Polar Form

### Definition

When a complex number is thought of as a vector in two dimensions, the $X$ coordinate $x$ and the $Y$ coordinate $y$ can be expressed in terms of the length of the vector $r$ and the angle made by this vector with the positive $X$-axis, namely $\theta$. Since $x = r \cos \theta$ and $y = r \sin \theta$, $z$ can be expressed as

(1)
\begin{align} z = r \cos\theta + j r \sin\theta \end{align}

where $\theta$ can be in degrees or radians (usually radians) and recall that $2 \pi \mbox{ rad } = 360\,^\circ$. $r$ is called the magnitude of $z$, denoted by $|z|$ and $\theta$ is called the phase of the complex number $z$, denoted by $\mbox{arg}{z}$ or $\angle z$.

Using Euler's identities $z$ can be written as

(2)
\begin{align} z = r \cos \theta + j r \sin \theta = r e^{j \theta} \end{align}

This is known as the polar form or exponential form and it is very important to be able to convert a complex number from cartesian form to exponential form and vice versa. It is easy to see that $x,y,r$ and $\theta$ are related according to

(3)
\begin{eqnarray} \nonumber x = r \ \cos \theta, & \ & y = r \ \sin \theta \\ r = \sqrt{x^2 + y^2}, & \ & \theta = \tan^{-1}\left(\frac{y}{x} \right) \end{eqnarray}