector of a circle. A sector is a region of a circle bounded by two radii and the intercepted arc, like a slice of pizza or pie. Recall that the area of a circle with radius r can be found using the formula A = πr 2. If the two radii form an angle of θ
, measured in radians, then θ 2π angle measure to the measure of a full rotation and is also, therefore, the ratio of the area of the sector to the area of the circle. Thus, the area of a sector is the fraction θ 2π multiplied by the entire area. (Always remember that this formula only is the ratio of the applies if θ is in radians.) Area of sector = = ⎞ ⎠πr 2 ⎛ θ ⎝ 2π θπr 2 2π θr 2 = 1 2 Area of a Sector The area of a sector of a circle with radius r subtended by an angle θ, measured in radians, is A = 1 2 θr 2 (7.2) See Figure 7.23. Figure 7.23 The area of the sector equals half the square of the radius times the central angle measured in radians. Given a circle of radius r, find the area of a sector defined by a given angle θ. 1. If necessary, convert θ to radians. 2. Multiply half the radian measure of θ by the square of the radius r : A = 1 2 θr 2. Example 7.9 Finding the Area of a Sector 828 Chapter 7 The Unit Circle: Sine and Cosine Functions An automatic lawn sprinkler sprays a distance of 20 feet while rotating 30 degrees, as shown in Figure 7.24. What is the area of the sector of grass the sprinkler waters? Figure 7.24 The sprinkler sprays 20 ft within an arc of 30°. Solution First, we need to convert the angle measure into radians. Because 30 degrees is one of our special angles, we already know the equivalent radian measure, but we can also convert: 30 degrees = 30 ⋅ π 180 radians = π 6 The area of the sector is then So the area is about 104.72 ft2. Area = 1 2 ≈ 104.72 π 6 ⎛ ⎝ ⎞ ⎠(20)2 7.9 In central pivot irrigation, a large irrigation pipe on wheels rotates around a center point. A farmer has a central pivot system with a radius of 400 meters. If water restrictions only allow her to water 150 thousand square meters a day, what angle should she set the system to cover? Write the answer in radian measure to two decimal places. Use
Linear and Angular Speed to Describe Motion on a Circular Path In addition to finding the area of a sector, we can use angles to describe the speed of a moving object. An object traveling in a circular path has two types of speed. Linear speed is speed along a straight path and can be determined by the distance it moves along (its displacement) in a given time interval. For instance, if a wheel with radius 5 inches rotates once a second, a point on the edge of the wheel moves a distance equal to the circumference, or 10π inches, every second. So the linear speed of the point is 10π in./s. The equation for linear speed is as follows where v is linear speed, s is displacement, and t is time. v = s t Angular speed results from circular motion and can be determined by the angle through which a point rotates in a given time interval. In other words, angular speed is angular rotation per unit time. So, for instance, if a gear makes a full rotation every 4 seconds, we can calculate its angular speed as 360 degrees 4 seconds in radians per second, rotations per minute, or degrees per hour for example. The equation for angular speed is as follows, where ω (read as omega) is angular speed, θ is the angle traversed, and t is time. = 90 degrees per second. Angular speed can be given ω = θ t This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 7 The Unit Circle: Sine and Cosine Functions 829 Combining the definition of angular speed with the arc length equation, s = rθ, we can find a relationship between angular and linear speeds. The angular speed equation can be solved for θ, giving θ = ωt. Substituting this into the arc length equation gives: Substituting this into the linear speed equation gives: s = rθ = rωt v = s t rωt = t = rω Angular and Linear Speed As a point moves along a circle of radius r, its angular speed, ω, is the angular rotation θ per unit time, t. ω = θ t The linear speed, v, of the point can be found as the distance traveled, arc length s, per unit time, t. v = s t (7.3) (7.4) When the angular speed is measured in radians
per unit time, linear speed and angular speed are related by the equation This equation states that the angular speed in radians, ω, representing the amount of rotation occurring in a unit of time, can be multiplied by the radius r to calculate the total arc length traveled in a unit of time, which is the definition of linear speed. v = rω (7.5) Given the amount of angle rotation and the time elapsed, calculate the angular speed. 1. If necessary, convert the angle measure to radians. 2. Divide the angle in radians by the number of time units elapsed: ω = θ t. 3. The resulting speed will be in radians per time unit. Example 7.10 Finding Angular Speed A water wheel, shown in Figure 7.25, completes 1 rotation every 5 seconds. Find the angular speed in radians per second. 830 Chapter 7 The Unit Circle: Sine and Cosine Functions Figure 7.25 Solution The wheel completes 1 rotation, or passes through an angle of 2π radians in 5 seconds, so the angular speed would be ω = 2π 5 ≈ 1.257 radians per second. An old vinyl record is played on a turntable rotating clockwise at a rate of 45 rotations per minute. Find 7.10 the angular speed in radians per second. Given the radius of a circle, an angle of rotation, and a length of elapsed time, determine the linear speed. 1. Convert the total rotation to radians if necessary. 2. Divide the total rotation in radians by the elapsed time to find the angular speed: apply ω = θ t. 3. Multiply the angular speed by the length of the radius to find the linear speed, expressed in terms of the length unit used for the radius and the time unit used for the elapsed time: apply v = rω. Example 7.11 Finding a Linear Speed A bicycle has wheels 28 inches in diameter. A tachometer determines the wheels are rotating at 180 RPM (revolutions per minute). Find the speed the bicycle is traveling down the road. Solution Here, we have an angular speed and need to find the corresponding linear speed, since the linear speed of the outside of the tires is the speed at which the bicycle travels down the road. This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 7 The Unit Circle: Sine and Cosine Functions 831 We begin
by converting from rotations per minute to radians per minute. It can be helpful to utilize the units to make this conversion: 180 rotations minute ⋅ 2π radians rotation = 360πradians minute Using the formula from above along with the radius of the wheels, we can find the linear speed: ⎛ ⎝360πradians v = (14 inches) minute ⎞ ⎠ = 5040π inches minute Remember that radians are a unitless measure, so it is not necessary to include them. Finally, we may wish to convert this linear speed into a more familiar measurement, like miles per hour. 5040π inches minute ⋅ 1 feet 12 inches ⋅ 1 mile 5280 feet ⋅ 60 minutes 1 hour = 14.99 miles per hour (mph) A satellite is rotating around Earth at 0.25 radian per hour at an altitude of 242 km above Earth. If the 7.11 radius of Earth is 6378 kilometers, find the linear speed of the satellite in kilometers per hour. Access these online resources for additional instruction and practice with angles, arc length, and areas of sectors. • Angles in Standard Position (http://openstaxcollege.org/l/standardpos) • Angle of Rotation (http://openstaxcollege.org/l/angleofrotation) • Coterminal Angles (http://openstaxcollege.org/l/coterminal) • Determining Coterminal Angles (http://openstaxcollege.org/l/detcoterm) • Positive and Negative Coterminal Angles (http://openstaxcollege.org/l/posnegcoterm) • Radian Measure (http://openstaxcollege.org/l/radianmeas) • Coterminal Angles in Radians (http://openstaxcollege.org/l/cotermrad) • Arc Length and Area of a Sector (http://openstaxcollege.org/l/arclength) 832 Chapter 7 The Unit Circle: Sine and Cosine Functions 21. − 4π 3 For the following exercises, refer to Figure 7.26. Round to two decimal places. 7.1 EXERCISES Verbal Draw an angle in standard position. Label the vertex, 1. initial side, and terminal side. Explain why there are an infinite number of angles that 2. are c
oterminal to a certain angle. State what a positive or negative angle signifies, and 3. explain how to draw each. 4. How does radian measure of an angle compare to the degree measure? Include an explanation of 1 radian in your paragraph. 5. Explain the differences between linear speed and angular speed when describing motion along a circular path. Graphical For the following exercises, draw an angle in standard position with the given measure. Figure 7.26 6. 7. 8. 9. 30° 300° −80° 135° 10. −150° 11. 12. 13. 14. 15. 2π 3 7π 4 5π 6 π 2 − π 10 16. 415° 17. −120° 18. −315° 19. 20. 22π 3 − π 6 This content is available for free at https://cnx.org/content/col11758/1.5 22. Find the arc length. 23. Find the area of the sector. For the following exercises, refer to Figure 7.27. Round to two decimal places. Figure 7.27 24. Find the arc length. 25. Find the area of the sector. Algebraic For the following exercises, convert angles in radians to degrees. 26. Chapter 7 The Unit Circle: Sine and Cosine Functions 833 3π 4 27. 28. 29. 30. 31. 32. radians radians π 9 − 5π 4 radians radians π 3 − 7π 3 − 5π 12 radians radians 11π 6 radians For the following exercises, convert angles in degrees to radians. 33. 90° 34. 100° 35. −540° 36. −120° 37. 180° 38. −315° 39. 150° For the following exercises, use the given information to find the length of a circular arc. Round to two decimal places. Find the length of the arc of a circle of radius 12 inches 40. subtended by a central angle of π 4. radians. Find the length of the arc of a circle of radius 5.02 41. miles subtended by the central angle of π 3. Find the length of the arc of a circle of diameter 14 42. meters subtended by the central angle of 5π 6. Find the length of the arc of a circle of radius 10 43. centimeters subtended by the central angle of 50°. Find the length of the arc of a circle
