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Cosine of the Difference of Two Angles

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 .

Angle Sum Formula for Cosine