Find the largest positive real number $r$ such that $|Im(\sin z)|\leq 1$ for all $z\in \mathbb{C}$ with $|z|\leq r$. Here $Im(z)$ denotes the imaginary part of $z$.
I know that $$\sin z=\sin x \cosh y+i\cos x\sinh y.$$ This means we need to find $r$ for which $$|\cos x\sinh y|\leq 1$$ where $x^2+y^2\leq r^2.$ How to proceed further? Please help.
Since $|\cos x|\le1$ for real $x$, the points in the infinite strip where $|\sinh y|\le1$ or $|y|\le\sinh^{-1}1=\ln(1+\sqrt2)=k$ satisfy $\operatorname{Im}(\sin(x+yi))\le1$. In particular the points with $|x+yi|\le k$ satisfy the inequality.
On the other hand, setting $x=0$ shows that $\operatorname{Im}(\sin(yi))\ge1$ if $|y|\ge k$, so the maximum $r$ for the original problem is $k$.