This episode is concerned with one of the most striking formulas in mathematics. Namely the relationship between sin, cos, and the complex exponential for purely imaginary arguments. To derive this formula, we require a closer look into the complex exponential function. The most important fact that we will derive is that no matter the modulus of the purely imaginary number put into the complex exponential the result will always be on the unit circle. Whether or not we reach all elements of the unit circle in the complex plane while running through all purely imaginary numbers remains to be seen in the next episode.

Picture taken from https://hyrodium.tumblr.com/post/106601751454/euler1