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Trigonometry: WJEC A‑Level Maths questions

Every WJEC A-Level Maths exam question on P.5 Trigonometry, 2018–2025, with its mark scheme and examiner’s report.

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314 questions

  1. Q2

    Summer 2025 · Unit 3

    12 marksP.5 Trigonometry
    The whole paper ↗
    Summer 2025 Unit 3, question 2: 2. (a) Find all values of x in the range 0XxX2π satisfying the equation cos2 sin 0 x  x  . Give your answers in terms of π. [6] Find all values of θ in the ra

  2. Q3

    Summer 2025 · Unit 3

    6 marksP.5 Trigonometry
    The whole paper ↗
    Summer 2025 Unit 3, question 3: cos  Given that θ is small, find an estimate for the solution of the equation  0 8  3. . 1 sin 2  [6]

  3. Q10

    Summer 2025 · Unit 3

    3 marksP.5 TrigonometryP.8 Integration
    The whole paper ↗
    Summer 2025 Unit 3, question 10: . for 0 X xX α, where 0 ~ α ~ 2 y sec x  10. (a) Sketch the graph of [1]   sec d x x   (b) Explain why . [2] 0

  4. Q7

    Summer 2025 · AS Unit 1

    6 marksP.5 Trigonometry
    The whole paper ↗
    Summer 2025 AS Unit 1, question 7: The triangle ABC is such that AB = 20 3 cm, AC = 20 cm, and angle ABC = 30°. Find the two 7. possible exact values of the area of the triangle ABC. [6]

  5. Q10

    Summer 2025 · AS Unit 1

    6 marksP.5 Trigonometry
    The whole paper ↗
    Summer 2025 AS Unit 1, question 10: 10. Solve the following equation for values of θ between 0° and 360°. [6] 15sin 2 2cos 14    

  6. Q13

    Summer 2025 · AS Unit 1

    4 marksP.5 Trigonometry
    The whole paper ↗
    Summer 2025 AS Unit 1, question 13: 13. Given that 3 sin2x – cos2x = 2, prove that sin2x = 3 cos2x. Hence, or otherwise, find the possible values of tanx. [4]

  7. Q2

    Summer 2024 · Unit 3

    11 marksP.5 Trigonometry
    The whole paper ↗
    Summer 2024 Unit 3, question 2: Find all values of θ in the range 0° ! θ ! 360° satisfying 2. (a) 2 3cot   4cosec   5 [5] .   , where R and α are constants with By writing 24cos x  7sin

  8. Q3

    Summer 2024 · Unit 3

    7 marksP.5 Trigonometry
    The whole paper ↗
    Summer 2024 Unit 3, question 3: 3. The diagram below shows a badge ODC. The shape OAB is a sector of a circle centre O and radius r cm. The shape ODC is a sector of a circle with the same cent

  9. Q12

    Summer 2024 · Unit 3

    6 marksP.5 Trigonometry
    The whole paper ↗
    Summer 2024 Unit 3, question 12: 2cos sin 1 1 2 Given that θ is small, show that        12. (a) . [2] Hence, when θ is small, show that (b) 1 2  1  a   b  , 2cos   sin   1 whe

  10. Q2

    Summer 2024 · AS Unit 1

    3 marksP.5 Trigonometry
    The whole paper ↗
    Summer 2024 AS Unit 1, question 2: Find all values of θ in the range 0° ! θ ! 180° that satisfy the equation 2. 2sin2 1   . [3]

  11. Q5

    Summer 2024 · AS Unit 1

    4 marksP.5 Trigonometry
    The whole paper ↗
    Summer 2024 AS Unit 1, question 5: 5. A triangle ABC has sides AB = 6 cm, BC = 11 cm and AC = 13 cm. Calculate the area of the triangle. [4]

  12. Q15

    Summer 2024 · AS Unit 1

    7 marksP.5 Trigonometry
    The whole paper ↗
    Summer 2024 AS Unit 1, question 15: y 2sin x 3cos 2 x 3    . 15. The diagram shows a sketch of part of the curve with equation The curve crosses the x-axis at the points O, A, B and C. y C O x

  13. Q6

    Summer 2023 · Unit 3

    15 marksP.5 Trigonometry
    The whole paper ↗
    Summer 2023 Unit 3, question 6:   cos cos cos sin sin 0 6 A  B  A B  A B a) Using the trigonometric identity , show that 6 − 2 the exact value of cos75° is . [3] 4 Solve the equation 2cot

  14. Q2

    Summer 2023 · AS Unit 1

    7 marksP.5 Trigonometry
    The whole paper ↗
    Summer 2023 AS Unit 1, question 2: Solve the following equation for values of θ between 0° and 360°. 0 2 2 2 3sin   5cos   2cos  [7]

