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Solving trigonometric equations

Past-paper questions pack

Name
Class
Date set
Date due
Score/ 134

Questions are numbered; the mark schemes (the answer key) follow on separate pages so the question pages can be handed out alone. www.mathsaleveltutor.com

Edexcel AS and A-level Maths past-paper questions since June 2018 · 37 parts, 134 marks · mark schemes follow the questions · www.mathsaleveltutor.com · Questions and mark schemes © Pearson Education Ltd

1. June 2025 · AS Paper 1 · Q8 · 5 marks
8. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. Solve, for 0 x 90°, the equation 2 3x = 4 cos 2 3x sin giving your answers, in degrees, to one decimal place. (5)
2. June 2025 · Paper 1 · Q14(b) · 4 marks
(b) Hence solve, for 0 theta < 180° sin( theta + 30°) + 3 cos( theta + 30°) = 3 sin 2 theta giving your answers to one decimal place, where appropriate. (4)
3. June 2025 · Paper 2 · Q13(ii) · 2 marks
(ii) deduce the number of solutions of the equation 2 (nx) = k sin in the interval 0 x < 5 pi , justifying your answer. (2)
4. June 2024 · Paper 1 · Q12(d) · 4 marks
The number of foxes, F, in the wood during the same year is modelled by the equation F = 100 + 70 sin(30t + 70)° The number of foxes is at its minimum value after T months. (d) Find, according to the models, the number of rabbits in the wood at time T months. (4)
5. June 2024 · AS Paper 1 · Q13(b) · 4 marks
(b) Hence solve, for 0 < x < 360° sin x (7 sin x – 4 cos x) = 4 giving your answers to one decimal place. (4)
6. June 2024 · AS Paper 1 · Q13(c) · 1 marks
(c) Hence find the smallest solution of the equation sin 4 alpha (7 sin 4 alpha – 4 cos 4 alpha ) = 4 in the range 720° < alpha < 1080°, giving your answer to one decimal place. (1)
7. June 2024 · Paper 2 · Q8(b) · 4 marks
(b) Hence solve, for 0 < x < 90°, the equation O cot x x sec x x cosec cosec Give each answer, in degrees, to one decimal place. (4)
8. June 2023 · AS Paper 1 · Q12(b) · 4 marks
(b) Hence solve, for tan x = cos x giving your answers to one decimal place. (4)
9. June 2023 · AS Paper 1 · Q12(c) · 2 marks
(c) Hence find the number of solutions of the equation x x tan = cos in the interval 0 N x 1800 ; , explaining briefly the reason for your answer. (2)
10. June 2023 · Paper 2 · Q14(b) · 4 marks
(b) Hence, solve for 360° x 540° 2 tan x (8 cos x + 23 sin 2 x) = 8 sin 2x(1 + tan 2 x) x ∈ ℝ x ≠ 450° (4)
11. June 2023 · Paper 3 (Mechanics) · Q5(c) · 3 marks
(c) find the greatest possible height, in metres, of the ball above the ground as the ball moves from O to A. (3)
12. June 2022 · AS Paper 1 · Q13(b) · 5 marks
Given that cos 2x ≠ 0 (b) solve for 0 < x < 90° cos2x + tan 2x = 3 cos 2x giving your answers to one decimal place. (5)
13. June 2022 · Paper 1 · Q14(b) · 4 marks
(b) Hence or otherwise solve, for 0 theta < 180° 2 sin 2 theta = cos (2 theta + 30°) giving your answers to one decimal place. (4)
14. June 2022 · Paper 2 · Q6(a) · 4 marks
6. y alpha x P Figure 2 Figure 2 shows a sketch of part of the curve with equation y = f(x) where f(x) = 8 sin 1 2 x - 3x + 9 x > 0 and x is measured in radians. The point P, shown in Figure 2, is a local maximum point on the curve. Using calculus and the sketch in Figure 2, (a) find the x coordinate of P, giving your answer to 3 significant figures. (4)
15. June 2022 · Paper 2 · Q15(b) · 2 marks
Given that theta is an obtuse angle measured in radians, (b) solve the equation in part (a) to find the exact value of theta (2)
16. Oct 2021 · Paper 1 · Q10(b) · 4 marks
(b) Hence solve, for 0 < x < 180° - x + x cos sin = x sin x x + + cos sin giving your answers to one decimal place where appropriate. (4)
17. Oct 2021 · AS Paper 1 · Q12(i) · 5 marks
12. In this question you should show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. (i) Solve, for 0 < theta 450°, the equation 2 theta = 6 sin theta 5 cos giving your answers to one decimal place. (5)
18. Oct 2021 · AS Paper 1 · Q12(ii)(a) · 2 marks
