Each point in the polar coordinate system can be described with the two polar coordinates, which are usually called $r$ (the radial coordinate) and $\theta$ (the angular coordinate, polar angle, or azimuth angle, sometimes represented as $\varphi$ or $t$). The $r$ coordinate represents the radial distance from the pole, and the $\theta$ coordinate represents the anticlockwise (counterclockwise) angle from the 0.

For example, the polar coordinates $(3, 6)$ would be plotted as a point 3 units from the pole on the 6 ray.

Converting between polar and Cartesian coordinates

From polar to Cartesian coordinates.

$$ \begin{aligned} x &= r \cos \theta \\ y &= r \sin \theta \end{aligned} $$

Example 1:

Convert $(3, \frac{\pi}{6})$ into polar coordinates

Solution:

$$ \begin{aligned} x &= r \cos \theta = 3 \cos \frac{\pi}{6} = 3 \frac{\sqrt{3}}{2} \\ y &= r \sin \theta = 3 \sin \frac{\pi}{6} = \frac{3}{2} \end{aligned} $$

From Cartesian to polar coordinates.

$$ r = \sqrt{x^2 + y^2} $$

$$ \theta = \left\{ {\begin{array}{*{20}{l}} {\arctan \frac{y}{x}}&{if}&{x > 0}\\ {\arctan \frac{y}{x} + \pi }&{if}&{x < 0 \ \ and \ \ y \ge 0}\\ {\arctan \frac{y}{x} - \pi }&{if}&{x < 0 \ \ and \ \ y < 0}\\ {\frac{\pi }{2}}&{if}&{x = 0 \ \ and \ \ y > 0}\\ { - \frac{\pi }{2}}&{if}&{x = 0 \ \ and \ \ y > 0} \end{array}} \right. $$

Example 2:

Convert $(-1,-1)$ into polar coordinates

Solution:

$$ r = \sqrt{(-1)^2 + (-1)^2} = \sqrt{2} \\ \theta = \arctan \left( \frac{-1}{-1} \right) - \pi = \arctan 1 - \pi = \frac{\pi}{4} - \pi = - \frac {3 \pi}{4} $$

Circle:

$r = 2a \cos \theta + 2b \sin \theta$ This is a circle of radius $\sqrt{a^2 + b^2}$ and center $(a, b)$.