<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] Your $z$ can be nicely written as
$$z=1-e^{-i \theta}$$
Now, 
\begin{align}
|z| &= |1-e^{-i \theta}| \\
&=\bigg |e^{-i \frac{\theta}{2}}\left(e^{i \frac{i\theta}{2}} -e^{-i \frac{\theta}{2}}  \right)\bigg | \\
&=\bigg |e^{-i \frac{i\theta}{2}} \left( \cos \left( \frac{\theta}{2}\right) + i \sin  \left( \frac{\theta}{2}\right)-\cos \left( \frac{-\theta}{2}\right) - i \sin  \left( \frac{-\theta}{2}\right) \right)\bigg |\\
&= \bigg |2i \sin \left( \frac{\theta}{2}\right)e^{i \frac{i\theta}{2}}\bigg|\\
&=  |2i|. \bigg |\sin \left( \frac{\theta}{2}\right)\bigg|.\bigg|e^{i \frac{-\theta}{2}}\bigg| \\
&= 2 \bigg |\sin \left(\frac{\theta}{2}\right)\bigg | 
\end{align}
Whence
\begin{align}
\angle z &= \angle (2i)+\angle\left( \sin \left(\frac{\theta}{2}\right) \right)+\angle (e^{-i \frac{\theta}{2}}) \\
&= \frac{\pi}{2}+0-\frac{\theta}{2}
\end{align}
as required.
</s>