Polar Coordinate System
Definition
The coordinates of a point on the plane are represented by “the distance of the point from the origin $r$” and “the angle $\theta$ between the line joining the point to the origin and the $x$ axis”; this is called the polar coordinate system. In this case, the ordered pair $(r, \theta)$ is called the polar coordinates.
Explanation
Useful when expressing functions that depend on the distance from the origin. Examples include the position of an object undergoing circular motion in physics and central forces such as gravity (a central force).
Relationship with the Cartesian coordinate system
If the polar coordinates of a point are $(r, \theta)$, expressing its $x$ coordinate and $y$ coordinate in polar form, by the definition of trigonometric functions,
$$ x = r\cos\theta \quad \text{ and } \quad y = r\sin \theta % (x, y) = (r\cos\theta, r\sin\theta) $$
Conversely, the polar coordinates $r$ and $\theta$ are expressed in the (two-dimensional) Cartesian coordinate system as follows. Since $\tan \theta = \dfrac{x}{y}$,
$$ r = \sqrt{x^{2}+y^{2}} \quad \text{ and }\quad \theta = \tan^{-1}\dfrac{x}{y} $$