What is polar cis form?

Question

14405 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2015-06-17T20:12:43

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

$r(cos theta + i sin theta)$

often abbreviated as

$r$ cis $theta$

A complex number $z$ is always expressible uniquely as $a+ib$ , where $a, b in RR$ . That is it is expressible as a point $(a, b)$ in $RR xx RR$ .

Any such point can also be represented using polar coordinates as $(r cos theta, r sin theta)$ for some radius $r >= 0$ and angle $theta in RR$ .

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

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

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

$r = sqrt(a^2 + b^2)$

$cos theta = a / r$

$sin theta = b / r$

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

$cos theta + i sin theta = e^(itheta)$

So polar cis form is equivalent to $re^(i theta)$