In a polar coordinate system, a point in a plane is located in reference to a fixed point, O, called the pole, and a fixed ray called the polar axis. The pole is the endpoint of the polar axis, which is usually horizontal and extends endlessly to the right.

Each point P in the plane can be assigned polar coordinates $(r, θ)$ as follows.

r = distance from O to P.

$θ =$ the angle, measured counterclockwise from the polar axis, whose initial side is the polar axis and whose terminal side is $\bar{O P}$

It might seem like r has to be positive. However, it is possible for r to be negative. When r is negative, then P is located |r| units from the pole on the ray opposite the terminal side of the angle $θ$ . The sketch of the two points $(4, \frac{π}{3})$ and $(- 4, \frac{π}{3})$ exemplifies this.

Note that in the rectangular coordinate system, each point is uniquely defined. This is not true for polar coordinates. For example, the coordinates $(r, θ)$ and $(r, θ + 2 k π)$ , for any integer k, represent the same point.

Coordinate Conversion The polar coordinate $(r, θ)$ are related to the rectangular coordinate $(x, y)$ as follows.
$x = r cos θ$	$r^{2} = x^{2} + y^{2}$
$y = r sin θ$	$tan θ = \frac{y}{x}$

NOTE: In order to find the measure of $θ$ , in radians, by the above formula, we need to know the quadrant in which the point lies.

If the point lies in $Q_{1}$ or $Q_{4}$ , then $θ = a r c tan (\frac{y}{x})$
If the point lies in $Q_{2}$ or $Q_{3}$ , then $θ = a r c tan (\frac{y}{x}) + π$