Compound Angle Formulae — Pure Mathematics 3 (9709)

Core Concept

The central idea is that $\sin(A + B) \neq \sin A + \sin B$. Instead, the correct expansion mixes sine and cosine of each angle. The formulae are derived geometrically (from right-triangle constructions) and hold for all real values of $A$ and $B$.

When you encounter an expression like $\cos(x - 30°) - \sqrt{3}\sin(x - 60°)$ (the syllabus example), the strategy is:

1. Expand each compound angle using the appropriate formula.
2. Collect like terms in $\sin x$ and $\cos x$.
3. Recognise a simplification — often the result collapses to a single term.

For exact evaluation, choose $A$ and $B$ from the standard angles $\{0°, 30°, 45°, 60°, 90°\}$ so that their sum or difference gives the target angle.

For solving equations, expand $\sin(x + \alpha) = k$ into $a\sin x + b\cos x = k$, or vice versa, then use a suitable technique (e.g. the $R$-form, or isolating the compound angle directly).