Question

If $arg\left(z\right)$ less than $0$, then $arg(-z)-argz$ equal

• A. $\pi$
• B. $-\pi$
• C. $-\pi /2$
• D. $\pi /2$

Answer 1

The correct option is A

$\pi$

Find the value of $\mathit{a}\mathit{r}\mathit{g}\mathbf{(}\mathbf{-}\mathit{z}\mathbf{)}\mathbf{}\mathbf{-}\mathbf{}\mathit{a}\mathit{r}\mathit{g}\mathbf{}\mathit{z}$:

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

Given that If $arg\left(z\right)$ less than $0$

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

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

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

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

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

$=\pi$

Hence, the correct option is A.