Fantastic post!
An important bit of context for those not familiar with signal processing/Fourier analysis concepts is that Euler's formula relates the complex exponential to sine and cosine: e^ix = cos(x) + i sin(x), and e^ix allows you to think about rotations in 2d dimensions on the complex plane.
The Taylor expansion of the complex exponential explains e^ix = cos(x) + i sin(x) and the alternating odd sine and even cosine in a bit more depth. Steven Brunton has an excellent walkthrough of the derivation here
https://www.youtube.com/watch?v=Rp-smPZLESc&list=PLMrJAkhIeNNQBRslPb7I0yTnES981R8Cg