Question

The argument of the complex number $\sin\left(\frac{6\pi}{5}\right) + i\left(1 +\cos \frac{6\pi}{5}\right) $ is

• A. $ \frac{\pi}{10} $
• B. $ \frac{5 \pi}{6} $
• C. $ \frac{ - \pi}{10} $
• D. $ \frac{2 \pi}{5} $

Answer 1

The Correct Option is C

Let $ z = \sin\frac{6\pi}{5} + i\left(1 +\cos \frac{6\pi}{5}\right) $
putting $\sin \frac{6\pi}{5} = r \cos \alpha $ ....(i)
and $ 1 + \cos \frac{6 \pi}{5} = r \sin \alpha $ .....(ii)
Dividing (ii) by (i), we get
$\tan \alpha = \frac{1+\cos \frac{6\pi}{5}}{\sin \frac{6\pi}{5}} = \frac{2 \cos^{2} \frac{3\pi}{5}}{2 \sin \frac{3\pi}{5} \cos \frac{3\pi}{5}} $
$\tan \alpha = \cot \frac{3\pi}{5} =\tan \left(\frac{\pi}{2} - \frac{3\pi}{5}\right) $
$\tan \alpha =\tan \left(\frac{-\pi}{10}\right)$
$\therefore$ argument of z is $ \alpha = \frac{-\pi}{10} $