Complex Inequality on hyperbolic functions How can proof $|\sinh(y)|\leq |\sin(z)|\leq |\cosh(y)|$ for every $z=x+iy\in\mathbb{C}$, any hinty or auxiliar result.

Hint: use $$\sin z=\frac{e^{iz}-e^{-iz}}{2i}$$ and $$|e^{iz}|=|e^{ix}e^{-y}|=e^{-y}$$ together with the triangle inequality.