Question

What will be the argument of the complex number $ z = 1 - cosp + isinp $ ?

• A. $ \pi - p $
• B. $ \pi /4 - p/2 $
• C. $ \pi /2 - p/2 $
• D. $ \pi - p/2 $

Answer 1

Answer: (C)

$arg\left(z\right) = tan^{-1}\left(\frac{sin\,p}{1-cos \,p}\right)$
$= tan^{-1}\left(\frac{2 \,sin\, p/2\, cos \,p/2}{1-\left(1-2\, sin^{2}\,p/2\right)}\right) $
$=tan^{-1}\left(\frac{cos\,p/2}{sin\, p/2}\right) = tan^{-1}\left(cot \frac{p}{2}\right)$
$ = tan^{-1}\left(tan\left(\frac{\pi}{2}-\frac{p}{2}\right)\right) = \frac{\pi}{2} - \frac{p}{2}$