TITLE: Complex cosine and sine
QUESTION [6 upvotes]: I would like to know what the mapping properties of the complex sine and cosine are. To start with one can say that $\sin(z)$ and $\cos(z)$ are conformal where their derivatives are nonzero, which means $\sin(z)$ preserves angles on $\mathbb{C}$ without $\frac{\pi}{2}+k\pi$ and $\cos(z)$ preserves angles on $\mathbb{C}$ without $\pi k$ for $\cos(z)$, with $k \in \mathbb{Z}$. What else can we say about these mappings? Thx.

REPLY [8 votes]: Lets consider the rectangle $-\pi/2<x<\pi/2$, $-3<y<3$ in the complex $z$-plane:

Its image under the map $\zeta=\sin(z)$ will look like

As you can see, the lines $y=\text{constant}$ are deformed to ellipses which approach circles for $|y|\to\infty$ and the lines $x=\text{constant}$ are mapped to hyperbolae. The angles between the two sets of lines are preserved. If you extend the rectangle in the $y$-direction you'll eventually cover the whole $\zeta$-plane (except for branch cuts from $\zeta=-\infty$ to $\zeta=-1$ and from $\zeta=1$ to $\zeta=\infty$).