# What is polar cis form?

Jun 17, 2015

Polar cis form is the polar form of a complex number:

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

often abbreviated as

$r$ cis $\theta$

#### Explanation:

A complex number $z$ is always expressible uniquely as $a + i b$, where $a , b \in \mathbb{R}$. That is it is expressible as a point $\left(a , b\right)$ in $\mathbb{R} \times \mathbb{R}$.

Any such point can also be represented using polar coordinates as $\left(r \cos \theta , r \sin \theta\right)$ for some radius $r \ge 0$ and angle $\theta \in \mathbb{R}$.

The point #(r cos theta, r sin theta) corresponds to the complex number:

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

Given $z = a + i b$, we can calculate a suitable $r$, $\cos \theta$ and $\sin \theta$ ...

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

$\cos \theta = \frac{a}{r}$

$\sin \theta = \frac{b}{r}$

One of the nice things about $\cos \theta + i \sin \theta$ is Euler's formula:

$\cos \theta + i \sin \theta = {e}^{i \theta}$

So polar cis form is equivalent to $r {e}^{i \theta}$