There is an easy way to show that the function $$\sin:\Bbb C\to\Bbb C,\quad z\mapsto \sin(z)\tag1$$
is surjective? My work so far: observe that
$$\sin(x+iy)=\sin(x)\cosh(y)+i\cos(x)\sinh(y)\tag2$$
Then its clear that $\sin$ is $2\pi$ periodic, as in the real case. Then it seems enough to restrict the study of $\sin$ to the domain $A:=[0,2\pi)\times[0,\infty)$ for $\Bbb C\ni z\equiv x+iy\in\Bbb R+i\Bbb R$.
However I can't see a clever (an elementary) way to show the surjectivity of $\sin|_A$. Some help will be appreciated.
EDIT: clearly $\sin(A)$ is connected because $\sin$ is continuous. Moreover: choosing $x_k=k\pi/2$ (for $k\in\Bbb Z$) we can see that $\Bbb R$ and $i\Bbb R$ are subsets of $\sin(A)$.