Question

If $\arg \left(z\right)<0,$ then $\arg \left(\frac{z-\overline{z}}{2}\right)$ is equal to

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

Answer 1

The correct option is C $-\frac{\pi }{2}$

$\because z-\overline{z}=2i\left(Im\right(z\left)\right)$
$\left(\frac{z-\overline{z}}{2}\right)$ is purely imaginary. Also, $\arg \left(z\right)<0$. So, $Im\left(z\right)<0$
$\Rightarrow \arg \left(\frac{z-\overline{z}}{2}\right)=-\frac{\pi }{2}$