What Are the Compound Angle Identities for Sine, Cosine, and Tangent?

The compound angle identities express the sine, cosine, or tangent of a sum or difference of two angles. In IB Mathematics: Analysis and Approaches HL, you must know and apply all six identities below.

For tangent, the denominator must be non-zero. These identities belong to AHL 3.10: Compound angle identities and are HL-only content.

The sign pattern is important. The sine identity keeps the original sign, while the cosine identity changes it. In the tangent identity, the numerator keeps the original sign and the denominator uses the opposite sign.

These formulas combine the horizontal and vertical coordinates of rotations on the unit circle. They are especially useful when an angle is not one of the standard exact angles but can be written as a sum or difference of them. The tangent formulas follow by dividing the sine identity by the cosine identity and using $\tan x=\frac{\sin x}{\cos x}$.

For example, calculate $\sin 75^\circ$ exactly by writing $75^\circ=45^\circ+30^\circ$:

$$\begin{aligned} \sin 75^\circ &=\sin(45^\circ+30^\circ)\\ &=\sin45^\circ\cos30^\circ+\cos45^\circ\sin30^\circ\\ &=\frac{\sqrt{2}}{2}\cdot\frac{\sqrt{3}}{2}+\frac{\sqrt{2}}{2}\cdot\frac{1}{2}\\ &=\frac{\sqrt{6}+\sqrt{2}}{4}. \end{aligned}$$

Here, both $45^\circ$ and $30^\circ$ have known exact ratios, so the sum identity converts an unfamiliar value into surds. The final answer should remain exact rather than being rounded to a decimal.

A common misconception is to write $\sin(A+B)=\sin A+\sin B$. This is false: trigonometric functions do not distribute over addition. Also check that every tangent denominator is non-zero before using it.

Exam technique: On Paper 1, split an unfamiliar angle into standard exact angles and show the identity before substituting values. These identities may also derive double-angle identities or simplify and solve trigonometric equations. In a "show that" question, begin from one side and transform it until it matches the other; do not assume the result you are asked to establish. On Paper 2, a GDC may check a value, but it does not replace the algebraic setup.