Solve $2\sin^2 x + 3\cos x - 3 = 0$ for $0^\circ \le x \le 360^\circ$.

M1 for substituting $\sin^2 x$ with $1 - \cos^2 x$ to get $2(1 - \cos^2 x) + 3\cos x - 3 = 0$. M1 for expanding and rearranging to form the quadratic $2\cos^2 x - 3\cos x + 1 = 0$. M1 for factorising the quadratic as $(2\cos x - 1)(\cos x - 1) = 0$. A1 for finding the solutions from $\cos x = 0.5$ giving $x = 60^\circ$ and $300^\circ$. A1 for finding the solutions from $\cos x = 1$ giving $x = 0^\circ$ and $360^\circ$.

Solve the equation $2\cos^2 x - \sin x - 1 = 0$ for $0^\circ \le x \le 360^\circ$.

M1 for using the identity $\cos^2 x = 1 - \sin^2 x$ to rewrite the equation: $2(1 - \sin^2 x) - \sin x - 1 = 0$. M1 for simplifying to a quadratic in $\sin x$: $2\sin^2 x + \sin x - 1 = 0$. M1 for factorising the quadratic to $(2\sin x - 1)(\sin x + 1) = 0$. A1 for finding the solutions from $\sin x = 0.5$, which are $x = 30^\circ, 150^\circ$. A1 for finding the solution from $\sin x = -1$, which is $x = 270^\circ$.

Solve $3\sin(2x - 15^\circ) = 2$ for $0^\circ \le x \le 180^\circ$. Give your answers to 1 decimal place.

M1 for rearranging to $\sin(2x - 15^\circ) = \frac{2}{3}$. M1 for finding the two primary solutions for $(2x - 15^\circ)$: $41.81^\circ$ and $180^\circ - 41.81^\circ = 138.19^\circ$. M1 for solving for $x$ by adding $15^\circ$ and dividing by $2$. A1 for stating the final values $x = 28.4^\circ$ and $x = 76.6^\circ$.

A Level Math Revision

Edexcel IAL P2 Trigonometric Identities and Equations: exam questions and answers, notes and worked examples

Q: Which trig identities do I need for IAL P2?

Two: tan θ = sin θ / cos θ and sin²θ + cos²θ = 1. Our IAL formula sheet marks them Learn it. Double angle and addition formulae come in P3.

Q: How do I find all the solutions of a trig equation?

Find the calculator's value, use the symmetry of the sin, cos or tan graph for the second value, then add or subtract the period (360° for sin and cos, 180° for tan) until you leave the interval.

Q: Is a calculator allowed in IAL P2?

Yes. A calculator is allowed in every Edexcel IAL maths unit, P2 included. Method marks are for the working you write down, so show it.

Q: How long is the IAL P2 exam?

Each IAL maths unit, P2 included, is one written paper of 1 hour 30 minutes worth 75 marks.

Q: How often does trigonometric identities and equations come up in P2?

In our mapping of the 4 Pearson P2 papers from June 2024 to June 2025, trigonometric identities and equations was tested in 4 of them (4 questions in all). Any topic can be examined in any series.

Updated October 2026 · By Pete Bromfield

P2 trigonometry uses two identities, $\tan θ = \frac{\sin θ}{\cos θ}$ and $\sin^{2} θ + \cos^{2} θ = 1$ , to prove other identities and to solve equations in a given interval, in degrees or radians. Find every solution in the interval: your calculator gives one, and the graph's symmetry gives the rest.

P2 · Pure Mathematics 2
Calculator allowed
3 worked examples
46 practice questions
Tested in 4 of the last 4 papers

Practise P2 trigonometric identities and equations →

Notes
Formulas
Worked examples
Mistakes
FAQ

P2 Trigonometric Identities and Equations notes

The two identities

$\tan θ \equiv \frac{\sin θ}{\cos θ}$ and $\sin^{2} θ + \cos^{2} θ \equiv 1$ . To prove an identity, start from one side and work to the other. For example $\frac{1 - \cos^{2} θ}{\cos θ} = \frac{\sin^{2} θ}{\cos θ} = \sin θ \tan θ$ .

Solving basic equations

Use the calculator value, then the symmetry of the graph. For $0^{\circ} \leq x < 360^{\circ}$ : $\sin x = 0.5$ gives $x = 30^{\circ}, 150^{\circ}$ ; $\cos x = - 0.5$ gives $x = 120^{\circ}, 240^{\circ}$ ; $\tan x = \sqrt{3}$ gives $x = 60^{\circ}, 240^{\circ}$ (tan repeats every $180^{\circ}$ ).

Equations like sin 2x = k

Change the interval first. For $\sin 2 x = \frac{\sqrt{3}}{2}$ with $0^{\circ} \leq x < 360^{\circ}$ , solve for $2 x$ in $0^{\circ} \leq 2 x < 720^{\circ}$ : $2 x = 60^{\circ}, 120^{\circ}, 420^{\circ}, 480^{\circ}$ , so $x = 30^{\circ}, 60^{\circ}, 210^{\circ}, 240^{\circ}$ .

Quadratics in sin, cos or tan

Use $\sin^{2} θ + \cos^{2} θ = 1$ to write everything in one function, then factorise. $2 \cos^{2} x + 3 \sin x = 3$ becomes $2 \sin^{2} x - 3 \sin x + 1 = 0$ , so $\sin x = 1$ or $\sin x = \frac{1}{2}$ , giving $x = 30^{\circ}, 90^{\circ}, 150^{\circ}$ for $0^{\circ} \leq x < 360^{\circ}$ . Reject any value outside $- 1 \leq \sin x \leq 1$ .

