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## Definitions of Sine and Cosine

The circle with center at the origin and radius of one unit is often called the unit circle. The equation of the unit circle is given by .

The trigonometric functions or ratios are often referred to as the circular functions. Let be an angle of rotation about the origin, measured from the positive x-axis, where a counterclockwise rotation produces a positive angle, as shown here. The point P(a,b) on the unit circle corresponds to .

angle2.epsDefinition: The cosine of , denoted , is the first, or x coordinate of the corresponding point P on the unit circle. In the figure above, .

Definition: The sine of , denoted , is the second, or y coordinate of the corresponding point P on the unit circle. In figure above, .

Since is a point on the unit circle, .

Note that the largest value of is 1 and is attained, for example, at , , and . The smallest value of is -1 and is attained, for example, at , , and .

Place your finger on the point (1,0), and trace around the unit circle counterclockwise. While doing this, how does the second coordinate of the point on the unit circle (the sine of the angle) change? It begins at 0 when , rises to 1 when , drops to 0 when , drops to -1 when , and rises to 0 when . How does the first coordinate of the point (the cosine of the angle). It begins at 1 when , drops to 0 when , drops to -1 when , rises to 0 when , and rises to 1 when . Sketch the graphs of sine and cosine.

Carl Lee
Wed Apr 21 08:17:28 EDT 1999