Angle Multiple Identities

Angle Sum and Difference Identities

These identities are formed when adding/subtracting two angles in a trigonometric function.

Formula
Angle Sum and Difference Identities

Given two angles and :

Multiple Angle Formulas

The double angle formulas can be derived from the application of the previous sum and difference identities.

Formula
Double Angle Identities

Proof

Proof

Variations of the cosine double angle formula by utilizing the Pythagorean identity for sine and cosine: .

Formula
Variations of the Cosine Double Angle Formula

When :

When :

The triple angle formulas can be derived the same way.

Formula
Triple Angle Formulas

Proof

Proof

For the quadruple and quintuple angle formulas, we can also derive them in the same manner.

Formula
Quadruple Angle Formulas

Proof

Proof

Since :

Like the double-angle identity, we could also get other angles in terms of sine and cosine counterparts.