Solve the equation $\cos 2x = \sin x$ for $-\pi \le x \le \pi$ . Give your answers as exact radian values.

$x = -\tfrac{\pi}{2}, \tfrac{\pi}{6}, \tfrac{5\pi}{6}$

Solve the equation $\tan 2x = 4\tan x$ for $0^\circ \le x \le 180^\circ$. Give your answers to 1 decimal place where appropriate.

M1 for substituting the identity $\tan 2x = \frac{2\tan x}{1 - \tan^2 x}$. M1 for rearranging to $2\tan x = 4\tan x (1 - \tan^2 x)$. M1 for grouping terms on one side and factorising to get $2\tan x(2\tan^2 x - 1) = 0$. A1 for finding solutions from $\tan x = 0$, giving $x = 0^\circ$ and $180^\circ$. A1 for finding solutions from $\tan^2 x = 0.5 \implies \tan x = \pm \frac{1}{\sqrt{2}}$, giving $x = 35.3^\circ$ and $x = 144.7^\circ$.

Edexcel IAL P3 Trig Addition Formulae

Q: Are the addition formulae in the IAL formula booklet?

On our IAL formula sheet the sin, cos and tan addition formulae are marked In the formula booklet, while the double angle formulae and the R form are marked Learn it.

Q: Why write a sin x + b cos x as R sin(x + α)?

One trig function is easier to work with: you can solve the equation in one step and read off the greatest value R and least value −R.

Q: Is a calculator allowed in IAL P3?

Yes. A calculator is allowed in every Edexcel IAL maths unit, P3 included. Method marks are for the working you write down, so show it.

Q: How long is the IAL P3 exam?

Each IAL maths unit, P3 included, is one written paper of 1 hour 30 minutes worth 75 marks.

Q: How often does the addition and double angle formulae come up in P3?

In our mapping of the 4 Pearson P3 papers from June 2024 to June 2025, the addition and double angle formulae was tested in 4 of them (6 questions in all). Any topic can be examined in any series.

P3 Trigonometric Addition Formulae notes

Compound angle formulae

$\sin (A \pm B) = \sin A \cos B \pm \cos A \sin B$ , $\cos (A \pm B) = \cos A \cos B \mp \sin A \sin B$ and $\tan (A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A \tan B}$ . They give exact values: $\sin 75^{\circ} = \sin (45^{\circ} + 30^{\circ}) = \frac{\sqrt{6} + \sqrt{2}}{4}$ .

Double angle formulae

$\sin 2 A = 2 \sin A \cos A$ , $\tan 2 A = \frac{2 \tan A}{1 - \tan^{2} A}$ and $\cos 2 A = \cos^{2} A - \sin^{2} A = 2 \cos^{2} A - 1 = 1 - 2 \sin^{2} A$ . Choose the form of $\cos 2 A$ that leaves one trig function in the equation. $\sin 2 x = \cos x$ becomes $\cos x (2 \sin x - 1) = 0$ , so for $0^{\circ} \leq x < 360^{\circ}$ : $x = 30^{\circ}, 90^{\circ}, 150^{\circ}, 270^{\circ}$ .

The R sin(θ + α) form

$a \sin θ + b \cos θ = R \sin (θ + α)$ with $R = \sqrt{a^{2} + b^{2}}$ and $\tan α = \frac{b}{a}$ . So $3 \sin x + 4 \cos x = 5 \sin (x + {53.1}^{\circ})$ . Its greatest value is $5$ , when $x + {53.1}^{\circ} = 90^{\circ}$ , that is $x = {36.9}^{\circ}$ . Expand $R \sin (θ + α)$ and compare coefficients rather than memorising sign rules.

Key formulas

Compound sin In the formula booklet: $\sin (A \pm B) = \sin A \cos B \pm \cos A \sin B$
Compound cos In the formula booklet: $\cos (A \pm B) = \cos A \cos B \mp \sin A \sin B$
Compound tan In the formula booklet: $\tan (A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A \tan B}$
Double angle Learn it: $\sin 2 A = 2 \sin A \cos A, \cos 2 A = 2 \cos^{2} A - 1 = 1 - 2 \sin^{2} A, \tan 2 A = \frac{2 \tan A}{1 - \tan^{2} A}$
$R$ form Learn it: $a \sin θ \pm b \cos θ = R \sin (θ \pm α), R = \sqrt{a^{2} + b^{2}}, \tan α = \frac{b}{a}$

In the formula booklet = printed in Pearson's IAL formulae booklet, so know where to find it. Learn it = not printed there, so learn it by heart. All P3 formulas on one page (free PDF) →

P3 Trigonometric Addition Formulae worked examples

Each mark scheme is checked line by line before it is published here. M marks are for method, A marks for accurate answers that follow from them, B marks for independent results.

Worked example 1 · 6 marks · hard

Solve the equation $\cos 2 x = \sin x$ for $- π \leq x \leq π$ . Give your answers as exact radian values.

