« Parallel and Perpendicular Lines 

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.
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:
$$ r = \sqrt{(1)^2 + (1)^2} = \sqrt{2} \\ \theta = \arctan \left( \frac{1}{1} \right)  \pi = \arctan 1  \pi = \frac{\pi}{4}  \pi =  \frac {3 \pi}{4} $$
$r = 2a \cos \theta + 2b \sin \theta$ This is a circle of radius $\sqrt{a^2 + b^2}$ and center $(a, b)$.