The Unit Circle
The unit circle is the circle whose center is at the
origin and whose radius is one. The
circumfrence of the unit circle is 2Π. An
arc of the unit circle has the same length as the
measure of the central angle that intercepts that
arc. Also, because the radius of the unit circle is one, the trigonometric
functions sine and cosine have special relevance for the unit circle.
If a point on the circle is on the terminal side
of an angle in standard
position, then the sine of such an angle is simply
the y-coordinate of the point, and the cosine of
the angle is the x-coordinate of the point.
Figure 5.1: The unit circle
This relationship has practical uses concerning the length of an arc on the unit
circle. If an arc has one endpoint at (1,0) and extends in the counterclockwise
direction, the other endpoint of the arc can be determined if the arc length is
known. Given an arc length s, the other endpoint of the arc is provided by the
coordinates (cos(s), sin(s)). This is a common alternative way to
plot the unit circle. Most often, the unit
circle can be drawn according to the equation x2 + y2 = 1. As we have seen
here, though, it can also be drawn according to the equations x = cos(s), y = sin(s), where s is the length of the arc starting at (1,0).
Figure 5.2: The unit circle
