AS & A Level Mathematics 9709 · Pure Mathematics 1 (for Paper 1) · 1.5

Trigonometry: past paper questions

279 real questionsasked in 10 of the last 10 exam series2020–2026usually: Show, Find, Solve, Prove, Give

Real questions on trigonometry

The newest 20 of 279 written questions. Open one to see it on the real paper.

9709/12 · May/June 2026 · Q6[4]
6 Solve the equation $cos^{- 1} (sin^{2} x + \frac{3}{2} sin x) = \frac{2}{3} π$ for $0 \leq x \leq 2 π$ .See it on the paper →
9709/32 · May/June 2026 · Q7b[3]
3-mark question — its maths and diagrams are best seen on the paper.
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9709/32 · May/June 2026 · Q8a[2]
2-mark question — its maths and diagrams are best seen on the paper.
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9709/32 · May/June 2026 · Q8b[2]
2-mark “Show” question — its maths and diagrams are best seen on the paper.
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9709/32 · May/June 2026 · Q8c[2]
2-mark “Show” question — its maths and diagrams are best seen on the paper.
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9709/32 · May/June 2026 · Q8d[3]
3-mark “Use” question — its maths and diagrams are best seen on the paper.
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9709/11 · October/November 2025 · Q2a[3]
3-mark “Find” question — its maths and diagrams are best seen on the paper.
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9709/11 · October/November 2025 · Q5a[3]
3-mark “Show” question — its maths and diagrams are best seen on the paper.
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9709/11 · October/November 2025 · Q5b[4]
4-mark question — its maths and diagrams are best seen on the paper.
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9709/12 · October/November 2025 · Q6b(i)[1]
1-mark question — its maths and diagrams are best seen on the paper.
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9709/12 · October/November 2025 · Q6b(ii)[1]
1-mark question — its maths and diagrams are best seen on the paper.
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9709/12 · October/November 2025 · Q6c[5]
5-mark “Solve” question — its maths and diagrams are best seen on the paper.
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9709/13 · October/November 2025 · Q2a[2]
2-mark “Solve” question — its maths and diagrams are best seen on the paper.
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9709/13 · October/November 2025 · Q2b[4]
4-mark “Solve” question — its maths and diagrams are best seen on the paper.
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9709/13 · October/November 2025 · Q5a[2]
2-mark “Find” question — its maths and diagrams are best seen on the paper.
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9709/21 · October/November 2025 · Q4[5]
5-mark “Solve” question — its maths and diagrams are best seen on the paper.
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9709/31 · October/November 2025 · Q3a[4]
4-mark question — its maths and diagrams are best seen on the paper.
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9709/31 · October/November 2025 · Q3b[4]
4-mark question — its maths and diagrams are best seen on the paper.
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9709/32 · October/November 2025 · Q6a[2]
2-mark question — its maths and diagrams are best seen on the paper.
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9709/32 · October/November 2025 · Q8a[3]
3-mark “Prove” question — its maths and diagrams are best seen on the paper.
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What examiners said

Most candidates, sensibly, started with the left-hand side of the given identity and immediately sin2 replaced tan2 with . They then either replaced cos2 with 1 – sin2 and collected the terms cos2 into a single fraction, or they converted to a single fraction first. To gain full marks in a ‘show that’ question it is necessary for candidates to show each step clearly and to be accurate in their alg...
— Examiner report, February/March 2025, 9709/12 Q7(a) (Cambridge)

Most candidates correctly used the result from part (a) and arrived at a quartic equation which they recognised as a quadratic in sin2. One of the solutions to the correct quadratic was greater than 1 and could therefore be discounted. The other solution for sin2 should have produced two values for sin, which many candidates did not realise and consequently did not find all three solutions.
— Examiner report, February/March 2025, 9709/12 Q7(b) (Cambridge)

Successful candidates wrote as and then multiplied by sin x. tan x sin x Most candidates who found that 4sin2 x + 5cos x + 2 = 0 then went on to use the substitution sin2 x = 1 − cos2 x to find 4cos2 x − 5cos x − 6 = 0 . Some candidates were unable to gain any marks due to errors in algebraic manipulation.
— Examiner report, October/November 2023, 9709/11 Q5(a) (Cambridge)

Practise this topic

Answer these questions online and get each one marked against the official mark scheme, with what you missed and how to fix it.

Practise trigonometry

In the syllabus

1.5 Trigonometry

sketch and use graphs of the sine, cosine and Including e.g. y = 3 sin x, y = 1 – cos 2x,
y = tan cx + rm .
use the exact values of the sine, cosine and e.g. cos 150c = − = . 2 3 , sin 4 r
use the notations sin–1x, cos–1x, tan–1x to No specialised knowledge of these functions is
required, but understanding of them as examples
of inverse functions is expected.
use the identities
e.g. in proving identities, simplifying expressions and solving equations.
find all the solutions of simple trigonometrical e.g. solve 3 sin 2x + 1 = 0 for − r 1 x 1 r ,
3 sin 2 i − 5 cos i − 1 = 0 for 0c G i G 360c.

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