  15. Q6

    Summer 2023 · AS Unit 1

    7 marksP.5 Trigonometry
    The whole paper ↗
    Summer 2023 AS Unit 1, question 6: 6 The diagram below shows a triangle ABC. B A C $ 3, 2 5, 4 3 Given that     , find the value of cosABC. Show all your AB BC AC working and give your answer

  16. Q12

    Summer 2023 · AS Unit 1

    8 marksP.10 VectorsP.5 Trigonometry
    The whole paper ↗
    Summer 2023 AS Unit 1, question 12: 2 The position vectors of the points A and B, relative to a fixed origin O, are given by a = –3i + 4j, b = 5i + 8j, respectively. a) Find the vector AB. [2] Fin

  17. Q1

    Summer 2022 · Unit 3

    6 marksP.5 Trigonometry
    The whole paper ↗
    Summer 2022 Unit 3, question 1: 1 Solve the equation 6sec2x – 8 = tanx for 0° X x X 360°. [6]

  18. Q3

    Summer 2022 · Unit 3

    4 marksP.5 Trigonometry
    The whole paper ↗
    Summer 2022 Unit 3, question 3: The diagram below shows a plan of the patio Eric wants to build. 0 3 4 m B A π 3 O C The walls OA and OC are perpendicular. The straight line AB is of length 4

  19. Q11

    Summer 2022 · Unit 3

    5 marksP.5 Trigonometry
    The whole paper ↗
    Summer 2022 Unit 3, question 11: ( ) , where R and α are constants Express 9cosx + 40sinx in the form R cos x −α a) 1 1 with R 1 0 and 0° ! α ! 90°. [3] 12 b) Find the maximum possible value of