of radius 5 inches 44. subtended by the central angle of 220°. Find the length of the arc of a circle of diameter 12 meters subtended by the central angle is 63°. For the following exercises, use the given information to find the area of the sector. Round to four decimal places. A sector of a circle has a central angle of 45° and a 46. radius 6 cm. A sector of a circle has a central angle of 30° and a 47. radius of 20 cm. A sector of a circle with diameter 10 feet and an angle radians. 48. of π 2 A sector of a circle with radius of 0.7 inches and an 49. angle of π radians. For the following exercises, find the angle between 0° and 360° that is coterminal to the given angle. 50. −40° 51. −110° 52. 700° 53. 1400° For the following exercises, find the angle between 0 and 2π in radians that is coterminal to the given angle. 54. 55. 56. 57. − π 9 10π 3 13π 6 44π 9 Real-World Applications A truck with 32-inch diameter wheels is traveling at 60 58. mi/h. Find the angular speed of the wheels in rad/min. How many revolutions per minute do the wheels make? A bicycle with 24-inch diameter wheels is traveling at 59. 15 mi/h. Find the angular speed of the wheels in rad/min. How many revolutions per minute do the wheels make? A wheel of radius 8 inches is rotating 15°/s. What is the the angular speed in RPM, and the angular 60. linear speed v, speed in rad/s? 45. 61. 834 Chapter 7 The Unit Circle: Sine and Cosine Functions A wheel of radius 14 inches is rotating 0.5 rad/s. What is the linear speed v, the angular speed in RPM, and the angular speed in deg/s? 72. A bicycle has wheels 28 inches in diameter. A tachometer determines that the wheels are rotating at 180 RPM (revolutions per minute). Find the speed the bicycle is travelling down the road. A car travels 3 miles. Its tires make 2640 revolutions. 73. What is the radius of a tire in inches? 74. A wheel on a tractor has a 24-inch diameter. How many revolutions does the wheel make if the tractor travels 4 miles? A CD
has diameter of 120 millimeters. When playing 62. audio, the angular speed varies to keep the linear speed constant where the disc is being read. When reading along the outer edge of the disc, the angular speed is about 200 RPM (revolutions per minute). Find the linear speed. When being burned in a writable CD-R drive, the 63. angular speed of a CD is often much faster than when playing audio, but the angular speed still varies to keep the linear speed constant where the disc is being written. When writing along the outer edge of the disc, the angular speed of one drive is about 4800 RPM (revolutions per minute). Find the linear speed if the CD has diameter of 120 millimeters. A person is standing on the equator of Earth (radius 64. 3960 miles). What are his linear and angular speeds? Find the distance along an arc on the surface of Earth. The radius of Earth is 3960 65. that subtends a central angle of 5 minutes ⎛ ⎝1 minute = 1 60 miles. ⎞ degree ⎠ 66. that subtends a central angle of 7 minutes ⎛ ⎝1 minute = 1 60 miles. ⎞ degree ⎠ Find the distance along an arc on the surface of Earth. The radius of Earth is 3960 67. Consider a clock with an hour hand and minute hand. What is the measure of the angle the minute hand traces in 20 minutes? Extensions 68. Two cities have the same longitude. The latitude of city A is 9.00 degrees north and the latitude of city B is 30.00 degree north. Assume the radius of the earth is 3960 miles. Find the distance between the two cities. 69. A city is located at 40 degrees north latitude. Assume the radius of the earth is 3960 miles and the earth rotates once every 24 hours. Find the linear speed of a person who resides in this city. 70. A city is located at 75 degrees north latitude. Assume the radius of the earth is 3960 miles and the earth rotates once every 24 hours. Find the linear speed of a person who resides in this city. 71. Find the linear speed of the moon if the average distance between the earth and moon is 239,000 miles, assuming the orbit of the moon is circular and requires about 28 days. Express answer in miles per hour. This content is available for free at https://cnx.org/
content/col11758/1.5 Chapter 7 The Unit Circle: Sine and Cosine Functions 835 7.2 \| Right Triangle Trigonometry Learning Objectives In this section you will: 7.2.1 Use right triangles to evaluate trigonometric functions. ⎛ 7.2.2 Find function values for 30° ⎝ π 6 ⎞ ⎛ ⎠, 45° ⎝ π 4 ⎛ ⎞ ⎠, and 60° ⎝ ⎞ ⎠. π 3 7.2.3 Use equal cofunctions of complementary angles. 7.2.4 Use the deﬁnitions of trigonometric functions of any angle. 7.2.5 Use right-triangle trigonometry to solve applied problems. Mt. Everest, which straddles the border between China and Nepal, is the tallest mountain in the world. Measuring its height is no easy task and, in fact, the actual measurement has been a source of controversy for hundreds of years. The measurement process involves the use of triangles and a branch of mathematics known as trigonometry. In this section, we will define a new group of functions known as trigonometric functions, and find out how they can be used to measure heights, such as those of the tallest mountains. Using Right Triangles to Evaluate Trigonometric Functions Figure 7.28 shows a right triangle with a vertical side of length y and a horizontal side has length x. Notice that the triangle is inscribed in a circle of radius 1. Such a circle, with a center at the origin and a radius of 1, is known as a unit circle. Figure 7.28 We can define the trigonometric functions in terms an angle t and the lengths of the sides of the triangle. The adjacent side is the side closest to the angle, x. (Adjacent means “next to.”) The opposite side is the side across from the angle, y. The hypotenuse is the side of the triangle opposite the right angle, 1. These sides are labeled in Figure 7.29. Figure 7.29 The sides of a right triangle in relation to angle t Given a right triangle with an acute angle of t, the first three trigonometric functions are listed. Sine sin t = Cosine cos t = opposite hypotenuse adjacent hypotenuse (7.6) (7.7) 836
Chapter 7 The Unit Circle: Sine and Cosine Functions Tangent tan t = opposite adjacent (7.8) A common mnemonic for remembering these relationships is SohCahToa, formed from the first letters of “Sine is opposite over hypotenuse, Cosine is adjacent over hypotenuse, Tangent is opposite over adjacent.” For the triangle shown in Figure 7.28, we have the following. sin t = cos t = sec t = y 1 x 1 y x Given the side lengths of a right triangle and one of the acute angles, find the sine, cosine, and tangent of that angle. 1. Find the sine as the ratio of the opposite side to the hypotenuse. 2. Find the cosine as the ratio of the adjacent side to the hypotenuse. 3. Find the tangent as the ratio of the opposite side to the adjacent side. Example 7.12 Evaluating a Trigonometric Function of a Right Triangle Given the triangle shown in Figure 7.30, find the value of cos α. Figure 7.30 Solution The side adjacent to the angle is 15, and the hypotenuse of the triangle is 17. cos(α) = adjacent hypotenuse = 15 17 7.12 Given the triangle shown in Figure 7.31, find the value of sin t. Figure 7.31 This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 7 The Unit Circle: Sine and Cosine Functions 837 Reciprocal Functions In addition to sine, cosine, and tangent, there are three more functions. These too are defined in terms of the sides of the triangle. Secant sec t = Cosecant csc t = Cotangent cot t = hypotenuse adjacent hypotenuse opposite adjacent opposite Take another look at these definitions. These functions are the reciprocals of the first three functions. sin t = cos t = tan t = 1 csc t 1 sec t 1 cot t csc t = sec t = cot t = 1 sin t 1 cos t 1 tan t (7.9) (7.10) (7.11) (7.12) When working with right triangles, keep in mind that the same rules apply regardless of the orientation of the triangle. In fact, we can evaluate the six trigonometric functions of either of the two acute angles in the triangle
in Figure 7.32. The side opposite one acute angle is the side adjacent to the other acute angle, and vice versa. Figure 7.32 The side adjacent to one angle is opposite the other angle. Many problems ask for all six trigonometric functions for a given angle in a triangle. A possible strategy to use is to find the sine, cosine, and tangent of the angles first. Then, find the other trigonometric functions easily using the reciprocals. Given the side lengths of a right triangle, evaluate the six trigonometric functions of one of the acute angles. 1. 2. If needed, draw the right triangle and label the angle provided. Identify the angle, the adjacent side, the side opposite the angle, and the hypotenuse of the right triangle. 3. Find the required function: ◦ sine as the ratio of the opposite side to the hypotenuse ◦ cosine as the ratio of the adjacent side to the hypotenuse ◦ ◦ tangent as the ratio of the opposite side to the adjacent side secant as the ratio of the hypotenuse to the adjacent side ◦ cosecant as the ratio of the hypotenuse to the opposite side ◦ cotangent as the ratio of the adjacent side to the opposite side Example 7.13 Evaluating Trigonometric Functions of Angles Not in Standard Position 838 Chapter 7 The Unit Circle: Sine and Cosine Functions Using the triangle shown in Figure 7.33, evaluate sin α, cos α, tan α, sec α, csc α, and cot α. Figure 7.33 Solution sin α = cos α = tan α = sec α = csc α = cot α = opposite α = 4 hypotenuse 5 adjacent to α = 3 hypotenuse 5 opposite α adjacent to α = 4 3 hypotenuse adjacent to α = 5 3 hypotenuse = 5 opposite α 4 adjacent to α opposite α = 3 4 Analysis Another approach would have been to find sine, cosine, and tangent first. Then find their reciprocals to determine the other functions. sec α = 1 cos α = 1 3 5 csc α = 1 csc α = 1 4 5 cot α = 1 tan.13 Using the triangle shown in Figure 7.34,evaluate sin t, cos t, tan t, sec t, csc t, and cot t. Figure 7.34 This content is available for free at https://
