# Sine and Cosine Functions

- Author:
- Albert Navetta

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.