15. Polar Coordinates
a.1. Polar Coordinate System
Rectangular coordinates are one way to specify a point, , in the plane, but they are not the only way. When we are studying circles, it is useful to use Polar Coordinates which identify the point, , using
- the radius and
- the polar angle .
Like rectangular coordinates , the polar coordinates can be written as an ordered pair, . So you need to be careful to say whether this ordered pair is rectangular or polar.

Most often, the radius measures the (positive) distance from the origin, , to the point, , and the polar angle measures an angle (usually measured in radians and usually with ) counterclockwise from the positive -axis to the ray .
However, frequently the angle is allowed to be bigger than or is allowed to be negative, in which case it is measured clockwise from the positve -axis. Consequently, is non-unique; we can always add or subtract an arbitrary multiple of or . Thus the point with polar coordinates can also be written as , or , or , or all of which are shown in the plot with blue angles positive and red angles negative.

Further, once in a while, we allow to be negative, for example in the context of solving equations or graphing. Then is the negative of the distance from to . When is negative, the point is obtained by going a distance along the ray at the angle . Think of this as going backwards along the ray at the angle . Thus the point with polar coordinates is actually the point or .

Identify the point with each of the following coordinates:
= | |||||
= | |||||
= | |||||
= | |||||
= | |||||

Notice that for the two points with negative radii, and , we have to go backwards along the angles and , respectively.
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