Show $\sin(z)=w$ can be solved $\forall w\in\Bbb C$

Question

I need help showing that the equation $\sin(z)=w$ can be solved for $z\in \mathbb{C}$ for every $w\in \mathbb{C}$. How would I be able to come to that conclusion and what would that mean about the function $f(z)=\sin(z)$?

Answer 1

You would need to write $w=a+bi$, where $a,b$ are real. Then, find $z$ in terms of $a$ and $b$ such that $\sin(z)=a+bi$.

What would this mean for the function $f(z)=\sin z$? Well, this would tell us that the function hits every $w\in\Bbb C$, meaning it is surjective on $\Bbb C$.