Compound Angle Formulae — CAIE A-Level Mathematics

Introduction

The compound angle formulae allow us to expand trigonometric expressions of the form $\sin(A \pm B)$ , $\cos(A \pm B)$ , and $\tan(A \pm B)$ into expressions involving $\sin A$ , $\cos A$ , $\sin B$ , and $\cos B$ separately. In the 9709 Paper 3 exam, these identities appear in three key ways: simplifying apparently complex expressions into a single trig function, evaluating exactly expressions at angles such as $75°$ or $\frac{5\pi}{12}$ that are not on the standard table, and solving equations that cannot be tackled directly. Mastery of these formulae is also the foundation for the double angle formulae and the $R\cos(\theta \pm \alpha)$ form studied later in the same unit.

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:

Expand each compound angle using the appropriate formula.
Collect like terms in $\sin x$ and $\cos x$ .
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).

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Introduction

Core Concept

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