Trigonometry: Trigonometric Functions

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 %: 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 x 2 + y 2 = 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 %: The unit circle

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