Question

If arg $z<0$, then $arg(-z)-arg\left(z\right)$ is equal to

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

Answer 1

The correct option is A $\pi$

Let $z=r(\cos \theta +i\sin \theta )$

$arg\left(z\right)=\theta <0$

$-z=-r(\cos \theta +i\sin \theta )$

$=r\left(\cos \right(\pi +\theta )+i\sin (\pi +\theta \left)\right)$

$arg(-z)=\pi +\theta$

$\therefore arg(-z)-arg\left(z\right)=\pi +\theta -\theta =\pi$

Hence, $\left(A\right)$