I have the following homework question for a course in complex analysis:
Determine all the $z\in\mathbb{C}$ with $|\sin z| ≤ 1$, and find an $n\in \mathbb{N}$ such that $|\sin(in)| > 10 000$.
I have however no clue how to do this, also because during the lecture the teacher said that is no ordering in the complex numbers. I tried substituting $\sin(z)=\frac{1}{2i}(e^{iz}$ - $e^{-iz})$ but this didn't bring my any further.
The substitution is a good hint: by letting $z=x+iy$ we get $$\sin(z)=\frac{e^{iz} - e^{-iz}}{2i}=\sin(x)\cosh(y)+i\cos(x)\sinh(y).$$ Hence $$\begin{align} \lvert\sin(z)\rvert^2&=\sin(x)^2\cosh(y)^2+\cos(x)^2\sinh(y)^2 \\&= \sin(x)^2\cosh(y)^2+(1-\sin(x)^2)\sinh(y)^2\\ &=\sin(x)^2+\sinh(y)^2. \end{align}$$ Can you take it from here?