Question

The amplitude of the complex number $z=\sin \alpha+i(1-\cos \alpha)$ is

(A) $2 \sin \frac{\alpha}{2}$

(B) $\frac{\alpha}{2}$

(C) $\alpha$

(D) None of these

Answer 1

$\therefore \operatorname{amp}(z)=\tan ^{-1}\left(\frac{1-\cos \alpha}{\sin \alpha}\right)$

$=\tan ^{-1}\left(\frac{2 \sin ^2 \frac{\alpha}{2}}{2 \sin \frac{\alpha}{2} \cos \frac{\alpha}{2}}\right)$

$=\tan ^{-1} \tan \left(\frac{\alpha}{2}\right)=\frac{\alpha}{2}$

So, The correct option will be (B).