TITLE: real and imaginary part in $\sin z$ where z is complex
QUESTION [5 upvotes]: I wanted to know, how can I determine the real and imaginary part in 

$\sin z$ where $z \in \Bbb{C}$?

Well, this is a part of a series of questions comprising the same in

$\log z$ and $\tan^{-1} z$

I was able to solve this but no idea on how to solve for $\sin z$.
Any help appreciated.
Thanks.

REPLY [11 votes]: If $z=x+iy, \sin(x+iy)=\sin x\cos (iy)+\cos x\sin(iy)=\sin x\cosh y+i\cos x\sinh y$  using the relationship between Hyperbolic & Trigonometric ratios