Definition:Complex Number/Polar Form

Definition
For any complex number $z = x + i y \ne 0$, let:

where $x, y \in \R$.

From the definition of $\arg \left({z}\right)$:
 * $(1): \quad \dfrac x r = \cos \theta$


 * $(2): \quad \dfrac y r = \sin \theta$

which implies that:
 * $x = r \cos \theta$
 * $y = r \sin \theta$

which in turn means that any number $z = x + i y \ne 0$ can be written as:
 * $z = x + i y = r \left({\cos \theta + i \sin \theta}\right)$

The pair $\left \langle {r, \theta} \right \rangle$ is called the polar form of the complex number $z \ne 0$.

The number $z = 0 + 0i$ is defined as $\left \langle {0, 0} \right \rangle$.

Also known as
Some sources refer to polar form as trigonometric form.

As $\cos \theta + i \sin \theta$ appears so often in complex analysis, the abbreviation $\operatorname{cis} \theta$ is frequently seen.