Complex cosine and sine 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.
 A: 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 $\zeta=\infty$).
A: If you want to see pictorially how these maps work it is enough to look at the function $z\mapsto \cosh z=(e^z +e^{-z})/2$, because $\cos z=\cosh(i z)$ and $\sin z=\cos\bigl(z-{\pi\over2}\bigr)$. 
Now $\cosh$ is the composition of the two maps
$\exp:\ z\mapsto w:=e^z$ and $j:\ w\mapsto \zeta:={1\over2}(w+w^{-1})$. Therefore we draw figures for these maps separately.
Presumably you have a mental image of $\exp$: It is periodic with period $2\pi i$, and it maps the strip $|y|\leq\pi$ of the $z$-plane essentially 1:1 onto the $w$-plane whereby lines $y=$const. are mapped onto rays emanating from the origin and the segments $x=$const are mapped onto concentric circles of radius $e^x$.
As for the so-called Joukowski function $j$ a detailed study shows that it maps circles $|w|=$const. onto confocal ellipses with foci $\pm1$ and rays $\arg w=$const. onto arcs of hyperbolae with the same foci. Since $j(w)=j(w^{-1})$ the map $j$ is essentially 2:1: The interior and the exterior of the unit disk in the $w$-plane each map onto the full $\zeta$-plane whereby the unit circle itself is mapped back and forth onto the segment $[{-1},1]$.
The map $j$ is explained in detail and with figures, e.g., in Peter Henrici: Applied and computational complex analysis, Vol. 1, pp. 294–298.
