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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 tex2html_wrap_inline713 .

The trigonometric functions or ratios are often referred to as the circular functions. Let tex2html_wrap_inline715 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 tex2html_wrap_inline715 .

angle2.epsDefinition: The cosine of tex2html_wrap_inline715 , denoted tex2html_wrap_inline725 , is the first, or x coordinate of the corresponding point P on the unit circle. In the figure above, tex2html_wrap_inline731 .

Definition: The sine of tex2html_wrap_inline715 , denoted tex2html_wrap_inline735 , is the second, or y coordinate of the corresponding point P on the unit circle. In figure above, tex2html_wrap_inline741 .

Since tex2html_wrap_inline743 is a point on the unit circle, tex2html_wrap_inline745 .

Note that the largest value of tex2html_wrap_inline735 is 1 and is attained, for example, at tex2html_wrap_inline751 , tex2html_wrap_inline753 , and tex2html_wrap_inline755 . The smallest value of tex2html_wrap_inline735 is -1 and is attained, for example, at tex2html_wrap_inline761 , tex2html_wrap_inline763 , and tex2html_wrap_inline765 .

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 tex2html_wrap_inline771 , rises to 1 when tex2html_wrap_inline751 , drops to 0 when tex2html_wrap_inline779 , drops to -1 when tex2html_wrap_inline761 , and rises to 0 when tex2html_wrap_inline787 . How does the first coordinate of the point (the cosine of the angle). It begins at 1 when tex2html_wrap_inline771 , drops to 0 when tex2html_wrap_inline751 , drops to -1 when tex2html_wrap_inline779 , rises to 0 when tex2html_wrap_inline761 , and rises to 1 when tex2html_wrap_inline787 . Sketch the graphs of sine and cosine.

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