cnx.org/content/col11758/1.5 Chapter 7 The Unit Circle: Sine and Cosine Functions 839 Finding Trigonometric Functions of Special Angles Using Side Lengths It is helpful to evaluate the trigonometric functions as they relate to the special angles—multiples of 30°, 60°, and 45°. Remember, however, that when dealing with right triangles, we are limited to angles between 0° and 90°. Suppose we have a 30°, 60°, 90° triangle, which can also be described as a π 6 relation s, s 3, 2s. The sides of a 45°, 45°, 90° triangle, which can also be described as a π 4 triangle. The sides have lengths in the triangle, have lengths π 2 π 3 π 4 π 2,,,, in the relation s, s, 2s. These relations are shown in Figure 7.35. Figure 7.35 Side lengths of special triangles We can then use the ratios of the side lengths to evaluate trigonometric functions of special angles. Given trigonometric functions of a special angle, evaluate using side lengths. 1. Use the side lengths shown in Figure 7.35 for the special angle you wish to evaluate. 2. Use the ratio of side lengths appropriate to the function you wish to evaluate. Example 7.14 Evaluating Trigonometric Functions of Special Angles Using Side Lengths Find the exact value of the trigonometric functions of π 3, using side lengths. Solution 840 Chapter 7 The Unit Circle: Sine and Cosine Functions 2s = 3 2 s 2s = 1 2 s = 3 ⎛ sin ⎝ ⎛ cos ⎝ ⎛ tan ⎝ ⎛ sec ⎝ ⎛ csc ⎝ cot ⎛ ⎝ ⎞ ⎠ = ⎞ ⎠ = ⎞ ⎠ = ⎞ ⎠ = ⎞ ⎠ = ⎞ ⎠ = = = 3s = 3s opp hyp adj hyp opp adj hyp = 2s s adj hyp opp = 2s 3s adj s opp = 3s =.14 Find the exact value of the trigonometric functions of π 4, using side lengths. Using Equal Cofunction of Complements If we look more closely at the relationship between the sine and cosine
of the special angles, we notice a pattern. In a right triangle with angles of π and π 3 6 is also the cosine of π 3, we see that the sine of π 3 is also the cosine of π 6, while the sine of π 6, namely 3 2 namely 1 2,,,. sin sin π 3 π 6 = cos = cos π 6 π 3 = 3s 2s = 3 2 s 2s = 1 2 = See Figure 7.36. Figure 7.36 The sine of π 3 equals the cosine of π 6 and vice versa. This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 7 The Unit Circle: Sine and Cosine Functions 841 This result should not be surprising because, as we see from Figure 7.36, the side opposite the angle of π 3 is also the side adjacent to π 6 ⎛, so sin ⎝ ⎛ ⎞ ⎠ and cos ⎝ π 6 ⎞ ⎠ are also the same ratio using the same two sides, s and 2s. π 3 ⎛ sin ⎝ π 6 ⎞ ⎛ ⎠ are exactly the same ratio of the same two sides, 3s and 2s. Similarly, cos ⎝ ⎞ ⎠ and π 3 The interrelationship between the sines and cosines of π 6 and π 3 also holds for the two acute angles in any right triangle, since in every case, the ratio of the same two sides would constitute the sine of one angle and the cosine of the other. Since the three angles of a triangle add to π, and the right angle is π. That 2 means that a right triangle can be formed with any two angles that add to π 2 the remaining two angles must also add up to π 2 —in other words, any two complementary, angles. So we may state a cofunction identity: If any two angles are complementary, the sine of one is the cosine of the other, and vice versa. This identity is illustrated in Figure 7.37. Figure 7.37 Cofunction identity of sine and cosine of complementary angles Using this identity, we can state without calculating, for instance, that the sine of �
� 12 sine of 5π 12 equals the cosine of π 12 well.. We can also state that if, for a given angle t, cos t = 5 13 equals the cosine of 5π 12 π ⎛ then sin ⎝ 2,, and that the − t⎞ ⎠ = 5 13 as Cofunction Identities The cofunction identities in radians are listed in Table 7.2. ⎛ cos t = sin ⎝ − t⎞ ⎠ π 2 ⎛ sin t = cos ⎝ − t⎞ ⎠ π 2 tan t = cot ⎛ ⎝ π 2 − t⎞ ⎠ ⎛ cot t = tan ⎝ − t⎞ ⎠ π 2 ⎛ sec t = csc ⎝ − t⎞ ⎠ π 2 ⎛ csc t = sec ⎝ − t⎞ ⎠ π 2 Table 7.2 842 Chapter 7 The Unit Circle: Sine and Cosine Functions Given the sine and cosine of an angle, find the sine or cosine of its complement. 1. To find the sine of the complementary angle, find the cosine of the original angle. 2. To find the cosine of the complementary angle, find the sine of the original angle. Example 7.15 Using Cofunction Identities If sin t = 5 12 ⎛, find cos ⎝ − t⎞ ⎠. π 2 Solution According to the cofunction identities for sine and cosine, we have the following. sin t = cos ⎛ ⎝ − t⎞ ⎠ π 2 cos ⎛ ⎝ π 2 − t⎞ ⎠ = 5 12 So 7.15 If csc ⎛ ⎝ π 6 ⎞ ⎠ = 2, find sec ⎛ ⎝ ⎞ ⎠. π 3 Using Trigonometric Functions In previous examples, we evaluated the sine and cosine in triangles where we knew all three sides. But the real power of right-triangle trigonometry emerges when we look at triangles in which we know an angle but do not know all the sides. Given a right triangle, the length
of one side, and the measure of one acute angle, find the remaining sides. 1. For each side, select the trigonometric function that has the unknown side as either the numerator or the denominator. The known side will in turn be the denominator or the numerator. 2. Write an equation setting the function value of the known angle equal to the ratio of the corresponding sides. 3. Using the value of the trigonometric function and the known side length, solve for the missing side length. Example 7.16 Finding Missing Side Lengths Using Trigonometric Ratios Find the unknown sides of the triangle in Figure 7.38. This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 7 The Unit Circle: Sine and Cosine Functions 843 Figure 7.38 Solution We know the angle and the opposite side, so we can use the tangent to find the adjacent side. We rearrange to solve for a. We can use the sine to find the hypotenuse. Again, we rearrange to solve for c. tan(30°) = 7 a a = 7 tan(30°) ≈ 12.1 sin(30°) = 7 c c = 7 sin(30°) ≈ 14 7.16 A right triangle has one angle of π 3 triangle. and a hypotenuse of 20. Find the unknown sides and angle of the Using Right Triangle Trigonometry to Solve Applied Problems Right-triangle trigonometry has many practical applications. For example, the ability to compute the lengths of sides of a triangle makes it possible to find the height of a tall object without climbing to the top or having to extend a tape measure along its height. We do so by measuring a distance from the base of the object to a point on the ground some distance away, where we can look up to the top of the tall object at an angle. The angle of elevation of an object above an observer relative to the observer is the angle between the horizontal and the line from the object to the observer's eye. The right triangle this position creates has sides that represent the unknown height, the measured distance from the base, and the angled line of sight from the ground to the top of the object. Knowing the measured distance to the base of the object and the angle of the line of sight, we can use trigonometric functions to calculate the unknown height. Similarly, we can form a triangle from
the top of a tall object by looking downward. The angle of depression of an object below an observer relative to the observer is the angle between the horizontal and the line from the object to the observer's eye. See Figure 7.39. 844 Chapter 7 The Unit Circle: Sine and Cosine Functions Figure 7.39 Given a tall object, measure its height indirectly. 1. Make a sketch of the problem situation to keep track of known and unknown information. 2. Lay out a measured distance from the base of the object to a point where the top of the object is clearly visible. 3. At the other end of the measured distance, look up to the top of the object. Measure the angle the line of sight makes with the horizontal. 4. Write an equation relating the unknown height, the measured distance, and the tangent of the angle of the line of sight. 5. Solve the equation for the unknown height. Example 7.17 Measuring a Distance Indirectly To find the height of a tree, a person walks to a point 30 feet from the base of the tree. She measures an angle of 57° between a line of sight to the top of the tree and the ground, as shown in Figure 7.40. Find the height of the tree. Figure 7.40 Solution We know that the angle of elevation is 57° and the adjacent side is 30 ft long. The opposite side is the unknown height. The trigonometric function relating the side opposite to an angle and the side adjacent to the angle is the tangent. So we will state our information in terms of the tangent of 57°, letting h be the unknown height. This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 7 The Unit Circle: Sine and Cosine Functions 845 tan θ = tan(57°) = opposite adjacent h 30 Solve for h. h = 30tan(57°) Multiply. h ≈ 46.2 Use a calculator. The tree is approximately 46 feet tall. How long a ladder is needed to reach a windowsill 50 feet above the ground if the ladder rests against the 7.17 building making an angle of 5π 12 with the ground? Round to the nearest foot. Access these online resources for additional instruction and practice with right triangle trigonometry. • Finding Trig Functions on Calculator (http://openstaxcollege.org/l/findtrigcal) • Finding
