Trigonometric values in the complex plane

Question

So one can take $\log(x)$ of a complex number and $n^{th}$ roots. However, how would you take $\sin(a+bi)$ or $\cos(a+bi)$ and other trig functions? This is my reasoning $\sin(a+bi)=\sin(a)\cos(bi)+\sin(bi)\cos(a)$ One can do that for all the other Trig functions. The Question is how do I find $\sin(bi),\cos(bi),\tan(bi)$?

Answer 1

Use that

$$\cos z=\frac{e^{iz}+e^{-iz}}{2} \quad \sin z=\frac{e^{iz}-e^{-iz}}{2i} $$