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  4. The amplitude of the complex number 1 + sinα - i cos α is

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Q. The amplitude of the complex number $ 1 + \sin\alpha - i \cos \alpha $ is

AMUAMU 2018

Solution:

Let $z=1+\sin \alpha-i \cos \alpha$
$\operatorname{amp}(z) =\tan ^{-1}\left(\frac{-\cos \alpha}{1+\sin \alpha}\right)$
$=-\tan ^{-1}\left(\frac{\cos \alpha}{1+\sin \alpha}\right)$
$=-\tan ^{-1}\left(\frac{\cos ^{2} \frac{\alpha}{2}-\sin ^{2} \frac{\alpha}{2}}{\left(\cos \frac{\alpha}{2}+\sin \frac{\alpha}{2}\right)}\right)$
$=-\tan ^{-1}\left(\frac{\cos \frac{\alpha}{2}-\sin \frac{\alpha}{2}}{\cos \frac{\alpha}{2}+\sin \frac{\alpha}{2}}\right) $
$=-\tan ^{-1}\left(\tan \frac{\pi}{4}-\frac{\alpha}{2}\right)$
$=-\left(\frac{\pi}{4}-\frac{\alpha}{2}\right)=\frac{\alpha}{2}-\frac{\pi}{4}$