Trig Functions Using a Right Triangle (http://openstaxcollege.org/l/trigrttri) • Relate Trig Functions to Sides of a Right Triangle (http://openstaxcollege.org/l/reltrigtri) • Determine Six Trig Functions from a Triangle (http://openstaxcollege.org/l/sixtrigfunc) • Determine Length of Right Triangle Side (http://openstaxcollege.org/l/rttriside) 846 Chapter 7 The Unit Circle: Sine and Cosine Functions 7.2 EXERCISES Verbal For the given right triangle, label the adjacent side, 75. opposite side, and hypotenuse for the indicated angle. sin B = 1 3, a = 2 89. a = 5, ∡ A = 60° 90. c = 12, ∡ A = 45° Graphical For the following exercises, use Figure 7.41 to evaluate each trigonometric function of angle A. When a right triangle with a hypotenuse of 1 is placed the triangle radius 1, which sides of 76. in a circle of correspond to the x- and y-coordinates? The tangent of an angle compares which sides of the 77. right triangle? Figure 7.41 91. sin A 92. cos A 93. tan A 94. csc A 95. sec A 96. cot A For the following exercises, use Figure 7.42 to evaluate each trigonometric function of angle A. What is the relationship between the two acute angles 78. in a right triangle? 79. Explain the cofunction identity. Algebraic the For complementary angles. following exercises, use cofunctions of 80. 81. 82. 83. cos(34°) = sin(___°) ⎛ cos ⎝ π 3 ⎞ ⎠ = sin(___) csc(21°) = sec(___°) ⎛ tan ⎝ π 4 ⎞ ⎠ = cot(___) For the following exercises, find the lengths of the missing sides if side a is opposite angle A, side b is opposite angle B, and side c is the hypotenuse. 84. 85. 86. cos B = 4 5, a = 10 sin B = 1 2, a = 20 tan A = 5 12, b = 6 87. tan A = 100, b = 100 88. This
content is available for free at https://cnx.org/content/col11758/1.5 Chapter 7 The Unit Circle: Sine and Cosine Functions 847 Figure 7.42 97. sin A 98. cos A 99. tan A 100. csc A 101. sec A 102. cot A For the following exercises, solve for the unknown sides of the given triangle. 103. 104. 105. Technology For the following exercises, use a calculator to find the length of each side to four decimal places. 106. 107. 108. 848 Chapter 7 The Unit Circle: Sine and Cosine Functions 109. 110. 111. b = 15, ∡ B = 15° 112. c = 200, ∡ B = 5° 113. c = 50, ∡ B = 21° 114. a = 30, ∡ A = 27° 115. b = 3.5, ∡ A = 78° Extensions 116. Find x. 117. Find x. This content is available for free at https://cnx.org/content/col11758/1.5 118. Find x. 119. Find x. A radio tower is located 400 feet from a building. 120. From a window in the building, a person determines that the angle of elevation to the top of the tower is 36°, and that the angle of depression to the bottom of the tower is 23°. How tall is the tower? A radio tower is located 325 feet from a building. 121. From a window in the building, a person determines that the angle of elevation to the top of the tower is 43°, and that the angle of depression to the bottom of the tower is 31°. How tall is the tower? 122. Chapter 7 The Unit Circle: Sine and Cosine Functions 849 A 200-foot tall monument is located in the distance. From a window in a building, a person determines that the angle of elevation to the top of the monument is 15°, and that the angle of depression to the bottom of the tower is 2°. How far is the person from the monument? A 400-foot tall monument is located in the distance. 123. From a window in a building, a person determines that the angle of elevation to the top of the monument is 18°, and that the angle of depression to the bottom of the tower is 3°. How far is the person from the monument? 124. There is an antenna on the top of a
building. From a location 300 feet from the base of the building, the angle of elevation to the top of the building is measured to be 40°. From the same location, the angle of elevation to the top of the antenna is measured to be 43°. Find the height of the antenna. 125. There is lightning rod on the top of a building. From a location 500 feet from the base of the building, the angle of elevation to the top of the building is measured to be 36°. From the same location, the angle of elevation to the top of the lightning rod is measured to be 38°. Find the height of the lightning rod. Real-World Applications 126. A 33-ft ladder leans against a building so that the angle between the ground and the ladder is 80°. How high does the ladder reach up the side of the building? 127. A 23-ft ladder leans against a building so that the angle between the ground and the ladder is 80°. How high does the ladder reach up the side of the building? 128. The angle of elevation to the top of a building in New York is found to be 9 degrees from the ground at a distance of 1 mile from the base of the building. Using this information, find the height of the building. 129. The angle of elevation to the top of a building in Seattle is found to be 2 degrees from the ground at a distance of 2 miles from the base of the building. Using this information, find the height of the building. 130. Assuming that a 370-foot tall giant redwood grows vertically, if I walk a certain distance from the tree and measure the angle of elevation to the top of the tree to be 60°, how far from the base of the tree am I? 850 Chapter 7 The Unit Circle: Sine and Cosine Functions 7.3 \| Unit Circle In this section you will: Learning Objectives ⎛ 7.3.1 Find function values for the sine and cosine of 30° or ⎝ π 6 ⎞ ⎛ ⎠, 45° or ⎝ π 4 ⎛ ⎞ ⎠, and 60∘ or ⎝ ⎞ ⎠. π 3 7.3.2 Identify the domain and range of sine and cosine functions. 7.3.3 Find reference angles. 7.3.4 Use reference angles to evaluate trigonometric functions. Figure 7.43
The Singapore Flyer is the world’s tallest Ferris wheel. (credit: ʺVibin JKʺ/Flickr) Looking for a thrill? Then consider a ride on the Singapore Flyer, the world’s tallest Ferris wheel. Located in Singapore, the Ferris wheel soars to a height of 541 feet—a little more than a tenth of a mile! Described as an observation wheel, riders enjoy spectacular views as they travel from the ground to the peak and down again in a repeating pattern. In this section, we will examine this type of revolving motion around a circle. To do so, we need to define the type of circle first, and then place that circle on a coordinate system. Then we can discuss circular motion in terms of the coordinate pairs. Finding Trigonometric Functions Using the Unit Circle We have already defined the trigonometric functions in terms of right triangles. In this section, we will redefine them in terms of the unit circle. Recall this a unit circle is a circle centered at the origin with radius 1, as shown in Figure 7.44. The angle (in radians) that t intercepts forms an arc of length s. Using the formula s = rt, and knowing that r = 1, we see that for a unit circle, s = t. The x- and y-axes divide the coordinate plane into four quarters called quadrants. We label these quadrants to mimic the direction a positive angle would sweep. The four quadrants are labeled I, II, III, and IV. For any angle t, we can label the intersection of the terminal side and the unit circle as by its coordinates, (x, y). The coordinates x and y will be the outputs of the trigonometric functions f (t) = cos t and f (t) = sin t, respectively. This means x = cos t and y = sin t. This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 7 The Unit Circle: Sine and Cosine Functions 851 Figure 7.44 Unit circle where the central angle is t radians Unit Circle A unit circle has a center at (0, 0) and radius 1. In a unit circle, the length of the intercepted arc is equal to the radian measure of the central angle t. Let (x, y) be the endpoint on the unit circle of an arc of arc length s. The
(x, y) coordinates of this point can be described as functions of the angle. Defining Sine and Cosine Functions from the Unit Circle The sine function relates a real number t to the y-coordinate of the point where the corresponding angle intercepts the unit circle. More precisely, the sine of an angle t equals the y-value of the endpoint on the unit circle of an arc of length t. In Figure 7.44, the sine is equal to y. Like all functions, the sine function has an input and an output. Its input is the measure of the angle; its output is the y-coordinate of the corresponding point on the unit circle. The cosine function of an angle t equals the x-value of the endpoint on the unit circle of an arc of length t. In Figure 7.45, the cosine is equal to x. Figure 7.45 Because it is understood that sine and cosine are functions, we do not always need to write them with parentheses: sin t is the same as sin(t) and cost is the same as cos(t). Likewise, cos2 t is a commonly used shorthand notation for (cos(t))2. Be aware that many calculators and computers do not recognize the shorthand notation. When in doubt, use the extra parentheses when entering calculations into a calculator or computer. 852 Chapter 7 The Unit Circle: Sine and Cosine Functions Sine and Cosine Functions If t is a real number and a point (x, y) on the unit circle corresponds to a central angle t, then cos t = x sin t = y (7.13) (7.14) Given a point P (x, y) on the unit circle corresponding to an angle of t, find the sine and cosine. 1. The sine of t is equal to the y-coordinate of point P : sin t = y. 2. The cosine of t is equal to the x-coordinate of point P : cos t = x. Example 7.18 Finding Function Values for Sine and Cosine Point P is a point on the unit circle corresponding to an angle of t, as shown in Figure 7.46. Find cos(t) and sin(t). Figure 7.46 Solution We know that cos t is the x-coordinate of the corresponding point on the unit circle and sin t is the y-coordinate of the corresponding point on the unit circle.
