The polar coordinate system is an alternative to the cartesian coordinate system. Whereas points in the cartesian system are denoted (x,y), points in the polar coordinate system are denoted (r,theta). This coordinate system defines points not on a grid, but rather more on a continuous set of circles. A point at (r,theta) would be located on the circle with radius r at the position displaced theta (degrees or radians, usually radians) from the right horizontal axis.

The right horizontal axis represents 0 degrees, and you would draw the circle counterclockwise from 0 degrees to find the position of the angle. Likewise, if the angle were negative, you would draw it clockwise. Each quadrant is a total of 90 degrees or pi/2 radians. When you reach 360 degrees or 2pi radians, you are back at the right horizontal axis. In this way, if theta is greater than 360 (or less than 0 for that matter), it can be converted to another number that fits in that range, which makes for easier graphing.

This coordinate system has only been useful, at least for me, in Calculus. Some integrals are only solvable if you use a change of variables, which sometimes involves exchanging x and y for r and theta. The equations for such are below:

r=(x2+y2)1/2 <-- Comes from our good friend Pythagorus

x=r cos theta
y=r sin theta