Question

If $\arg \left(z\right)$ denotes the principal argument of a complex number $z$, then the value of the expression $\arg (-i{e}^{\frac{i\pi }{9}}.z^2)+2\arg (2i{e}^{\frac{-i\pi }{18}}.\overline{z})$ is

• A. $0$
• B. $\frac{\pi }{2}$
• C. $\pi$
• D. $\arg \left(z\right)$

Answer 1

The correct option is B $\frac{\pi }{2}$

$\arg (-i{e}^{\frac{i\pi }{9}}.z^2)+2\arg (2i{e}^{\frac{-i\pi }{18}}.\overline{z})$
$=\arg (-i)+\frac{\pi }{9}+2\arg \left(z\right)+2[\arg \left(2i\right)-\frac{\pi }{18}+\arg \left(\overline{z}\right)]$
$=\frac{-\pi }{2}+\frac{\pi }{9}+\pi -\frac{\pi }{9}+2[\arg \left(z\right)+\arg \left(\overline{z}\right)]$
$=\frac{\pi }{2}+2\left(0\right)$
$=\frac{\pi }{2}$