  20. Q17

    Summer 2022 · Unit 3

    5 marksP.5 Trigonometry
    The whole paper ↗
    Summer 2022 Unit 3, question 17: 7 a) Prove that ( ). ( ) + sin α + β ( ) ≡ cosα + sinα ( ) cosβ + sinβ cos α −β [2] cos3θ + sin5θ cosθ + sinθ b) i) Hence show that can be expressed as . cos4θ
All 34 questions on Trigonometry, as a list
  1. Summer 2025, Unit 3, Q2 (12 marks): 2. (a) Find all values of x in the range 0XxX2π satisfying the equation cos2 sin 0 x  x  . Give your answers in terms of π. [6] Find all values of θ in the ra
  2. Summer 2025, Unit 3, Q3 (6 marks): cos  Given that θ is small, find an estimate for the solution of the equation  0 8  3. . 1 sin 2  [6]
  3. Summer 2025, Unit 3, Q10 (3 marks): . for 0 X xX α, where 0 ~ α ~ 2 y sec x  10. (a) Sketch the graph of [1]   sec d x x   (b) Explain why . [2] 0
  4. Summer 2025, AS Unit 1, Q7 (6 marks): The triangle ABC is such that AB = 20 3 cm, AC = 20 cm, and angle ABC = 30°. Find the two 7. possible exact values of the area of the triangle ABC. [6]
  5. Summer 2025, AS Unit 1, Q10 (6 marks): 10. Solve the following equation for values of θ between 0° and 360°. [6] 15sin 2 2cos 14    
  6. Summer 2025, AS Unit 1, Q13 (4 marks): 13. Given that 3 sin2x – cos2x = 2, prove that sin2x = 3 cos2x. Hence, or otherwise, find the possible values of tanx. [4]
  7. Summer 2024, Unit 3, Q2 (11 marks): Find all values of θ in the range 0° ! θ ! 360° satisfying 2. (a) 2 3cot   4cosec   5 [5] .   , where R and α are constants with By writing 24cos x  7sin
  8. Summer 2024, Unit 3, Q3 (7 marks): 3. The diagram below shows a badge ODC. The shape OAB is a sector of a circle centre O and radius r cm. The shape ODC is a sector of a circle with the same cent
  9. Summer 2024, Unit 3, Q12 (6 marks): 2cos sin 1 1 2 Given that θ is small, show that        12. (a) . [2] Hence, when θ is small, show that (b) 1 2  1  a   b  , 2cos   sin   1 whe
  10. Summer 2024, AS Unit 1, Q2 (3 marks): Find all values of θ in the range 0° ! θ ! 180° that satisfy the equation 2. 2sin2 1   . [3]
  11. Summer 2024, AS Unit 1, Q5 (4 marks): 5. A triangle ABC has sides AB = 6 cm, BC = 11 cm and AC = 13 cm. Calculate the area of the triangle. [4]
  12. Summer 2024, AS Unit 1, Q15 (7 marks): y 2sin x 3cos 2 x 3    . 15. The diagram shows a sketch of part of the curve with equation The curve crosses the x-axis at the points O, A, B and C. y C O x
  13. Summer 2023, Unit 3, Q6 (15 marks):   cos cos cos sin sin 0 6 A  B  A B  A B a) Using the trigonometric identity , show that 6 − 2 the exact value of cos75° is . [3] 4 Solve the equation 2cot
  14. Summer 2023, AS Unit 1, Q2 (7 marks): Solve the following equation for values of θ between 0° and 360°. 0 2 2 2 3sin   5cos   2cos  [7]
  15. Summer 2023, AS Unit 1, Q6 (7 marks): 6 The diagram below shows a triangle ABC. B A C $ 3, 2 5, 4 3 Given that     , find the value of cosABC. Show all your AB BC AC working and give your answer
  16. Summer 2023, AS Unit 1, Q12 (8 marks): 2 The position vectors of the points A and B, relative to a fixed origin O, are given by a = –3i + 4j, b = 5i + 8j, respectively. a) Find the vector AB. [2] Fin
  17. Summer 2022, Unit 3, Q1 (6 marks): 1 Solve the equation 6sec2x – 8 = tanx for 0° X x X 360°. [6]
  18. Summer 2022, Unit 3, Q3 (4 marks): The diagram below shows a plan of the patio Eric wants to build. 0 3 4 m B A π 3 O C The walls OA and OC are perpendicular. The straight line AB is of length 4
  19. Summer 2022, Unit 3, Q11 (5 marks): ( ) , where R and α are constants Express 9cosx + 40sinx in the form R cos x −α a) 1 1 with R 1 0 and 0° ! α ! 90°. [3] 12 b) Find the maximum possible value of
  20. Summer 2022, Unit 3, Q17 (5 marks): 7 a) Prove that ( ). ( ) + sin α + β ( ) ≡ cosα + sinα ( ) cosβ + sinβ cos α −β [2] cos3θ + sin5θ cosθ + sinθ b) i) Hence show that can be expressed as . cos4θ
  21. Summer 2022, AS Unit 1, Q12 (9 marks): 2 a) Solve the equation 2x3 – x2 – 5x – 2 = 0. [6] Find all values of θ in the range 0° < θ < 180° satisfying b) cos(2θ– 51°) = 0.891. [3]
  22. Summer 2019, Unit 3, Q4 (9 marks): Express 5sinx – 12cosx in the form Rsin(x – α ), where R > 0 and 0° < α< 90°. a) 0 4 [3] 4 b) Find the minimum value of . [2] 5sinx −12cosx +15 c) Solve the equ
  23. Summer 2019, Unit 3, Q9 (8 marks): 9 a) Given that α and β are two angles such that tanα= 2cotβ, show that tan(α + β ) = – (tanα + tanβ ). [2] Find all values of θ in the range 0° X θX 360° satis
  24. Summer 2019, Unit 3, Q12 (10 marks): A chord AB subtends an angle θ radians at the centre of a circle. The chord divides the 1 2 circle into two segments whose areas are in the ratio 1 : 2. B A θ O
  25. Summer 2019, AS Unit 1, Q1 (6 marks): 1 Solve the following equation for values of θ between 0° and 360°. 3tanθ + 2cosθ = 0 [6]
  26. Summer 2019, AS Unit 1, Q14 (6 marks): 4 The diagram below shows a triangle ABC with AC = 5 cm, AB = x cm, BC = y cm and angle BAC = 120°. The area of the triangle ABC is 14 cm2. A 120° 5 cm x C y B
  27. Summer 2018, Unit 3, Q2 (4 marks): The diagram below shows a circle centre O, radius 4 cm. Points A and B lie on the 0 2 circumference such that arc AB is 5 cm. A 5 cm B O Calculate the angle sub
  28. Summer 2018, Unit 3, Q4 (5 marks): 4 Solve the equation 2tan2θ + 2tanθ– sec2θ = 2, for values of θbetween 0° and 360°. [5]
  29. Summer 2018, Unit 3, Q7 (3 marks): 7 Use small angle approximations to find the small negative root of the equation sinx + cosx = 0.5. [3]
  30. Summer 2018, Unit 3, Q11 (4 marks): π 1 1 Prove by contradiction that, for every real number x such that 0 X x X , 2 sinx + cosx x 1. [4]
  31. Summer 2018, Unit 3, Q13 (8 marks): Express 8sinθ– 15cosθ in the form Rsin(θ– α), where R and α are constants 1 3 a) with R > 0 and 0° < α < 90°. [3] Find all values of θ in the range 0° < θ < 360
  32. Summer 2018, AS Unit 1, Q3 (6 marks): 3 Solve the following equation for values of θ between 0° and 360°. 2 – 3cos2θ = 2sinθ. [6]
  33. Summer 2018, AS Unit 1, Q7 (3 marks): Prove that 0 7 sin3θ + sinθ cos2θ ≡ tanθ [3] . cosθ
  34. Summer 2018, AS Unit 1, Q9 (5 marks): $ The triangle ABC is such that AC = 16 cm, AB = 25 cm and ABC = 32°. Find two possible 0 9 values for the area of the triangle ABC. [5]

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