(ii) (a) A student’s attempt to solve the question “ Solve, for –90° < x < 90°, the equation 3 tan x – 5 sin x = 0 ” is set out below. 3 tan x – 5 sin x = 0 x 3 sin x – 5 sin x = 0 cos 3 sin x – 5 sin x cos x = 0 3 – 5 cos x = 0 cos x = 3 x = 53.1° Identify two errors or omissions made by this student, giving a brief explanation of each. (2)
19. Oct 2021 · AS Paper 1 · Q12(ii)(b) · 2 marks
The first four positive solutions, in order of size, of the equation cos (5 alpha + 40°) = 3 are alpha 1 , alpha 2 , alpha 3 and alpha 4 (b) Find, to the nearest degree, the value of alpha 4 (2)
20. Oct 2021 · Paper 2 · Q15(c) · 4 marks
In a single revolution of the wheel, P is below the water level for a total of T seconds. According to the model, (c) find the value of T giving your answer to 3 significant figures. (Solutions based entirely on calculator technology are not acceptable.) (4)
21. Oct 2020 · Paper 1 · Q6(c) · 3 marks
(c) find the time of day when the maximum temperature occurs, giving your answer to the nearest minute. (3)
22. Oct 2020 · AS Paper 1 · Q9(c) · 5 marks
(c) Solve, for 450° theta < 720°, 3 cos theta = 8 tan theta giving your solution to one decimal place. In part (c) you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. (5)
23. Oct 2020 · Paper 1 · Q12(b) · 5 marks
(b) Hence, or otherwise, solve for 0 < x < 180° cosec x - sin x = cos x cot (3x - 50°) (5)
24. Oct 2020 · Paper 2 · Q10(b) · 4 marks
(b) Hence solve, for -90° x 180°, the equation 1 - cos 3x = sin2 x (4)
25. Oct 2020 · Paper 2 · Q12(b) · 5 marks
y M N 4.2 x Figure 4 Part of the curve is used to model the profile of a small dam, shown shaded in Figure 4. Using the model and given that ● x and y are in metres ● the vertical wall of the dam is 4.2 metres high ● there is a horizontal walkway of width MN along the top of the dam (b) calculate the width of the walkway. (5)
26. June 2019 · Paper 1 · Q6(a) · 6 marks
6. (a) Solve, for –180° - ș - 180°, the equation 5 sin 2ș = 9 tan ș giving your answers, where necessary, to one decimal place. [Solutions based entirely on graphical or numerical methods are not acceptable.] (6)
27. June 2019 · Paper 1 · Q6(b) · 2 marks
(b) Deduce the smallest positive solution to the equation 5 sin (2x – 50°) = 9 tan (x – 25°) (2)
28. June 2019 · AS Paper 1 · Q12(b) · 3 marks
(b) Hence, or otherwise, solve, for 0 - x : 360q, the equation x x sin - cos + = 4 + 3 sin x x + cos (3)
29. June 2019 · Paper 1 · Q12(c) · 3 marks
The function H(t) is used to model the height, in metres, of a ball above the ground t seconds after it has been kicked. Using this model, find (c) the maximum height of the ball above the ground between the first and second bounce. (3)
30. June 2019 · Paper 2 · Q12(b) · 3 marks
(b) Hence solve, for 90° : ș : 180°, the equation cos3 theta + sin3 theta theta theta = 4 sin cos giving any solutions to one decimal place. (3)
31. June 2018 · Paper 1 · Q8(b) · 4 marks
(b) Find, to the nearest minute, the earliest time the boat can leave the harbour. (Solutions based entirely on graphical or numerical methods are not acceptable.) (4)
32. June 2018 · Paper 1 · Q10(c) · 2 marks
The passenger reaches the maximum height, for the second time, T seconds after the start of the ride. (c) Find the value of T. (2)
33. June 2018 · AS Paper 1 · Q12(b) · 4 marks
(b) Hence solve, for 0 - x : 90° 4 cos 3x í 1 = 2 sin 3x tan 3x giving your answers, where appropriate, to one decimal place. (Solutions based entirely on graphical or numerical methods are not acceptable.) (4)
34. June 2018 · Paper 2 · Q6(c) · 1 marks
(c) deduce the number of real solutions, for 7 pi theta < 10 pi , to the equation 3 tan3 theta - 8 tan2 theta + 9 tan theta - 10 = 0 (1)
35. June 2018 · Paper 2 · Q7(i) · 4 marks
pi , the equation 7. (i) Solve, for 0 x < 2 4 sin x = sec x (4)
36. June 2018 · Paper 2 · Q7(ii) · 5 marks
(ii) Solve, for 0 theta < 360°, the equation 5 sin theta - 5 cos theta = 2 giving your answers to one decimal place. (Solutions based entirely on graphical or numerical methods are not acceptable.) (5)
37. June 2018 · Paper 2 · Q12(b) · 6 marks
pi < x < 2 pi , the equation (b) Hence solve, for -2 (sec2 x - 5)(1 - cos 2x) = 3 tan2 x sin 2x Give any non-exact answer to 3 decimal places where appropriate. (6)

