$\sin(z)$ in polar coordinates

Question

the following formula can be found in the literature: $\vert \sin(z) \vert^2 = \sin(x)^2 + \sinh(y)^2$, $z=x+iy;$ $x,y\in\mathbb{R}$.

I am wondering if there is a similar formular in polar coordinates, i.e. $\vert \sin(r\cdot e^{i\theta}) \vert = ...?$, for $r>0, \theta\in\mathbb{R}$.

Best wishes

Answer 1

$r e^{i\theta} = r \cos(\theta) + i r \sin(\theta)$, so $$|\sin(r e^{i\theta})|^2 = \sin(r \cos(\theta))^2 + \sinh(r \sin(\theta))^2$$