Does there exist an analytic function such that $|f(z)|=|\sin z|$? Does there exist an analytic function $f(z)$ defined on $\Omega$ such that $|f(z)|=|\sin z|$ for all $z\in\Omega\subseteq\mathbb{C}$? Well I guess if there is a constant $c$ with $|c|=1$ and $f(z)=c\sin(z)$ for all $z\in\Omega$, am I right?

Well $\sin z$ itself is analytic, so you can just let $f(z)=\sin z$. But maybe that's too easy...but yes, take $c=e^{i\theta}$, $\theta\in\mathbb{R}$, and your answer is correct. Depending on what $\Omega$ is, there could be other solutions.
