Question

If arg $z<0$ then arg $(-z)-$ arg z is equal to

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

Answer 1

The correct option is A $\pi$

Let
$z=e^{-i\theta }$
$=cos\left(\theta \right)-isin\left(\theta \right)$
$arg\left(z\right)=-\theta$
$arg\left(z\right)<0$.
Now
$-z=-cos\left(\theta \right)+isin\left(\theta \right)$
$arg(-z)=\pi -\theta$
Hence
$arg(-z)-arg\left(z\right)$
$=\pi -\theta -(-\theta )$
$=\pi$
Hence, option 'A' is correct.