# Cosine of the Difference of Two Angles

- Author:
- Brian Sterr

## Cosine of the Sum/Difference of Two Angles

Using the diagram above, we can show two angles, and in standard position. Let and be the points where their terminal sides meet the unit circle.

*Show Alpha and Beta using the checkboxes.**Show the difference*Now, we can put this angle into standard position, by rotating the triangle.*Check "Rotation" and use the slider to rotate the triangle into standard position. Check "Rotated Labels"*If is the point where the terminal side of , now in standard position, meets the unit circle, then it has coordinates:Why is ?

Use the coordinates of and the distance formula to rewrite this equation. Square both sides. Expand everything and simplify. Finally solve for .

## Angle Difference Formula for Cosine

Now apply this formula using the angles and . Use even and odd properties to simplify so everything is expressed in terms of functions of and .