Mark scheme

Substitute the double angle identity $\cos 2 x = 1 - 2 \sin^{2} x$ . M1
$1 - 2 \sin^{2} x = \sin x ⟹ 2 \sin^{2} x + \sin x - 1 = 0$ . A1
Factorise the quadratic: $(2 \sin x - 1) (\sin x + 1) = 0$ . M1A1
Case 1: $2 \sin x - 1 = 0 ⟹ \sin x = \frac{1}{2}$ .
Within the domain $- π \leq x \leq π$ , solutions are $x = \frac{π}{6}$ and $x = \frac{5 π}{6}$ . A1
Case 2: $\sin x + 1 = 0 ⟹ \sin x = - 1$ .
Within the domain $- π \leq x \leq π$ , the solution is $x = - \frac{π}{2}$ . A1

Worked example 2 · 5 marks · hard

Solve the equation $\tan 2 x = 4 \tan x$ for $0^{\circ} \leq x \leq 180^{\circ}$ . Give your answers to 1 decimal place where appropriate.

Mark scheme

M1 for substituting the identity $\tan 2 x = \frac{2 \tan x}{1 - \tan^{2} x}$ .
M1 for rearranging to $2 \tan x = 4 \tan x (1 - \tan^{2} x)$ .
M1 for grouping terms on one side and factorising to get $2 \tan x (2 \tan^{2} x - 1) = 0$ .
A1 for finding solutions from $\tan x = 0$ , giving $x = 0^{\circ}$ and $180^{\circ}$ .
A1 for finding solutions from $\tan^{2} x = 0.5 ⟹ \tan x = \pm \frac{1}{\sqrt{2}}$ , giving $x = {35.3}^{\circ}$ and $x = {144.7}^{\circ}$ .

5 steps · 1 to fill in

Worked example 3 · 4 marks · medium

Given that $\sin A = \frac{4}{5}$ and $\cos B = \frac{12}{13}$ , where both $A$ and $B$ are acute angles, find the exact value of $\sin (A + B)$ .

Mark scheme

M1 for using a right-angled triangle or identity to find $\cos A = \frac{3}{5}$ .
M1 for finding $\sin B = \frac{5}{13}$ .
M1 for applying the formula $\sin (A + B) = \sin A \cos B + \cos A \sin B$ .
A1 for calculating $(\frac{4}{5}) (\frac{12}{13}) + (\frac{3}{5}) (\frac{5}{13}) = \frac{48}{65} + \frac{15}{65} = \frac{63}{65}$ .

4 steps · 1 to fill in

Common P3 the addition and double angle formulae mistakes

Splitting sin(A + B). $\sin (A + B)$ is not $\sin A + \sin B$ : use the formula.
The sign in cos(A + B). $\cos (A + B) = \cos A \cos B - \sin A \sin B$ : the sign flips.
Dividing by cos x. In $\sin 2 x = \cos x$ , dividing by $\cos x$ loses $x = 90^{\circ}$ and $270^{\circ}$ . Factorise instead.
The quadrant of α. With $R \cos α = a$ and $R \sin α = b$ both positive, $α$ is acute; check the signs before taking $\tan^{- 1}$ .

Every P3 mistake from the examiner reports, with the paper it came from →

P3 Trigonometric Addition Formulae FAQ

Are the addition formulae in the IAL formula booklet?

On our IAL formula sheet the sin, cos and tan addition formulae are marked In the formula booklet, while the double angle formulae and the R form are marked Learn it.

Why write a sin x + b cos x as R sin(x + α)?

One trig function is easier to work with: you can solve the equation in one step and read off the greatest value R and least value −R.

Is a calculator allowed in IAL P3?

Yes. A calculator is allowed in every Edexcel IAL maths unit, P3 included. Method marks are for the working you write down, so show it.

How long is the IAL P3 exam?

Each IAL maths unit, P3 included, is one written paper of 1 hour 30 minutes worth 75 marks.

How often does the addition and double angle formulae come up in P3?

In our mapping of the 4 Pearson P3 papers from June 2024 to June 2025, the addition and double angle formulae was tested in 4 of them (6 questions in all). Any topic can be examined in any series.

Next steps

More P3: Functions · Numerical Methods · Further Differentiation · Further Integration · Algebraic Methods · Trigonometric Functions · Exponentials and Logarithms

Edexcel IAL P3 Trigonometric Addition Formulae: exam questions and answers, notes and worked examples

Try it first

P3 Trigonometric Addition Formulae notes

Compound angle formulae

Double angle formulae

The R sin(θ + α) form

Key formulas

P3 Trigonometric Addition Formulae worked examples

Worked example 1 · 6 marks · hard

Mark scheme

Worked example 2 · 5 marks · hard

Mark scheme

Worked example 3 · 4 marks · medium

Mark scheme

Common P3 the addition and double angle formulae mistakes

P3 Trigonometric Addition Formulae FAQ

Are the addition formulae in the IAL formula booklet?

Why write a sin x + b cos x as R sin(x + α)?

Is a calculator allowed in IAL P3?

How long is the IAL P3 exam?

How often does the addition and double angle formulae come up in P3?

Next steps

Your P3 revision kit