In my book analysis they argue that $e^{ix}$ covers the whole unit circle as follows.
Suppose that $w = u +iv$, such that $|w| = 1$. Then if $v \geq 0$ we pick $x \in [0,\pi]$ with $\cos x = u$. else if $v < 0$ we pick $x \in [\pi,2\pi]$, with $\cos x = u$.
I really do not see why this works. I know we have to show that there exists $x$ and $y$ such that $\cos x = u$ and $\sin y = v$. Furthermore I know that due to the intermediate value theorem, since cosine is continuous on $[0,\pi]$ that if $u$ is in between $\cos 0=1$ and $\cos \pi = -1$, there exists $x \in ]0,\pi[$ such that $\cos x = u$.