Question:
If $z=1-\cos \theta+i \sin \theta$, then $|z|=$
(a) $2 \sin \frac{\theta}{2}$
(b) $2 \cos \frac{\theta}{2}$
(c) $2\left|\sin \frac{\theta}{2}\right|$
(d) $2\left|\cos \frac{\theta}{2}\right|$
Solution:
(c) $2\left|\sin \frac{\theta}{2}\right|$
$\because z=1-\cos \theta+i \sin \theta$
$\Rightarrow|z|=\sqrt{(1-\cos \theta)^{2}+\sin ^{2} \theta}$
$\Rightarrow|z|=\sqrt{1+\cos ^{2} \theta-2 \cos \theta+\sin ^{2} \theta}$
$\Rightarrow|z|=\sqrt{1+1-2 \cos \theta}$
$\Rightarrow|z|=\sqrt{2(1-\cos \theta)}$
$\Rightarrow|z|=\sqrt{4 \sin ^{2} \frac{\theta}{2}}$
$\Rightarrow|z|=2\left|\sin \frac{\theta}{2}\right|$
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