Why do trig functions appear in Euler’s formula? This was the question I had when I first saw Euler’s formula. This connection between trigonometry and exponents seems so unexpected, especially along with complex numbers. To answer this question, we must journey into the intricate and beautiful mathematical relationship between trig functions, e, and complex numbers. We will look at two different ways to approach this question: one using dynamics, geometry, and the complex plane, and the other using Taylor and Maclaurin series. Both are equally fascinating, and both reach the same, amazing result by using a lot of beautiful math.

0:00 - Intro
0:38 - Unit circle on complex plane approach
7:30 - Taylor and Maclaurin series approach
12:39 - Conclusion

Additional Resources:
Geometric proof of the derivatives of sin(x) and cos(x)
https://ocw.mit.edu/courses/mathemati...

Animations were made by Manim, an open-source python-based animation program by 3Blue1Brown.
https://github.com/3b1b/manim

This video was submitted to 3Blue1Brown's SoME (Summer of Math Exposition).
https://3b1b.co/some1