# Solve the following

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

$\therefore z=2\left(\cos \frac{2 \pi}{3}+i \sin \frac{2 \pi}{3}\right)$

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