TITLE: Is it true that $\lvert \sin z \rvert \leq 1$ for all $z\in \mathbb{C}$?
QUESTION [12 upvotes]: Is it true that $\left\lvert \sin z \right \rvert \leq 1$ for all $z \in \mathbb{C}$ ?
I think that is not true, can anyone help me?

REPLY [5 votes]: $$\sin z=z-\frac{z^3}{3!}+\frac{z^5}{5!}- \dots $$
$$\sin i=i\left( 1+\frac{1}{3!}+\frac{1}{5!}+ \cdots \right)$$
$$|\sin { i}|= 1+\frac{1}{3!}+\frac{1}{5!}+ \cdots >1$$