Question

The principal argument of
$\sqrt{2}[\cos \frac{5\pi }{3}+i\sin \frac{5\pi }{3}]$ is

• A. $\frac{5\pi }{3}$
• B. $\frac{\pi }{3}$
• C. $-\frac{\pi }{3}$
• D. $-\frac{\pi }{2}$

Answer 1

Answer: (C)

$-\frac{\pi }{3}$

$\sqrt{2}\left[cos\right(\frac{5\pi }{3})+isin(\frac{5\pi }{3}\left)\right]$
$=\sqrt{2}\left[cos\right(-2\pi +\frac{5\pi }{3})+isin(-2\pi +\frac{5\pi }{3}\left)\right]$
$=\sqrt{2}\left[cos\right(\frac{-\pi }{3})+isin(\frac{-\pi }{3}\left)\right]$
$=\sqrt{2}.e^{-i\frac{\pi }{3}}$.
Hence principal argument is $-\frac{\pi }{3}$.