Question

If arg$\left(z\right)<0$ then find arg$(-z)-$arg$\left(z\right)$

Answer 1

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

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

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

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

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

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