


Polar Coordinates
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 xaxis
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 xaxis 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 xaxis
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.
Conversions
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°).
