# What are the real and imaginary parts of the complex function?

So, it is asked to find the real and imaginary parts of the specific complex function:

$f(z)=sin(z)+i(3z+2)$ So I use $z$ as $z=x+iy$

everything seemed clear till I met Mr. Sinus:

$u+iv= sin(x+iy)+i(3(x+iy)+2)$

and I don't really know how to seperate the imaginary and real parts of $sin(x+iy)$ argument.

Need hints...

• You probably know the addition theorem for the sine, $\sin (a+b) = \dotsc$. Set $a = x$ and $b = iy$. What are $\sin (iy)$ and $\cos (iy)$? – Daniel Fischer Apr 4 '15 at 20:14

use addition formula: $\sin(x+iy)=\sin x \cos(iy)+\cos x \sin(iy)$ and remember that: $\cos(iy)=\cosh(y)\;$ and $\sin(iy)=i\sinh y$.