Question

The principal argument of the complex number $z=\frac{1+\sin \frac{\pi }{3}+i\cos \frac{\pi }{3}}{1+\sin \frac{\pi }{3}-i\cos \frac{\pi }{3}}$ is?

• A. $\frac{\pi }{3}$
• B. $\frac{\pi }{6}$
• C. $\frac{2\pi }{3}$
• D. $\frac{\pi }{2}$
• E. $\frac{\pi }{4}$

Answer 1

The correct option is B $\frac{\pi }{6}$

$arg\left(z\right)=arg\left(Numerator\right)-arg\left(Denominator\right)$
$={\tan }^{-1}\left|\frac{\cos \frac{\pi }{3}}{1+\sin \frac{\pi }{3}}\right|+{\tan }^{-1}\left|\frac{\cos \frac{\pi }{3}}{1+\sin \frac{\pi }{3}}\right|$
$=2{\tan }^{-1}\left|\frac{\cos \frac{\pi }{3}}{1+\sin \frac{\pi }{3}}\right|$
$=2{\tan }^{-1}(2-\sqrt{3})=2\times {15}^o=\frac{\pi }{6}$