The polar coordinate system is a two-dimensional coordinate system that uses distances and angles to define the position of each point on a plane. Specifically, the polar coordinates of a point $P$ are $(r,\theta ),$ where $r$ is its distance from a fixed point $O,$ called the pole, and $\theta$ is the angle measured counterclockwise from an axis known as the the polar axis.

This representation is different from rectangular coordinates, where each point has a unique $(x,y)$ pair. In contrast, polar coordinates can have multiple representations for the same point. The angle $\theta ,$ typically given in radians, repeats every $2\pi$ radians. As a result, $(r,\theta )$ and $(r,\theta +2\pi )$ always represent the same point.

Conversion between polar and rectangular coordinates uses specific formulas. These conversions allow easy switching between the two systems as needed.

Rectangular to Polar	Polar to Rectangular
$r=\sqrt{x^2+y^2}$ $\theta =\arctan \left(\frac{y}{x}\right)$	$x=r\cos \theta$ $y=r\sin \theta$

Polar coordinates are fundamental for describing complex numbers. When a complex number is in polar form, its modulus is the distance from the pole, and its argument is the angle. This makes operations like multiplication and exponentiation easier.