Is restriction of $exp:\mathbb{C} \to \mathbb{C}^*$ to $A = \{ x+iy : x \in R, y \in ]1, 1+2\pi]\} \subset \mathbb{C}$ a bijection?

Question

I have this question:

Is the restriction of exp function $exp:\mathbb{C} \to \mathbb{C}^*$ to $A = \{ x+iy : x \in R, y \in ]1, 1+2\pi]\} \subset \mathbb{C}$ a bijection?

Here's what I tried:

True because :

$e^z = e^x(\cos y + i\sin y)$

$e^x$ is bijective in $\mathbb{R}$

There isn't any number differing of $2k\pi$ in $]1, 1+2\pi]$. So $i\sin y$ is bijective and $\cos x$ is bijective. We can then say that:

$e^z:\mathbb{C} \to \mathbb{C}^*$ is a bijection.

What do you think about this answer?