Double Angle Formulae — CAIE A-Level Mathematics

Introduction

The double angle formulae are among the most frequently tested identities in 9709 Paper 3. They arise whenever an expression or equation involves $\sin 2A$ , $\cos 2A$ , or $\tan 2A$ — or, crucially, whenever a product such as $\sin A \cos A$ or a squared trigonometric term needs to be rewritten. Mastery of these formulae unlocks exact evaluation of trigonometric expressions, allows equations in mixed angles (e.g. one term in $2x$ and another in $x$ ) to be reduced to a single variable, and underpins later work on integration of $\sin^2 x$ and $\cos^2 x$ .

Core Concept

The double angle formulae are special cases of the compound angle formulae with both angles equal to $A$ . Starting from:

\sin(A + B) = \sin A \cos B + \cos A \sin B

set $B = A$ to obtain $\sin 2A$ . Similarly for cosine and tangent. This derivation is something examiners expect candidates to be able to reproduce or apply fluently.

The key insight for solving equations is that every formula converts between a function of $2A$ and functions of $A$ alone, allowing a single-variable polynomial (or factorised) equation to emerge. The three forms of $\cos 2A$ are especially powerful: the choice of which form to use is dictated by what other terms appear in the expression or equation.

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Introduction

Core Concept

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