Is it valid to write $\sin(x+iy)=\sin (x)\cos(iy)+\cos(x)\sin(iy)$

Question

I'm asked to sketch the set $\{z \in \mathbb{C}:\sin z$ is a real number $\}$

Here's what I did:

$$ \sin(x + iy) = \sin(x) \cos(iy) + \cos(x) \sin(iy) $$

But since we only want the real part, then this is equal $\cos x + \sin x$. Is this valid?

Answer 1

Hint. Continuing what you did, and using the comment from Pedro Tamaroff, $$\sin(x+iy)=\sin x\cosh y+i\cos x\sinh y\ .$$ But now read your question carefully: you are not asked for the real part of $\sin z$, but for the values of $z$ such that $\sin z$ is real. So, what can you say about $x$ and $y$ if the above expression is a real number?