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A-Level Maths · Pure · Trigonometry · last updated

Trigonometry practice questions (A-Level Maths)

This pure topic covers radians, arcs and sectors, the sine and cosine rules, exact values, trigonometric identities, the reciprocal and inverse functions, addition and double-angle formulae, the R-alpha form, small-angle approximations, and solving equations in an interval. Below is the one free A-Level Maths question on it from our free exams, with its worked answer behind a toggle, then 7 key terms with the formulas typeset, and a door to make an A-Level Maths-style paper on this topic from your own notes. The free exams are typeset PDFs with an answer key; the exam downloads without a signup.

Make a practice exam from your notes →Upload your notesA-Level Maths style preselected; tick "Pure · Trigonometry" on the next screen. Free to try, no card.

Sample questions

The one free A-Level Maths question on this topic, from the free A-Level Maths Practice Paper. Each answer is behind its own toggle; the whole paper, its PDF and the full answer key are on the exam page.

  1. Question 16 marks · Trigonometric equations
    Solve the equation 2sin⁡2θ+3cos⁡θ=32\sin^2\theta + 3\cos\theta = 3 for 0∘≤θ≤360∘0^\circ \leq \theta \leq 360^\circ. Show each step of your working.
    2sin⁡2θ+3cos⁡θ=3,0∘≤θ≤360∘2\sin^2\theta + 3\cos\theta = 3, \qquad 0^\circ \leq \theta \leq 360^\circ
    From the free A-Level Maths Practice Paper →

Key terms

The definitions and formulas A-Level Maths expects for trigonometry, typeset as they appear on the paper.

Radians, arc length and sector area
An angle in radians is the arc length divided by the radius. Arc length and sector area take their simple forms only in radians.
s=rθ,A=12r2θs = r\theta, \qquad A = \frac{1}{2} r^2 \theta
Pythagorean identities
The basic identity and the two obtained by dividing it by cos⁡2θ\cos^2\theta and sin⁡2θ\sin^2\theta, used to turn an equation into a quadratic in a single ratio.
sin⁡2θ+cos⁡2θ=1,1+tan⁡2θ=sec⁡2θ,1+cot⁡2θ=cosec⁡2θ\sin^2\theta + \cos^2\theta = 1, \qquad 1 + \tan^2\theta = \sec^2\theta, \qquad 1 + \cot^2\theta = \cosec^2\theta
Addition and double-angle formulae
The compound-angle results, in the formulae booklet, and the double-angle results derived from them, which are not.
sin⁡(A±B)=sin⁡Acos⁡B±cos⁡Asin⁡B,cos⁡2A=cos⁡2A−sin⁡2A=2cos⁡2A−1=1−2sin⁡2A\sin(A \pm B) = \sin A\cos B \pm \cos A\sin B, \qquad \cos 2A = \cos^2 A - \sin^2 A = 2\cos^2 A - 1 = 1 - 2\sin^2 A
R-alpha form
Writing acos⁡θ+bsin⁡θa\cos\theta + b\sin\theta as a single cosine or sine, which gives the maximum and minimum values and solves equations of that form.
acos⁡θ+bsin⁡θ=Rcos⁡(θ−α),R=a2+b2,tan⁡α=baa\cos\theta + b\sin\theta = R\cos(\theta - \alpha), \qquad R = \sqrt{a^2 + b^2}, \quad \tan\alpha = \frac{b}{a}
Small-angle approximations
Valid for θ\theta in radians close to zero, and used to find limits and approximate values.
sin⁡θ≈θ,cos⁡θ≈1−θ22,tan⁡θ≈θ\sin\theta \approx \theta, \qquad \cos\theta \approx 1 - \frac{\theta^2}{2}, \qquad \tan\theta \approx \theta
Solving trigonometric equations in an interval
Find the principal value, use the symmetry of the graph or the CAST diagram for the other solutions, and adjust the interval when the argument is 2x2x or x−30°x - 30° before solving.
Sine rule, cosine rule and area of a triangle
For any triangle; the ambiguous case of the sine rule can give two possible angles.
asin⁡A=bsin⁡B=csin⁡C,a2=b2+c2−2bccos⁡A,Area=12absin⁡C\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}, \qquad a^2 = b^2 + c^2 - 2bc\cos A, \qquad \text{Area} = \frac{1}{2}ab\sin C

Frequently asked questions

Are there free trigonometry practice questions for A-Level Maths?

Yes. 1 question on trigonometry appears in the free A-Level Maths Practice Paper, and the sample questions on this page come from it with worked answers. Each exam can be read online or downloaded as a typeset PDF without signing up, and its full answer key is emailed on request.

How do I make an A-Level Maths practice paper on trigonometry from my notes?

Open the create page with the A-Level Maths style preselected, tick "Pure · Trigonometry" under the units to cover (or upload your notes and slides on it), choose the question mix and time limit, and generate. You get a typeset PDF in the A-Level format with a separate answer key, in about a minute. It is free to try, no card needed.

What does A-Level Maths expect you to know about trigonometry?

This pure topic covers radians, arcs and sectors, the sine and cosine rules, exact values, trigonometric identities, the reciprocal and inverse functions, addition and double-angle formulae, the R-alpha form, small-angle approximations, and solving equations in an interval. The terms to have cold: Radians, arc length and sector area, Pythagorean identities, Addition and double-angle formulae, R-alpha form, Small-angle approximations. Each is defined below with its formula typeset where one applies.

What is the A-Level Maths exam format?

The A-Level Maths guide explains the papers, their timing and mark allocation, with a sample question and mark scheme and the common mistakes, and links to the generator with the A-Level Maths style preselected.

Other A-Level Maths topics

All A-Level Maths topics · A-Level Maths exam guide · All free practice exams

Make an A-Level Maths paper on trigonometry from your notes

Open "Pick your exam" with the A-Level Maths style preselected and tick "Pure · Trigonometry", or upload your notes on it. You get a typeset paper in the A-Level format with a separate answer key.

Make a practice exam from your notes →Upload your notes