What is the value of $|\sin(\cos\theta + i \sin\theta)| $ in complex analysis?

Question

How would I compute the value to/simplify the following expression?

$$\left|\sin\left( \cos\theta + i \sin\theta \vphantom{M^M} \right) \right| $$

Can I use the fact that $\cos\theta + i \sin\theta = e^{i\theta}$ and work from there?

Answer 1

The value of $|\sin(\cos\theta + i \sin\theta)|$ is indeed the image of mapping $f(z)=|\sin z|$ under unit circle $|z|=1$, which is a real number. One may write $$\sin(\cos\theta + i \sin\theta) $$ $$= \sin(\cos\theta)\cos(i\sin\theta) + \cos(\cos\theta)\sin(i\sin\theta)$$ $$ = \sin(\cos\theta)\cosh(\sin\theta) + i\cos(\cos\theta)\sinh(\sin\theta)$$