In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. When we think about plotting points in the plane, we usually think of rectangular coordinates (x, y) in the Cartesian coordinate plane.

The polar grid is scaled as the unit circle with the positive x –axis now viewed as the polar axis and the origin as the pole. The first coordinate r is the radius or length of the directed line segment from the pole. The angle θ, measured in radians, indicates the direction of r. We move counterclockwise from the polar axis by an angle of θ, and measure a directed line segment the length of r in the direction of θ. Even though we measure θ first and then r, the polar point is written with the r-coordinate first. For example, to plot the point (2, π/4), we would move π/4 units in the counterclockwise direction and then a length of 2 from the pole.

When given a set of polar coordinates, we may need to convert them to rectangular coordinates. To do so, we can recall the relationships that exist among the variables x, y, r, and θ, from the definitions of cos θ and sin θ. Solving for the variables x and y yields the following formulas:

Cosθ = x/r ⇒ x= rcosθ

Sinθ= y/r ⇒ y = rsinθ

An easy way to remember the equations above is to think of cosθ as the adjacent side over the hypotenuse and sinθ as the opposite side over the hypotenuse.

To convert polar coordinates (r, θ) to rectangular coordinates (x, y) follow these steps:

1) Write cosθ=xr ⇒ x=rcosθ and sinθ=yr ⇒ y=rsinθ.

2) Evaluate cosθ and sinθ.

3) Multiply cosθ by r to find the x-coordinate of the rectangular form.

4) Multiply sinθ by r to find the y-coordinate of the rectangular form.