The Polar Coordinate System
The most familiar graphing plane is the one using the Cartesian coordinates. You find two perpendicular lines called axes. The horizontal axis (or x-axis) and vertical axis (or y-axis) intersect at the origin, the point labeled (0, 0). The points on this Cartesian plane are labeled with an ordered pair (x,y) where the two coordinates represent the positive or negative distance from the origin, parallel to their respective axis.
Another way of representing points is to use polar coordinates. One major difference between this graphing plane and the Cartesian plane is that you need only one axis, the polar axis. The pole is a point on the polar axis somewhat corresponding to the origin on the Cartesian plane.
To use the polar coordinates, first recall that an angle sketched in standard position on the Cartesian plane has its vertex at the origin, its initial side along the positive x-axis, and its terminal side consisting of a ray with its endpoint at the origin. Positive angles are measured in a counterclockwise direction.
Points are graphed, in polar coordinates, using the ordered pair (r, θ). The r represents a radius or distance from the pole, and the θ is the measure of an angle. When graphing points using these coordinates, first find the terminal ray of the angle in standard position. (Yes, you usually graph the first coordinate first, when graphing points (x,y) on the Cartesian plane. But this works better.) After you’ve found the terminal ray, find the point representing r (a distance) on that ray. If r is a positive number, you start at the pole and move along the ray for that distance and plot the point. If r is a negative, you find the ray that goes in the opposite direction, along the same straight line, and graph that point. Another way of saying this is: When r is negative, move to the ray that is 180 degrees greater than θ and graph the absolute value of r.
Polar coordinates have a special quality not found with Cartesian coordinates. You may think this is a problem, or you may think it’s a bonus. This quality or property is that the points in the polar plane can be represented by more than one set of coordinates. For example, the point
could also be labeled
Two labels for the same point.
The polar coordinates allow you to graph many very interesting curves that use trigonometric functions. You can graph circles, of course. But you can also graph cardioids (heart-shaped figures), lemniscates (flower-petal shapes), and many more.