<s>[INST] Find the amplitude of a complex number 
If 
$$z = (1-\cos{\theta}) + i \sin{\theta},$$
then the amplitude of $z$ is:
$$a) \frac{\pi}{4}-\frac{\theta}{2}$$ $$b)
 \frac{\pi}{2}-\frac{\theta}{2}$$ $$c) \frac{\pi}{2}-\theta$$ $$d)
 \frac{\pi}{2}-\frac{\theta}{4}$$ where 0<$\theta$ <$\pi$.

I tried :
$x= 1-\cos{\theta}, y= \sin{\theta}$ then 
$\tan{\phi}=\frac{\sin{\theta}}{1-\cos{\theta}}$.
However I could not solve the last equation.
 [/INST] $\sin\theta = 2\sin\frac{\theta}{2}\cdot \cos\frac{\theta}{2}$
$1-\cos\theta = 2\sin^2\frac{\theta}{2}$
So, $\tan\phi = \frac{\sin\theta}{1-\cos\theta} = \frac{2\sin\frac{\theta}{2}\cdot \cos\frac{\theta}{2}}{2\sin^2\frac{\theta}{2}}$

$\tan\phi = \cot\frac{\theta}{2} = \tan(\frac{\pi}{2} - \frac{\theta}{2})$
$\implies\phi =\frac{\pi}{2} - \frac{\theta}{2} $

For half-angle formulas: https://www.intmath.com/analytic-trigonometry/4-half-angle-formulas.php
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