📂Mathematical Physics

Polar Coordinate System

Definition

The coordinates of a point on the coordinate plane are defined as “the distance $r$ from the origin” and “the angle $\theta$ made by the line connecting the point to the origin with the $x$ axis,” which is referred to as the polar coordinate system.

Explanation

It is useful for expressing functions that depend on the distance from the origin. For example, the position of an object performing circular motion in physics, and central forces such as gravity.

Relationship with Cartesian Coordinate System

For a point with polar coordinates $(r, \theta)$, its $x$ coordinate and $y$ coordinate can be expressed in polar coordinates as follows, according to the definition of trigonometric functions:

$$x = r\cos\theta \quad \text{ and } \quad y = r\sin \theta$$

Conversely, the polar coordinates $r$ and $\theta$ can be expressed in the (2-dimensional) Cartesian coordinate system as follows, since $\tan \theta = \dfrac{x}{y}$:

$$r = \sqrt{x^{2}+y^{2}} \quad \text{ and }\quad \theta = \tan^{-1}\dfrac{x}{y}$$