Mathematics 9709 · syllabus 1.5, 2.3, 3.3
Cambridge A Level Mathematics 9709 — Trigonometry topical past papers
Trigonometric graphs, exact values, identities and equations, from sin²θ + cos²θ = 1 in Paper 1 to compound angles and R sin(θ + α) later on.
- Questions
- 289
- Years
- 2009–2026
- With mark scheme
- 99%
- Questions 2022–2026
- 94
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What Trigonometry covers
In Paper 1 (section 1.5) you sketch y = sin x, y = cos x and y = tan x and transformations of them, in degrees or radians, use the exact values for 30°, 45° and 60°, and work with the principal values given by sin⁻¹, cos⁻¹ and tan⁻¹. The two identities are tan θ ≡ sin θ / cos θ and sin²θ + cos²θ ≡ 1, used to prove other identities and to solve equations for every solution in a stated interval.
Papers 2 and 3 (sections 2.3 and 3.3, which have the same content) add sec, cosec and cot with their graphs, the identities sec²θ ≡ 1 + tan²θ and cosec²θ ≡ 1 + cot²θ, the compound-angle formulae for sin(A ± B), cos(A ± B) and tan(A ± B), the double-angle formulae, and writing a sin θ + b cos θ as R sin(θ ± α) or R cos(θ ± α). What is tested is choosing the identity that turns an equation into one you can solve.
Paper 1's version is assumed in Papers 2 and 3, so a later equation may still turn on sin²θ + cos²θ ≡ 1 or a principal value. Radians from circular measure are used throughout, and the transformations from functions explain graphs such as y = 3 − 2 sin 2x. In turn, trigonometry feeds the rest of the course: the trigonometric derivatives and integrals of Papers 2 and 3, integrating sin²x with a double-angle formula, the polar form of complex numbers in Paper 3, and resolving forces in Mechanics.
In the syllabus
The headings this topic has in the Cambridge International 2026–2027 syllabus, as the board prints them.
- 1.5 Trigonometry
- 2.3 Trigonometry
- 3.3 Trigonometry
What you should be able to do
- Sketch trigonometric graphs and their transformations, in degrees or radians
- Use exact values of sin, cos and tan for 30°, 45°, 60° and related angles
- Prove identities by working from one side to the other
- Solve trigonometric equations for every root in a given interval, including equations quadratic in sin θ, cos θ or tan θ
- Use the compound-angle and double-angle formulae (Papers 2 and 3)
- Express a sin θ + b cos θ as R sin(θ ± α) or R cos(θ ± α), and use it to solve equations or find greatest and least values (Papers 2 and 3)
Key ideas
- tan θ ≡ sin θ / cos θ and sin²θ + cos²θ ≡ 1
- sec²θ ≡ 1 + tan²θ and cosec²θ ≡ 1 + cot²θ (Papers 2 and 3)
- sin(A ± B) ≡ sin A cos B ± cos A sin B; cos(A ± B) ≡ cos A cos B ∓ sin A sin B
- sin 2A ≡ 2 sin A cos A; cos 2A ≡ cos²A − sin²A ≡ 2cos²A − 1 ≡ 1 − 2sin²A; tan 2A ≡ 2 tan A / (1 − tan²A)
- a sin θ + b cos θ ≡ R sin(θ + α), where R = √(a² + b²) and tan α = b/a (for a, b > 0)
- Principal values: −90° ≤ sin⁻¹x ≤ 90°, 0° ≤ cos⁻¹x ≤ 180°, −90° < tan⁻¹x < 90°
How Trigonometry has been examined
From 2022 to 2026, Trigonometry was on Paper 3 · Pure Mathematics 3 in 10 of 11 exam series, worth about 14% of that paper’s marks.
| Paper | Questions | Share of marks | Series |
|---|---|---|---|
| Paper 1 · Pure Mathematics 1 | 49 | 13% | 11 of 11 |
| Paper 3 · Pure Mathematics 3 | 45 | 14% | 10 of 11 |
Momentum’s own count, over the questions in its bank for these years: a series is one exam session, every zone included, and a question filed under two chapters counts in both. Median marks per question: 6.
Filed in Momentum under: Trigonometry.
A worked example
Express 3 sin θ + 4 cos θ in the form R sin(θ + α), where R > 0 and 0° < α < 90°. Hence solve 3 sin θ + 4 cos θ = 2 for 0° ≤ θ ≤ 360°.
- R sin(θ + α) = R sin θ cos α + R cos θ sin α, so R cos α = 3 and R sin α = 4.
- R = √(3² + 4²) = 5 and tan α = 4/3, so α = 53.13°.
- The equation becomes 5 sin(θ + 53.13°) = 2, so sin(θ + 53.13°) = 0.4.
- θ + 53.13° lies between 53.13° and 413.13°. sin⁻¹ 0.4 = 23.58°, which is too small, so θ + 53.13° = 180° − 23.58° = 156.42° or 360° + 23.58° = 383.58°.
- Subtract 53.13° from each: θ = 103.29° or 330.45°.
Answer: 3 sin θ + 4 cos θ = 5 sin(θ + 53.1°); θ = 103.3° or 330.4°
Written by Momentum for this page, with its own numbers; it is not taken from a past paper.
Mistakes that cost marks
- Stopping at the calculator's value: sin θ = 0.6 has the second solution 180° − 36.9° = 143.1° in 0° ≤ θ ≤ 360°, and missing it loses the final mark.
- Dividing both sides by sin θ or cos θ, which throws away the solutions where that factor is zero; factorise it out instead.
- Solving an equation in 2θ over the original interval only: if 0° ≤ θ ≤ 360°, then 2θ runs from 0° to 720°, and every solution in that range is needed before halving.
- Proving an identity by working on both sides at once, or by starting from the result: a proof must move from one side to the other in valid steps.
- Giving answers in degrees when the interval is in radians, or the reverse: the answers must match the units of the interval.
Exam technique
- Before solving, use an identity to rewrite the equation in a single trigonometric function; a quadratic in sin θ, cos θ or tan θ is the usual target.
- Sketch the graph over the whole interval to find every solution, and give angles in degrees to 1 decimal place unless told otherwise.
- For R sin(θ + α), find R and α separately and show tan α = b/a; keep α unrounded in your calculator, since later parts depend on it.
Practising in Momentum
- Every question in a chapter as one list, from every paper and year in the bank, newest first or shuffled.
- Each question turns over to its own part of the mark scheme, where the board published one.
- Narrow by paper, year, session, zone and marks, and star the questions worth keeping.
- Pick chapters, a number of questions and a time, and the exam builder draws a mock to sit and mark.
Past-paper questions and mark schemes are the copyright of Cambridge International. Momentum shows them to signed-in learners for revision and does not publish them on these pages.