So: x = cos t = 1 2 y = sin t = 3 2 This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 7 The Unit Circle: Sine and Cosine Functions 853 7.18 A certain angle t corresponds to a point on the unit circle at ⎛ ⎝− 2 2, 2 2 ⎞ as shown in Figure 7.47. Find ⎠ cos t and sin t. Figure 7.47 Finding Sines and Cosines of Angles on an Axis For quadrantral angles, the corresponding point on the unit circle falls on the x- or y-axis. In that case, we can easily calculate cosine and sine from the values of x and y. Example 7.19 Calculating Sines and Cosines along an Axis Find cos(90°) and sin(90°). Solution Moving 90° counterclockwise around the unit circle from the positive x-axis brings us to the top of the circle, where the (x, y) coordinates are (0, 1), as shown in Figure 7.48. 854 Chapter 7 The Unit Circle: Sine and Cosine Functions Figure 7.48 We can then use our definitions of cosine and sine. x = cos t = cos(90°) = 0 y = sin t = sin(90°) = 1 The cosine of 90° is 0; the sine of 90° is 1. 7.19 Find cosine and sine of the angle π. The Pythagorean Identity Now that we can define sine and cosine, we will learn how they relate to each other and the unit circle. Recall that the equation for the unit circle is x2 + y2 = 1. Because x = cos t and y = sin t, we can substitute for x and y to get cos2 t + sin2 t = 1. This equation, cos2 t + sin2 t = 1, is known as the Pythagorean Identity. See Figure 7.49. Figure 7.49 We can use the Pythagorean Identity to find the cosine of an angle if we know the sine, or vice versa. However, because the equation yields two solutions, we need additional knowledge of the angle to choose the solution with the correct sign. If we know the quadrant where the angle is, we can easily choose the correct solution. This content is available for free at
ERROR: type should be string, got " https://cnx.org/content/col11758/1.5 Chapter 7 The Unit Circle: Sine and Cosine Functions 855 Pythagorean Identity The Pythagorean Identity states that, for any real number t, cos2 t + sin2 t = 1 (7.15) Given the sine of some angle t and its quadrant location, find the cosine of t. 1. Substitute the known value of sin t into the Pythagorean Identity. 2. Solve for cos t. 3. Choose the solution with the appropriate sign for the x-values in the quadrant where t is located. Example 7.20 Finding a Cosine from a Sine or a Sine from a Cosine If sin(t) = 3 7 and t is in the second quadrant, find cos(t). Solution If we drop a vertical line from the point on the unit circle corresponding to t, we create a right triangle, from which we can see that the Pythagorean Identity is simply one case of the Pythagorean Theorem. See Figure 7.50. Figure 7.50 Substituting the known value for sine into the Pythagorean Identity, cos2(t) + sin2(t) = 1 cos2(t) + 9 49 = 1 cos2(t) = 40 49 cos(t) = ± 40 49 = ± 40 7 = ± 2 10 7 856 Chapter 7 The Unit Circle: Sine and Cosine Functions Because the angle is in the second quadrant, we know the x-value is a negative real number, so the cosine is also negative. cos(t) = − 2 10 7 7.20 If cos(t) = 24 25 and t is in the fourth quadrant, find sin(t). Finding Sines and Cosines of Special Angles We have already learned some properties of the special angles, such as the conversion from radians to degrees, and we found their sines and cosines using right triangles. We can also calculate sines and cosines of the special angles using the Pythagorean Identity. Finding Sines and Cosines of 45° Angles First, we will look at angles of 45° or π 4 so the x- and y-coordinates of the corresponding point on the circle are the same. Because the x- and y-values are the same, the sine and cosine values will also be equal., as shown in Figure 7"
.51. A 45° – 45° – 90° triangle is an isosceles triangle, Figure 7.51 At t = π 4, which is 45 degrees, the radius of the unit circle bisects the first quadrantal angle. This means the radius lies along the line y = x. A unit circle has a radius equal to 1 so the right triangle formed below the line y = x has sides x and y (y = x), and radius = 1. See Figure 7.52. Figure 7.52 This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 7 The Unit Circle: Sine and Cosine Functions 857 From the Pythagorean Theorem we get We can then substitute y = x. Next we combine like terms. And solving for x, we get In quadrant I, x = 1 2. At t = π 4 or 45 degrees, x2 + y2 = 1 x2 + x2 = 1 2x2 = 1 x2 = cos t = 1 2 If we then rationalize the denominators, we get (x, y) = (x, x, sin t = 1 2 2 2 2 2 cos t = 1 2 = 2 2 sin t = 1 2 = 2 2 Therefore, the (x, y) coordinates of a point on a circle of radius 1 at an angle of 45° are ⎛ ⎝ 2 2, 2 2 ⎞ ⎠. Finding Sines and Cosines of 30° and 60° Angles Next, we will find the cosine and sine at an angle of 30°, or π 6 at an angle of 30°, and another at an angle of −30°, as shown in Figure 7.53. If the resulting two right triangles are combined into one large triangle, notice that all three angles of this larger triangle will be 60°, as shown in Figure 7.54.. First, we will draw a triangle inside a circle with one side 858 Chapter 7 The Unit Circle: Sine and Cosine Functions Figure 7.53 Figure 7.54 Because all the angles are equal, the sides are also equal. The vertical line has length 2y, and since the sides are all equal, we can also conclude that r = 2y or y = 1 2 r. Since sin t = y, And since r = 1 in our unit circle, ⎛
sin ⎝ π 6 ⎞ ⎠ = 1 2 r ⎛ sin ⎝ π 6 ⎞ ⎠ = 1 2 (1) = 1 2 Using the Pythagorean Identity, we can find the cosine value. ⎠ + sin2 ⎛ ⎞ ⎝ cos2 ⎛ ⎝ π 6 cos2 ⎛ ⎝ cos2 ⎛ π ⎝ 6 π 6 ⎛ cos ⎝ = Use the square root property. Since y is positive, choose the positive root. The (x, y) coordinates for the point on a circle of radius 1 at an angle of 30° are ⎛ ⎝ 3 2, 1 2 ⎞ ⎠. At t = π 3 (60°), the radius of the unit circle, 1, serves as the hypotenuse of a 30-60-90 degree right triangle, BAD, as shown in Figure 7.55. Angle A has measure 60°. At point B, we draw an angle ABC with measure of 60°. We know the angles in a triangle sum to 180°, so the measure of angle C is also 60°. Now we have an equilateral triangle. Because each side of the equilateral triangle ABC is the same length, and we know one side is the radius of the unit circle, all sides must be of length 1. This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 7 The Unit Circle: Sine and Cosine Functions 859 Figure 7.55 The measure of angle ABD is 30°. Angle ABC is double angle ABD, so its measure is 60°. BD is the perpendicular bisector of AC, so it cuts AC in half. This means that AD is 1 2. Notice that AD is the x-coordinate of the radius, or 1 2 point B, which is at the intersection of the 60° angle and the unit circle. This gives us a triangle BAD with hypotenuse of 1 and side x of length 1 2. From the Pythagorean Theorem, we get Substituting x = 1 2, we get Solving for y, we get x2 + y2 = 1 2 ⎞ ⎠ ⎛ ⎝ 1 2 + y2 = 1 + y2 = 1 1 4 y2
= 1 − 1 4 y2 = 3 4 y = ± 3 2 Since t = At t = π 3 has the terminal side in quadrant I where the y-coordinate is positive, we choose y = 3 2 π 3 (60°), the (x, y) coordinates for the point on a circle of radius 1 at an angle of 60° are ⎛ ⎝ the positive value., 1 2, 3 2 ⎞ ⎠, so we can find the sine and cosine. ⎛ ⎝ ⎞ ⎠ (x, y, sin t = 3 cos t = 1 2 2 We have now found the cosine and sine values for all of the most commonly encountered angles in the first quadrant of the unit circle. Table 7.3 summarizes these values. 860 Chapter 7 The Unit Circle: Sine and Cosine Functions, or 60°, or 90° π 2, or 30° π 6, or 45 Angle Cosine Sine Table 7.3 π 3 1 2 3 2 0 1 Figure 7.56 shows the common angles in the first quadrant of the unit circle. Figure 7.56 Using a Calculator to Find Sine and Cosine To find the cosine and sine of angles other than the special angles, we turn to a computer or calculator. Be aware: Most calculators can be set into “degree” or “radian” mode, which tells the calculator the units for the input value. When we evaluate cos(30) on our calculator, it will evaluate it as the cosine of 30 degrees if the calculator is in degree mode, or the cosine of 30 radians if the calculator is in radian mode. Given an angle in radians, use a graphing calculator to find the cosine. 1. If the calculator has degree mode and radian mode, set it to radian mode. 2. Press the COS key. 3. Enter the radian value of the angle and press the close-parentheses key ")". 4. Press ENTER. This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 7 The Unit Circle: Sine and Cosine Functions 861 Example 7.21 Using a Graphing Calculator to Find Sine and Cosine ⎛ Evaluate cos ⎝ 5π 3 ⎞ ⎠ using
