Polar coordinates make rare appearances on the Math IIC. They offer an alternate way of expressing the location of a point in the coordinate plane. Instead of measuring a given point’s distance from the y and x axes, as rectangular coordinates do, polar coordinates measure the distance between a point and the origin, and the angle between the positive x-axis and the ray whose endpoint is the origin that contains the point. The distance from a point to the origin is the first coordinate, r, and the angle between the positive x-axis and the ray containing the point is the second coordinate, .

The distance r and the angle are both directed—meaning that they represent the distance and angle in a given direction. A positive angle starts at the positive x-axis and rotates counterclockwise, whereas a negative angle rotates clockwise. Once you have rotated through an angle of degrees (or radians), a positive value of r means that the point lies on the ray at which the angle terminated. If r is negative, however, then the point lies r units from the origin in the opposite direction of the specified angle. It is possible, therefore, to have negative values for both r and .

In the rectangular coordinate system, each point is specified by exactly one ordered pair. This is not true in the polar coordinate system. A point can be specified by many ordered pairs. To express the same point using different polar coordinates, simply add or subtract 360° to the measure of . The point (7, 45°), for example, can also be expressed as (7, 405°) or (7, –315°). Another way to express the same point using different polar coordinates is to add or subtract 180° and reverse the sign of r. The point (7, 45°), for example, is the same as (–7, 225°) and (–7, –135°).

Generally speaking, any point (r, ) is also given by the coordinates (r, + 2nπ) and (–r, + (2n + 1)π), where n is an integer.

However, the usual way to express a point in polar coordinates is with a positive r and between 0° and 360°. A given point has only one set of polar coordinates that satisfies these conditions.

For the Math IIC, you should know how to convert polar coordinates into rectangular coordinates and back.

To make these conversions, you have to have some knowledge of trigonometry, which we will cover in the next chapter. To find the normal rectangular coordinates of the point (r, ), use the following two formulas:

To find the polar coordinates of the point (x, y), use these formulas:

The diagram below might help you see these relationships:

For example, the point (12, 60°) can be expressed by rectangular coordinates as the point (12 cos 60°, 12 sin 60°) = (6,¹²⁄₂). Practice this conversion by finding the polar coordinates of (–2, –2).

So the polar coordinates of (–2, –2) are (2, 45°).