Question

If $z=1+i\sqrt{3}$, then $|arg\left(z\right)|+|arg\left(\overline{z}\right)|=$

• A. $\frac{\pi }{3}$
• B. $\frac{2\pi }{3}$
• C. 0
• D. $\frac{\pi }{2}$

Answer 1

Answer: (B)

$\frac{2\pi }{3}$

$z=1+i\sqrt{3}$
$\Rightarrow arg\left(z\right)=ta{n}^{-1}\left(\frac{\sqrt{3}}{1}\right)=\frac{\pi }{3}$
Now, $z=1+i\sqrt{3}\Rightarrow \overline{z}=1-i\sqrt{3}$
$\Rightarrow arg\left(\overline{z}\right)=-arg\left(z\right)=-\frac{\pi }{3}$
Thus, $|arg\left(z\right)|+|arg\left(\overline{z}\right)|=\frac{\pi }{3}+\frac{\pi }{3}=\frac{2\pi }{3}$
Hence, option B.