Question about Strang's Proof of Euler's Identity

Question

Strang proves Euler's Identity by making the following assumption:

Any complex number $z$ is such that $$z = r(\cos(\theta)+i \sin(\theta))$$ for some choice of $\theta, r$. How would one justify this claim? I is s equivalent to say that $$\mathcal{C} = \{ r(cos(\theta)+i sin(\theta)) \ : r,\theta \in \mathbb{R} \}$$ But this is the claim that $\{\sin, \cos\}$ in some sense span the complex plane.

This is intuitively correct; but how can we prove this assumption more rigerously?

Answer 1

Given any complex number $a+bi$, the numbers $a/\sqrt{a^2+b^2}$ and $b/\sqrt{a^2+b^2}$ are the cosine (resp. the sine) of some angle. This is because the sum of their squares is $=1$. Therefore, $$ a+bi=\sqrt{a^2+b^2}\left(\frac a{\sqrt{a^2+b^2}}+i\frac b{\sqrt{a^2+b^2}}\right)=\sqrt{a^2+b^2}(\cos \phi+i\sin\phi)=r(\cos \phi+i\sin\phi), $$ where $r=\sqrt{a^2+b^2}$.