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?


closed as off-topic by Eevee Trainer, Kavi Rama Murthy, max_zorn, Cesareo, Shailesh May 12 at 15:20

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    $\begingroup$ By definition, $\sin(z)=\frac{e^{iz}-e^{-iz}}{2i}$ for any $z\in\mathbb{C}$. You may want to use this for $z=\cos\theta+i\sin\theta=e^{i\theta}$ $\endgroup$ – Julian Mejia May 12 at 0:39
  • $\begingroup$ @JulianMejia Ok, so then this would simplify to |sin(z)|, correct? Out of curiosity, is this the same as max |sin(z)| about the unit circle? $\endgroup$ – bigsbylp May 12 at 0:45
  • $\begingroup$ To get $|\sin z|$ you need to use $|\sin(z)|^2=\sin(z)\overline{\sin(z)}$, and you may want to use the fact that $\overline{\sin(z)}=\sin(\overline{z})$. Now, your other question, I didn't understand it what do you mean by max? You are trying to compute $|\sin(z)| $ not a max. $\endgroup$ – Julian Mejia May 12 at 0:54
  • $\begingroup$ @JulianMejia I was asking if the above express would, after simplification, result in |sinz| and if the value of this function is the same as the max |sinz| about the unit circle. $\endgroup$ – bigsbylp May 12 at 0:59
  • $\begingroup$ No, these are two different things. $|\sin z|$ is just the value at some particular point of the unit circle, namely at $z=e^{i\theta}$. $\endgroup$ – Julian Mejia May 12 at 1:02

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 $$\sin(\cos\theta + i \sin\theta) $$ $$= \sin(\cos\theta)\cos(i\sin\theta) + \cos(\cos\theta)\sin(i\sin\theta)$$ $$ = \sin(\cos\theta)\cosh(\sin\theta) + i\cos(\cos\theta)\sinh(\sin\theta)$$


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