# How do you convert -1+i to polar form?

May 22, 2016

Polar form of $- 1 + i$ is $\left(\sqrt{2} , \frac{3 \pi}{4}\right)$

#### Explanation:

A complex number $a + i b$ in polar form is written as

$r \cos \theta + i r \sin \theta$, $\cos \theta = \frac{a}{r}$ and $\sin \theta = \frac{b}{r}$

Hence $r = \sqrt{{a}^{2} + {b}^{2}}$

As in $- 1 + i$ $a = - 1$ and $b = 1$

$r = \sqrt{{\left(- 1\right)}^{2} + {1}^{2}} = \sqrt{2}$ and hence

$\cos \theta = - \frac{1}{\sqrt{2}}$ and $\sin \theta = \frac{1}{\sqrt{2}}$

Hence, $\theta = \frac{3 \pi}{4}$ and

Polar form of $- 1 + i$ is $\left(\sqrt{2} , \frac{3 \pi}{4}\right)$