Question:
Write $-1+i \sqrt{3}$ in polar form
Solution:
Let $z=-1+\sqrt{3} i$. Then,
$r=|z|=\sqrt{[-1]^{2}+[\sqrt{3}]^{2}}=2$
Let $\tan \alpha=\left|\frac{\operatorname{Im}(z)}{\operatorname{Re}(z)}\right|$
$=\sqrt{3}$
$\Rightarrow \alpha=\frac{\pi}{3}$
Since the point representing $z$ lies in the second quadrant. Therefore, the argument of $z$ is given by $\theta=\pi-\alpha$
$=\pi-\frac{\pi}{3}$
$=\frac{2 \pi}{3}$
So, the polar form is $r(\cos \theta+i \sin \theta)$
$\therefore z=2\left(\cos \frac{2 \pi}{3}+i \sin \frac{2 \pi}{3}\right)$