Where does this equation come from?

Question

Since I study 3 years i ask myself very often where does this equation come from? $$e^{i\theta} = \cos(\theta)+i \sin(\theta)$$ Is it found by series expansion?

Answer 1

This result is commonly shown via Taylor series, as explained in the comments, and is well-known. I'd like to offer a different sort of proof, for those who are interested, that I believe is easier yet less well-known.

Consider the second order linear differential equation $$y''=-y$$ We know the most general solution is: $$y = A\cos{x}+B\sin{x}$$ But $$y = e^{ix}$$ is also a solution, and by existence and uniqueness theorems, that means $$e^{ix} = A\cos{x}+B\sin{x}$$ for some $A,B$. Plugging in $x=0$ for the expression and its first derivative, we see that $A = 1, B = i$.

Thus, $$e^{ix} = \cos{x}+i\sin{x}$$