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Sine and Cosine Functions

On the unit circle, we define the sine of an angle , denoted , as the -coordinate of the terminal point of  and the cosine of  , denoted as the -coordinate of the terminal point of .  Recall that the unit circle is the graph of the equation . By substituting and  we immediately have the Pythagorean identity

Since the arc length is the same as the angle subtended on the unit circle,  can be considered an angle or a real number corresponding to the arc length on the unit circle starting from the point .  Counter-clockwise rotation is considered a positive direction of rotation, and clockwise is negative.  

Sine and Cosine in the Unit Circle

In circles of arbitrary radius the arc length,  and coordinates all grow or shrink proportionally to  So the coordinates of the terminal point are and .   Dividing both sides of both equations by gives the more general definitions for sine and cosine for a circle of any radius.

Sine and Cosine in Circle of Radius r

The circle of radius centered at the origin is defined by the equation . Substituting for and we have . Dividing both sides by , we see that the Pythagorean identity holds for a circle of any radius.