Classify the set of all analytic functions $f$ on $\Omega$ such that $\lvert f(z)\rvert = \lvert \sin z\rvert$ for all $z\in \Omega$

Question

let $\Omega$ be a domain in $\mathbb C$. Classify the set of all analytic functions $f$ on $\Omega$ such that $\lvert f(z)\rvert = \lvert \sin z\rvert$ for all $z\in \Omega$.

I can see that if $f(z)=c\sin z$ for any constant $c$ such that $\lvert c\rvert =1$, then $f$ satisfies the required property. But, I cannot figure out how to classify the given set.