Key formulas

Trig Learn it: $\tan θ = \frac{\sin θ}{\cos θ}, \sin^{2} θ + \cos^{2} θ = 1$

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 P2 formulas on one page (free PDF) →

P2 Trigonometric Identities and Equations 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 · 5 marks · hard

Solve $2 \sin^{2} x + 3 \cos x - 3 = 0$ for $0^{\circ} \leq x \leq 360^{\circ}$ .

Mark scheme

M1 for substituting $\sin^{2} x$ with $1 - \cos^{2} x$ to get $2 (1 - \cos^{2} x) + 3 \cos x - 3 = 0$ .
M1 for expanding and rearranging to form the quadratic $2 \cos^{2} x - 3 \cos x + 1 = 0$ .
M1 for factorising the quadratic as $(2 \cos x - 1) (\cos x - 1) = 0$ .
A1 for finding the solutions from $\cos x = 0.5$ giving $x = 60^{\circ}$ and $300^{\circ}$ .
A1 for finding the solutions from $\cos x = 1$ giving $x = 0^{\circ}$ and $360^{\circ}$ .

5 steps

Worked example 2 · 5 marks · hard

Solve the equation $2 \cos^{2} x - \sin x - 1 = 0$ for $0^{\circ} \leq x \leq 360^{\circ}$ .

Mark scheme

M1 for using the identity $\cos^{2} x = 1 - \sin^{2} x$ to rewrite the equation: $2 (1 - \sin^{2} x) - \sin x - 1 = 0$ .
M1 for simplifying to a quadratic in $\sin x$ : $2 \sin^{2} x + \sin x - 1 = 0$ .
M1 for factorising the quadratic to $(2 \sin x - 1) (\sin x + 1) = 0$ .
A1 for finding the solutions from $\sin x = 0.5$ , which are $x = 30^{\circ}, 150^{\circ}$ .
A1 for finding the solution from $\sin x = - 1$ , which is $x = 270^{\circ}$ .

5 steps · 1 to fill in

Worked example 3 · 4 marks · hard

Solve $3 \sin (2 x - 15^{\circ}) = 2$ for $0^{\circ} \leq x \leq 180^{\circ}$ . Give your answers to 1 decimal place.

Mark scheme

M1 for rearranging to $\sin (2 x - 15^{\circ}) = \frac{2}{3}$ .
M1 for finding the two primary solutions for $(2 x - 15^{\circ})$ : ${41.81}^{\circ}$ and $180^{\circ} - {41.81}^{\circ} = {138.19}^{\circ}$ .
M1 for solving for $x$ by adding $15^{\circ}$ and dividing by $2$ .
A1 for stating the final values $x = {28.4}^{\circ}$ and $x = {76.6}^{\circ}$ .

4 steps · 1 to fill in

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Common P2 trigonometric identities and equations mistakes

Dividing by a function that can be zero. In $\sin x \cos x = \frac{1}{2} \cos x$ , dividing by $\cos x$ loses the solutions of $\cos x = 0$ . Move everything to one side and factorise instead.
Keeping the original interval. For $\sin 2 x$ or $\cos (x + 30^{\circ})$ , adjust the interval for the new angle first, or solutions go missing.
Accepting impossible values. $\sin x = 2$ has no solutions. Say so and reject it.
Degrees and radians mixed. If the interval is given in radians, give the answers in radians (often as multiples of $π$ ).

Every P2 mistake from the examiner reports, with the paper it came from →

P2 Trigonometric Identities and Equations FAQ

Which trig identities do I need for IAL P2?

Two: tan θ = sin θ / cos θ and sin²θ + cos²θ = 1. Our IAL formula sheet marks them Learn it. Double angle and addition formulae come in P3.

How do I find all the solutions of a trig equation?

Find the calculator's value, use the symmetry of the sin, cos or tan graph for the second value, then add or subtract the period (360° for sin and cos, 180° for tan) until you leave the interval.

Is a calculator allowed in IAL P2?

Yes. A calculator is allowed in every Edexcel IAL maths unit, P2 included. Method marks are for the working you write down, so show it.

How long is the IAL P2 exam?

Each IAL maths unit, P2 included, is one written paper of 1 hour 30 minutes worth 75 marks.

How often does trigonometric identities and equations come up in P2?

In our mapping of the 4 Pearson P2 papers from June 2024 to June 2025, trigonometric identities and equations was tested in 4 of them (4 questions in all). Any topic can be examined in any series.

Next steps

More P2: Sequences and Series · Logarithms and Exponentials · Circles · Algebraic Methods · Proof · The Binomial Expansion · Differentiation (stationary points) · Integration (definite integrals and area)

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Edexcel IAL P2 Trigonometric Identities and Equations: exam questions and answers, notes and worked examples

Try it first

P2 Trigonometric Identities and Equations notes

The two identities

Solving basic equations

Equations like sin 2x = k

Quadratics in sin, cos or tan

Key formulas

P2 Trigonometric Identities and Equations worked examples

Worked example 1 · 5 marks · hard

Mark scheme

Worked example 2 · 5 marks · hard

Mark scheme

Worked example 3 · 4 marks · hard

Mark scheme

Common P2 trigonometric identities and equations mistakes

P2 Trigonometric Identities and Equations FAQ

Which trig identities do I need for IAL P2?

How do I find all the solutions of a trig equation?

Is a calculator allowed in IAL P2?

How long is the IAL P2 exam?

How often does trigonometric identities and equations come up in P2?

Next steps

Your P2 revision kit