a graphing calculator or computer. Solution Enter the following keystrokes: COS( 5 × π ÷ 3 ) ENTER ⎛ cos ⎝ 5π 3 ⎞ ⎠ = 0.5 Analysis We can find the cosine or sine of an angle in degrees directly on a calculator with degree mode. For calculators or software that use only radian mode, we can find the sign of 20°, for example, by including the conversion factor to radians as part of the input: SIN( 20 × π ÷ 180 ) ENTER 7.21 ⎛ Evaluate sin ⎝ ⎞ ⎠. π 3 Identifying the Domain and Range of Sine and Cosine Functions Now that we can find the sine and cosine of an angle, we need to discuss their domains and ranges. What are the domains of the sine and cosine functions? That is, what are the smallest and largest numbers that can be inputs of the functions? Because angles smaller than 0 and angles larger than 2π can still be graphed on the unit circle and have real values of x, y, and r, there is no lower or upper limit to the angles that can be inputs to the sine and cosine functions. The input to the sine and cosine functions is the rotation from the positive x-axis, and that may be any real number. What are the ranges of the sine and cosine functions? What are the least and greatest possible values for their output? We can see the answers by examining the unit circle, as shown in Figure 7.57. The bounds of the x-coordinate are [−1, 1]. The bounds of the y-coordinate are also [−1, 1]. Therefore, the range of both the sine and cosine functions is [−1, 1]. Figure 7.57 862 Chapter 7 The Unit Circle: Sine and Cosine Functions Finding Reference Angles We have discussed finding the sine and cosine for angles in the first quadrant, but what if our angle is in another quadrant? For any given angle in the first quadrant, there is an angle in the second quadrant with the same sine value. Because the sine value is the y-coordinate on the unit circle, the other angle with the same sine will share the same y-value, but have the opposite x-value. Therefore, its cosine value
will be the opposite of the first angle’s cosine value. Likewise, there will be an angle in the fourth quadrant with the same cosine as the original angle. The angle with the same cosine will share the same x-value but will have the opposite y-value. Therefore, its sine value will be the opposite of the original angle’s sine value. As shown in Figure 7.58, angle α has the same sine value as angle t; the cosine values are opposites. Angle β has the same cosine value as angle t; the sine values are opposites. sin(t) = sin(α) sin(t) = − sin(β) and and cos(t) = − cos(α) cos(t) = cos(β) Figure 7.58 Recall that an angle’s reference angle is the acute angle, t, formed by the terminal side of the angle t and the horizontal axis. A reference angle is always an angle between 0 and 90°, or 0 and π radians. As we can see from Figure 7.59, for 2 any angle in quadrants II, III, or IV, there is a reference angle in quadrant I. Figure 7.59 This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 7 The Unit Circle: Sine and Cosine Functions 863 Given an angle between 0 and 2π, find its reference angle. 1. An angle in the first quadrant is its own reference angle. 2. For an angle in the second or third quadrant, the reference angle is \|π − t\| or \|180° − t\|. 3. For an angle in the fourth quadrant, the reference angle is 2π − t or 360° − t. 4. If an angle is less than 0 or greater than 2π, add or subtract 2π as many times as needed to find an equivalent angle between 0 and 2π. Example 7.22 Finding a Reference Angle Find the reference angle of 225° as shown in Figure 7.60. Figure 7.60 Solution Because 225° is in the third quadrant, the reference angle is \|(180° − 225°)\| = \|−45°\| = 45° 7.22 Find the reference angle of 5π 3. Using Reference Angles Now let’s take a moment to reconsider the Ferris wheel introduced at the
beginning of this section. Suppose a rider snaps a photograph while stopped twenty feet above ground level. The rider then rotates three-quarters of the way around the circle. What is the rider’s new elevation? To answer questions such as this one, we need to evaluate the sine or cosine functions at angles that are greater than 90 degrees or at a negative angle. Reference angles make it possible to evaluate trigonometric functions for angles outside the first quadrant. They can also be used to find (x, y) coordinates for those angles. We will use the reference angle of the angle of rotation combined with the quadrant in which the terminal side of the angle lies. 864 Chapter 7 The Unit Circle: Sine and Cosine Functions Using Reference Angles to Evaluate Trigonometric Functions We can find the cosine and sine of any angle in any quadrant if we know the cosine or sine of its reference angle. The absolute values of the cosine and sine of an angle are the same as those of the reference angle. The sign depends on the quadrant of the original angle. The cosine will be positive or negative depending on the sign of the x-values in that quadrant. The sine will be positive or negative depending on the sign of the y-values in that quadrant. Using Reference Angles to Find Cosine and Sine Angles have cosines and sines with the same absolute value as their reference angles. The sign (positive or negative) can be determined from the quadrant of the angle. Given an angle in standard position, find the reference angle, and the cosine and sine of the original angle. 1. Measure the angle between the terminal side of the given angle and the horizontal axis. That is the reference angle. 2. Determine the values of the cosine and sine of the reference angle. 3. Give the cosine the same sign as the x-values in the quadrant of the original angle. 4. Give the sine the same sign as the y-values in the quadrant of the original angle. Example 7.23 Using Reference Angles to Find Sine and Cosine a. Using a reference angle, find the exact value of cos(150°) and sin(150°). b. Using the reference angle, find cos 5π 4 and sin 5π 4. Solution a. 150° is located in the second quadrant. The angle it makes with the x-axis is
180° − 150° = 30°, so the reference angle is 30°. This tells us that 150° has the same sine and cosine values as 30°, except for the sign. cos(30°) = 3 2 and sin(30°) = 1 2 Since 150° is in the second quadrant, the x-coordinate of the point on the circle is negative, so the cosine value is negative. The y-coordinate is positive, so the sine value is positive. cos(150°) = 3 2 and sin(150°) = 1 2 b. is in the third quadrant. Its reference angle is 5π 5π 4 4 In the third quadrant, both x and y are negative, so: − π = π 4. The cosine and sine of π 4 are both 2 2. cos5π 4 = − 2 2 and sin5π 4 = − 2 2 7.23 a. Use the reference angle of 315° to find cos(315°) and sin(315°). b. Use the reference angle of − ⎛ to find cos ⎝− π 6 ⎛ ⎞ ⎠ and sin ⎝− π 6 ⎞ ⎠. π 6 This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 7 The Unit Circle: Sine and Cosine Functions 865 Using Reference Angles to Find Coordinates Now that we have learned how to find the cosine and sine values for special angles in the first quadrant, we can use symmetry and reference angles to fill in cosine and sine values for the rest of the special angles on the unit circle. They are shown in Figure 7.61. Take time to learn the (x, y) coordinates of all of the major angles in the first quadrant. Figure 7.61 Special angles and coordinates of corresponding points on the unit circle In addition to learning the values for special angles, we can use reference angles to find (x, y) coordinates of any point on the unit circle, using what we know of reference angles along with the identities x = cos t y = sin t First we find the reference angle corresponding to the given angle. Then we take the sine and cosine values of the reference angle, and give them the signs corresponding to the y- and x-values of the quadrant