Mark schemes

Each mark scheme shows the whole question; the parts on this topic are the ones listed above.

June 2025 · AS Paper 1 · Q8
Mark scheme June 2025 Paper 1 Q8
June 2025 · Paper 1 · Q14
Mark scheme June 2025 Paper 1 Q14(b)
June 2025 · Paper 2 · Q13
Mark scheme June 2025 Paper 2 Q13(ii)
June 2024 · Paper 1 · Q12
Mark scheme June 2024 Paper 1 Q12(d)
June 2024 · AS Paper 1 · Q13
Mark scheme June 2024 Paper 1 Q13(c)
June 2024 · Paper 2 · Q8
Mark scheme June 2024 Paper 2 Q8(b)
June 2023 · AS Paper 1 · Q12
Mark scheme June 2023 Paper 1 Q12(c)
June 2023 · Paper 2 · Q14
Mark scheme June 2023 Paper 2 Q14(b)
June 2023 · Paper 3 (Mechanics) · Q5
Mark scheme June 2023 Paper 3 (Mechanics) Q5(c)
June 2022 · AS Paper 1 · Q13
Mark scheme June 2022 Paper 1 Q13(b)
June 2022 · Paper 1 · Q14
Mark scheme June 2022 Paper 1 Q14(b)
June 2022 · Paper 2 · Q6
Mark scheme June 2022 Paper 2 Q6(a)
June 2022 · Paper 2 · Q15
Mark scheme June 2022 Paper 2 Q15(b)
Oct 2021 · Paper 1 · Q10
Mark scheme Oct 2021 Paper 1 Q10(b)
Oct 2021 · AS Paper 1 · Q12
Mark scheme Oct 2021 Paper 1 Q12(ii)(b)
Oct 2021 · Paper 2 · Q15
Mark scheme Oct 2021 Paper 2 Q15(c)
Oct 2020 · Paper 1 · Q6
Mark scheme Oct 2020 Paper 1 Q6(c)
Oct 2020 · AS Paper 1 · Q9
Mark scheme Oct 2020 Paper 1 Q9(c)
Oct 2020 · Paper 1 · Q12
Mark scheme Oct 2020 Paper 1 Q12(b)
Oct 2020 · Paper 2 · Q10
Mark scheme Oct 2020 Paper 2 Q10(b)
Oct 2020 · Paper 2 · Q12
Mark scheme Oct 2020 Paper 2 Q12(b)
June 2019 · Paper 1 · Q6
Mark scheme June 2019 Paper 1 Q6(b)
June 2019 · AS Paper 1 · Q12
Mark scheme June 2019 Paper 1 Q12(b)
June 2019 · Paper 1 · Q12
Mark scheme June 2019 Paper 1 Q12(c)
June 2019 · Paper 2 · Q12
Mark scheme June 2019 Paper 2 Q12(b)
June 2018 · Paper 1 · Q8
Mark scheme June 2018 Paper 1 Q8(b)
June 2018 · Paper 1 · Q10
Mark scheme June 2018 Paper 1 Q10(c)
June 2018 · AS Paper 1 · Q12
Mark scheme June 2018 Paper 1 Q12(b)
June 2018 · Paper 2 · Q6
Mark scheme June 2018 Paper 2 Q6(c)
June 2018 · Paper 2 · Q7
Mark scheme June 2018 Paper 2 Q7(ii)
June 2018 · Paper 2 · Q12
Mark scheme June 2018 Paper 2 Q12(b)