Expressing a sinθ + b cosθ in R Form (R sin/cos) — CAIE A-Level Mathematics

Introduction

Many exam questions in 9709 Paper 3 involve expressions of the form $a\sin\theta + b\cos\theta$ , where $a$ and $b$ are real constants. On their own, these combined expressions are awkward to work with — they cannot be solved or optimised directly. The R-form (or harmonic form) rewrites them as a single sinusoidal function with amplitude $R$ and a phase shift $\alpha$ . This unlocks two powerful applications that appear repeatedly in exam questions:

Solving equations such as $3\cos\theta + 2\sin\theta = 1$ .
Finding maximum and minimum values of expressions involving $a\sin\theta + b\cos\theta$ .

The method rests entirely on the compound angle formulae you already know, so no new identities need to be memorised beyond the form itself.

Core Concept

The idea is to match $a\sin\theta + b\cos\theta$ to one of four expanded compound-angle expressions. The most commonly used forms are:

R\sin(\theta + \alpha) = R\sin\theta\cos\alpha + R\cos\theta\sin\alpha

R\sin(\theta - \alpha) = R\sin\theta\cos\alpha - R\cos\theta\sin\alpha

R\cos(\theta - \alpha) = R\cos\theta\cos\alpha + R\sin\theta\sin\alpha

R\cos(\theta + \alpha) = R\cos\theta\cos\alpha - R\sin\theta\sin\alpha

Choosing the correct form: Match the signs in the target expression to the signs produced by the expansion.

Target expression	Natural form to use
$a\sin\theta + b\cos\theta$ ( $a,b > 0$ )	$R\sin(\theta + \alpha)$
$a\sin\theta - b\cos\theta$ ( $a,b > 0$ )	$R\sin(\theta - \alpha)$
$a\cos\theta + b\sin\theta$ ( $a,b > 0$ )	$R\cos(\theta - \alpha)$
$a\cos\theta - b\sin\theta$ ( $a,b > 0$ )	$R\cos(\theta + \alpha)$

Finding R and α:

Expanding $R\sin(\theta + \alpha)$ and comparing coefficients of $\sin\theta$ and $\cos\theta$ with $a\sin\theta + b\cos\theta$ :

R\cos\alpha = a \qquad \text{and} \qquad R\sin\alpha = b

Squaring and adding eliminates $\alpha$ ; dividing gives $\tan\alpha$ .

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Introduction

Core Concept

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