. Given the angle of a point on a circle and the radius of the circle, find the (x, y) coordinates of the point. 1. Find the reference angle by measuring the smallest angle to the x-axis. 2. Find the cosine and sine of the reference angle. 3. Determine the appropriate signs for x and y in the given quadrant. Example 7.24 Using the Unit Circle to Find Coordinates Find the coordinates of the point on the unit circle at an angle of 7π 6. 866 Chapter 7 The Unit Circle: Sine and Cosine Functions Solution We know that the angle 7π 6 is in the third quadrant. First, let’s find the reference angle by measuring the angle to the x-axis. To find the reference angle of an angle whose terminal side is in quadrant III, we find the difference of the angle and π. Next, we will find the cosine and sine of the reference angle. 7π 6 − π = π 6 ⎛ cos ⎝ π 6 ⎞ ⎠ = 3 2 ⎛ sin ⎝ π 6 ⎞ ⎠ = 1 2 We must determine the appropriate signs for x and y in the given quadrant. Because our original angle is in the third quadrant, where both x and y are negative, both cosine and sine are negative. ⎛ cos ⎝ 7π 6 ⎞ ⎠ = − 3 2 sin(7π) = − 1 2 Now we can calculate the (x, y) coordinates using the identities x = cos θ and y = sin θ. The coordinates of the point are ⎛ ⎝− 3 2, − 1 2 ⎞ on the unit circle. ⎠ 7.24 Find the coordinates of the point on the unit circle at an angle of 5π 3. Access these online resources for additional instruction and practice with sine and cosine functions. • Trigonometric Functions Using the Unit Circle (http://openstaxcollege.org/l/trigunitcir) • Sine and Cosine from the Unit (http://openstaxcollege.org/l/sincosuc) • Sine and Cosine from the Unit Circle and Multiples of Pi Divided by Six (http://openstaxcollege.org/l/sincosmult) • Sine and Cosine
from the Unit Circle and Multiples of Pi Divided by Four (http://openstaxcollege.org/l/sincosmult4) • Trigonometric Functions Using Reference Angles (http://openstaxcollege.org/l/trigrefang) This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 7 The Unit Circle: Sine and Cosine Functions 867 7.3 EXERCISES Verbal 131. Describe the unit circle. What do the x- and y-coordinates of the points on the 132. unit circle represent? Discuss the difference between a coterminal angle 133. and a reference angle. 134. Explain how the cosine of an angle in the second quadrant differs from the cosine of its reference angle in the unit circle. 135. Explain how the sine of an angle in the second quadrant differs from the sine of its reference angle in the unit circle. Algebraic For the following exercises, use the given sign of the sine and cosine functions to find the quadrant in which the terminal point determined by t lies. 136. sin(t) < 0 and cos(t) < 0 137. sin(t) > 0 and cos(t) > 0 138. sin(t) > 0 and cos(t) < 0 139. sin(t) > 0 and cos(t) > 0 For the following exercises, find the exact value of each trigonometric function. 140. 141. 142. 143. 144. 145. 146. sin π 2 sin π 3 cos π 2 cos π 3 sin π 4 cos π 4 sin π 6 147. sin π 148. sin 3π 2 149. cos π 150. cos 0 151. cos π 6 152. sin 0 Numeric For the following exercises, state the reference angle for the given angle. 153. 240° 154. −170° 155. 100° 156. −315° 157. 135° 158. 159. 160. 5π 4 2π 3 5π 6 161. −11π 3 162. 163. −7π 4 −π 8 For the following exercises, find the reference angle, the quadrant of the terminal side, and the sine and cosine of each angle. If the angle is not one of the angles on the unit circle, use a calculator and round to three decimal
places. 164. 225° 165. 300° 166. 320° 167. 135° 168. 210° 169. 120° 868 170. 250° 171. 150° 172. 173. 174. 175. 176. 177. 178. 179. 5π 4 7π 6 5π 3 3π 4 4π 3 2π 3 5π 6 7π 4 Chapter 7 The Unit Circle: Sine and Cosine Functions 188. State the domain of the sine and cosine functions. 189. State the range of the sine and cosine functions. Graphical For the following exercises, use the given point on the unit circle to find the value of the sine and cosine of t. 190. 191. For the following exercises, find the requested value. 180. If cos(t) = 1 7 sin(t). 181. If cos(t) = 2 9 sin(t). and t is in the fourth quadrant, find and t is in the first quadrant, find 182. If sin(t) = 3 8 cos(t). and t is in the second quadrant, find 183. If sin(t) = − 1 4 cos(t). and t is in the third quadrant, find 192. Find the coordinates of the point on a circle with 184. radius 15 corresponding to an angle of 220°. Find the coordinates of the point on a circle with 185. radius 20 corresponding to an angle of 120°. Find the coordinates of the point on a circle with 186. radius 8 corresponding to an angle of 7π 4. Find the coordinates of the point on a circle with 187. radius 16 corresponding to an angle of 5π 9. 193. This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 7 The Unit Circle: Sine and Cosine Functions 869 194. 197. 195. 198. 196. 199. 870 Chapter 7 The Unit Circle: Sine and Cosine Functions 200. 203. 201. 204. 202. 205. This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 7 The Unit Circle: Sine and Cosine Functions 871 206. 209. 207. 208. Technology For the following exercises, use a graphing calculator to evaluate. 210. 211. 212. 213. 214. 215. sin 5π 9 cos 5π 9 sin π 10 cos π
10 sin 3π 4 cos 3π 4 216. sin 98° 217. cos 98° 218. Chapter 7 The Unit Circle: Sine and Cosine Functions 872 cos 310° 219. sin 310° Extensions For the following exercises, evaluate. 220. 221. 222. 223. 224. 225. 226. 227. 228. 229. ⎛ sin ⎝ 11π 3 ⎛ ⎞ ⎠ cos ⎝ −5π 6 ⎞ ⎠ ⎛ sin ⎝ 3π 4 ⎛ ⎞ ⎠ cos ⎝ ⎞ ⎠ 5π 3 ⎛ ⎝− 4π sin 3 ⎞ ⎛ ⎠ cos ⎝ ⎞ ⎠ π 2 ⎛ sin ⎝ −9π 4 ⎛ ⎞ ⎠ cos ⎝ −π 6 ⎞ ⎠ ⎛ sin ⎝ π 6 ⎞ ⎛ ⎠ cos ⎝ −π 3 ⎞ ⎠ ⎛ sin ⎝ 7π 4 ⎛ ⎞ ⎠cos ⎝ −2π 3 ⎞ ⎠ ⎛ cos ⎝ 5π 6 ⎞ ⎛ ⎠ cos ⎝ ⎞ ⎠ 2π 3 ⎛ cos ⎝ −π 3 ⎞ ⎛ ⎠cos ⎝ ⎞ ⎠ π 4 ⎛ sin ⎝ −5π 4 ⎛ ⎞ ⎠ sin ⎝ 11π 6 ⎞ ⎠ ⎛ sin(π)sin ⎝ ⎞ ⎠ π 6 Real-World Applications For the following exercises, use this scenario: A child enters a carousel that takes one minute to revolve once around. The child enters at the point (0, 1), that is, on the due north position. Assume the carousel revolves counter clockwise. What are the coordinates of 230. seconds? the child after 45 What are the coordinates of 231. seconds? the child after 90 What are the coordinates of the child after 125 232. seconds? will When coordinates 233. (0.707, –0.707) if the ride lasts 6 minutes? (There are multiple answers.) child
have the When have 234. (–0.866, –0.5) if the ride lasts 6 minutes? child will the coordinates This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 7 The Unit Circle: Sine and Cosine Functions 873 7.4 \| The Other Trigonometric Functions Learning Objectives In this section you will: 7.4.1 Find exact values of the trigonometric functions secant, cosecant, tangent, and cotangent of π 3, and π 6 π 4., 7.4.2 Use reference angles to evaluate the trigonometric functions secant, cotangent. 7.4.3 Use properties of even and odd trigonometric functions. 7.4.4 Recognize and use fundamental identities. 7.4.5 Evaluate trigonometric functions with a calculator. tangent, and A wheelchair ramp that meets the standards of the Americans with Disabilities Act must make an angle with the ground whose tangent is 1 12 or less, regardless of its length. A tangent represents a ratio, so this means that for every 1 inch of rise, the ramp must have 12 inches of run. Trigonometric functions allow us to specify the shapes and proportions of objects independent of exact dimensions. We have already defined the sine and cosine functions of an angle. Though sine and cosine are the trigonometric functions most often used, there are four others. Together they make up the set of six trigonometric functions. In this section, we will investigate the remaining functions. Finding Exact Values of the Trigonometric Functions Secant, Cosecant, Tangent, and Cotangent We can also define the remaining functions in terms of the unit circle with a point (x, y) corresponding to an angle of t, as shown in Figure 7.62. As with the sine and cosine, we can use the (x, y) coordinates to find the other functions. Figure 7.62 The first function we will define is the tangent. The tangent of an angle is the ratio of the y-value to the x-value of the corresponding point on the unit circle. In Figure 7.62, the tangent of angle t is equal to y x, x ≠ 0. Because the ythe tangent of angle t can also be defined value is equal to the sine
of t, and the x-value is equal to the cosine of t, as sin t cos t, cos t ≠ 0. The tangent function is abbreviated as tan. The remaining three functions can all be expressed as reciprocals of functions we have already defined. • The secant function is the reciprocal of the cosine function. In Figure 7.62, the secant of angle t is equal to cos t = 1 1 x, x ≠ 0. The secant function is abbreviated as sec. • The cotangent function is the reciprocal of the tangent function. In Figure 7.62, the cotangent of angle t is equal to cos t sin t = x y, y ≠ 0. The cotangent function is abbreviated as cot. 874 Chapter 7 The Unit Circle: Sine and Cosine Functions • The cosecant function is the reciprocal of the sine function. In Figure 7.62, the cosecant of angle t is equal to 1 sin t = 1 y, y ≠ 0. The cosecant function is abbreviated as csc. Tangent, Secant, Cosecant, and Cotangent Functions If t is a real number and (x, y) is a point where the terminal side of an angle of t radians intercepts the unit circle, then y x, x ≠ 0 tan t = sec t = 1 x, x ≠ 0 csc t = 1 y, y ≠ 0 x y, y ≠ 0 cot t = Example 7.25 Finding Trigonometric Functions from a Point on the Unit Circle The point ⎛ ⎝− 3 2, 1 2 cot t. ⎞ is on the unit circle, as shown in Figure 7.63. Find sin t, cos t, tan t, sec t, csc t, and ⎠ Figure 7.63 Solution Because we know the (x, y) coordinates of the point on the unit circle indicated by angle t, we can use those coordinates to find the six functions: This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 7 The Unit Circle: Sine and Cosine Functions 875 sin t = y = 1 2 cos t = x = − 3 2 tan t = sec t = 1 csc t = 1 cot ⎛ �
��−.25 The point ⎛ ⎝ sin t, cos t, tan t, sec t, csc t, and cot t., − 2 2 ⎞ is ⎠ 2 2 on the unit circle, as shown in Figure 7.64. Find Figure 7.64 Example 7.26 Finding the Trigonometric Functions of an Angle Find sin t, cos t, tan t, sec t, csc t, and cot t. when t = π 6. Solution We have previously used the properties of equilateral triangles to demonstrate that sin π 6 = 1 2 and cos π 6 = 3 2. We can use these values and the definitions of tangent, secant, cosecant, and cotangent as functions of sine and cosine to find the remaining function values. 876 Chapter 7 The Unit Circle: Sine and Cosine Functions tan π 6 sec π 6 csc π 6 cot π 6 = sin π 6 cos π 6 1 2 3 2 = 1 = cos sin π 6 cos π 6 sin π 6 = = 3 2 1 2 7.26 Find sin t, cos t, tan t, sec t, csc t, and cot t. when t = π 3. Because we know the sine and cosine values for the common first-quadrant angles, we can find the other function values for those angles as well by setting x equal to the cosine and y equal to the sine and then using the definitions of tangent, secant, cosecant, and cotangent. The results are shown in Table 7.4. This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 7 The Unit Circle: Sine and Cosine Functions 877 Angle 0 π 6, or 30° π 4, or 45° π 3, or 60° π 2, or 90° Cosine Sine Tangent Secant 1 0 0 1 Cosecant Undefined Cotangent Undefined Table 7. Undefined Undefined 1 0 Using Reference Angles to Evaluate Tangent, Secant, Cosecant, and Cotangent We can evaluate trigonometric functions of angles outside the first quadrant using reference angles as we have already done with the sine and cosine functions. The procedure is the
same: Find the reference angle formed by the terminal side of the given angle with the horizontal axis. The trigonometric function values for the original angle will be the same as those for the reference angle, except for the positive or negative sign, which is determined by x- and y-values in the original quadrant. Figure 7.65 shows which functions are positive in which quadrant. To help remember which of the six trigonometric functions are positive in each quadrant, we can use the mnemonic phrase “A Smart Trig Class.” Each of the four words in the phrase corresponds to one of the four quadrants, starting with quadrant I and rotating counterclockwise. In quadrant I, which is “A,” all of the six trigonometric functions are positive. In quadrant II, “Smart,” only sine and its reciprocal function, cosecant, are positive. In quadrant III, “Trig,” only tangent and its reciprocal function, cotangent, are positive. Finally, in quadrant IV, “Class,” only cosine and its reciprocal function, secant, are positive. 878 Chapter 7 The Unit Circle: Sine and Cosine Functions Figure 7.65 The trigonometric functions are each listed in the quadrants in which they are positive. Given an angle not in the first quadrant, use reference angles to find all six trigonometric functions. 1. Measure the angle formed by the terminal side of the given angle and the horizontal axis. This is the reference angle. 2. Evaluate the function at the reference angle. 3. Observe the quadrant where the terminal side of the original angle is located. Based on the quadrant, determine whether the output is positive or negative. Example 7.27 Using Reference Angles to Find Trigonometric Functions Use reference angles to find all six trigonometric functions of − 5π 6. Solution The angle between this angle’s terminal side and the x-axis is π 6, so that is the reference angle. Since − 5π 6 is in the third quadrant, where both x and y are negative, cosine, sine, secant, and cosecant will be negative, while tangent and cotangent will be positive. ⎛ ⎝− 5π cos 6 ⎛ ⎝− 5π sec ⎛ �
�− 5π, sin 6 ⎛ ⎝− 5π, csc 2, cot ⎛ 5π, tan ⎝ 6 ⎛ ⎝− 5π.27 Use reference angles to find all six trigonometric functions of − 7π 4. Using Even and Odd Trigonometric Functions To be able to use our six trigonometric functions freely with both positive and negative angle inputs, we should examine how each function treats a negative input. As it turns out, there is an important difference among the functions in this regard. This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 7 The Unit Circle: Sine and Cosine Functions 879 Consider the function f (x) = x2, shown in Figure 7.66. The graph of the function is symmetrical about the y-axis. All along the curve, any two points with opposite x-values have the same function value. This matches the result of calculation: (4)2 = (−4)2, (−5)2 = (5)2, and so on. So f (x) = x2 is an even function, a function such that two inputs that are opposites have the same output. That means f (−x) = f (x). Figure 7.66 The function f (x) = x2 is an even function. Now consider the function f (x) = x3, shown in Figure 7.67. The graph is not symmetrical about the y-axis. All along the graph, any two points with opposite x-values also have opposite y-values. So f (x) = x3 is an odd function, one such that two inputs that are opposites have outputs that are also opposites. That means f (−x) = − f (x). 880 Chapter 7 The Unit Circle: Sine and Cosine Functions Figure 7.67 The function f (x) = x3 is an odd function. We can test whether a trigonometric function is even or odd by drawing a unit circle with a positive and a negative angle, as in Figure 7.68. The sine of the positive angle is y. The sine of the negative angle is −y. The sine function, then, is an odd function. We can test each of the six trigonometric functions in this fashion. The results are shown in Table 7.5. Figure

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