plumiTrigonometry: OCR MEI A‑Level Maths questions
Every OCR MEI A-Level Maths exam question on P.5 Trigonometry, 2018–2025, with its mark scheme and examiner’s report.
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![June 2025 Paper 1, question 4: The diagram shows part of a circle with centre O and radius 5 cm. The circle passes through the points A and B. The length AB is 5 cm. O A B 5cm [4] Calculate t](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/h64001-jun25-04-qp.webp?v=3)
![June 2025 Paper 2, question 12: In this question you must show detailed reasoning 2i Solve the equation sec 3 2 tan 4 0 for 180 ° 180 ° . [6] + - = - 1 1 i i](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/h64002-jun25-12-qp.webp?v=3)
![June 2025 Paper 3, question 14: (a) Draw and label a suitable triangle on the diagram in the Printed Answer Booklet to show y that tan , as given in line 30. [1] i = x 2 a ( 3 t ) 2 4 at - (b)](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/h64003-jun25-14-qp.webp?v=3)
![June 2025 AS Paper 1, question 6: In this question you must show detailed reasoning. 2 = Solve the equation cos 6 x 5 sin x giving all the roots in the interval ° 0 x 360 ° . [4] - 1 1](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/h63001-jun25-06-qp.webp?v=3)
![June 2025 AS Paper 2, question 2: Triangle LMS is shown in the diagram. M 5.9cm 112° S L 16.2cm LS = 16.2 cm, LM = 5.9 cm and angle MLS = 112°. Determine the area of triangle LMS. [2]](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/h63002-jun25-02-qp.webp?v=3)
![June 2025 AS Paper 2, question 3: You are given that sin 5 9 cos and ° 0 360 ° . = 1 1 i i i Determine the possible values of i correct to 3 significant figures. [2]](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/h63002-jun25-03-qp.webp?v=3)
![June 2024 Paper 1, question 8: The equation of a curve is y sin 4 x 2 cos 2 x , where x is in radians. = + 4 2 (a) Show that, for small values of x, y 2 x 2 x . [2] . + - The diagram shows th](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/h64001-jun24-08-qp.webp?v=3)
![June 2024 Paper 2, question 4: (a) On the axes in the Printed Answer Booklet, sketch the graph of y sin 2i for 0 2 r. = i G G [2] 1 (b) Solve the equation sin2 i =- for 0 2 . [3] i r G G 2](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/h64002-jun24-04-qp.webp?v=3)
![June 2024 Paper 3, question 6: In this question you must show detailed reasoning. Solve the equation tan x 3 cot x 2 for values of x in the interval 0 ° G x 360 ° . [5] - = G](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/h64003-jun24-06-qp.webp?v=3)
![June 2024 Paper 3, question 7: Prove that sin 8 tan 4 cos 8 = . 1 [3] i i + i](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/h64003-jun24-07-qp.webp?v=3)
![June 2024 AS Paper 2, question 14: In this question you must show detailed reasoning. 2 Solve the equation 5 cos 6 sin 0 for ° 0 1 1 360 ° . [6] i i i - - = for question 15](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/h63002-jun24-14-qp.webp?v=3)
![June 2023 Paper 1, question 6: a k a k can be written in the form 1 1 6 (a) Show that the equation sin x cos x + = - r r 6 4 2 1 - tan x . [4] = 3 2 - a k a k for 1 1 (b) Hence solve the equa](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/h64001-jun23-06-qp.webp?v=3)
All 73 questions on Trigonometry, as a list
- June 2025, Paper 1, Q4 (4 marks): The diagram shows part of a circle with centre O and radius 5 cm. The circle passes through the points A and B. The length AB is 5 cm. O A B 5cm [4] Calculate t
- June 2025, Paper 2, Q3 (2 marks): The diagram shows triangle ABC. C Not to scale A B AB = 4.7 cm, AC = 6.9 cm and BC = 11.2 cm. Find the size of angle CAB. Give your answer correct to 3 signific
- June 2025, Paper 2, Q6 (2 marks): The diagram shows part of the graph of f( ) x cosec 2 x . = y 1 x 0 – 1 1 (a) State which of the following is the equation of the curve y . = f( ) x y cos 2 x y
- June 2025, Paper 2, Q12 (6 marks): In this question you must show detailed reasoning 2i Solve the equation sec 3 2 tan 4 0 for 180 ° 180 ° . [6] + - = - 1 1 i i
- June 2025, Paper 3, Q13 (3 marks): tan A tan B + 13 Use tan ( A B ) , with suitable choices for A and B, to show that + = 1 tan A tan B - 2 t ( 3 t ) - tan 3 , where t tan , as given in line 25.
- June 2025, Paper 3, Q14 (5 marks): (a) Draw and label a suitable triangle on the diagram in the Printed Answer Booklet to show y that tan , as given in line 30. [1] i = x 2 a ( 3 t ) 2 4 at - (b)
- June 2025, AS Paper 1, Q2 (2 marks): Triangle ABC is shown in the diagram. The length of AC is 2p cm and the length of BC is 3p cm, where p is a positive constant. The angle ABC is 25°. C 3p cm 2pc
- June 2025, AS Paper 1, Q6 (4 marks): In this question you must show detailed reasoning. 2 = Solve the equation cos 6 x 5 sin x giving all the roots in the interval ° 0 x 360 ° . [4] - 1 1
- June 2025, AS Paper 2, Q2 (2 marks): Triangle LMS is shown in the diagram. M 5.9cm 112° S L 16.2cm LS = 16.2 cm, LM = 5.9 cm and angle MLS = 112°. Determine the area of triangle LMS. [2]
- June 2025, AS Paper 2, Q3 (2 marks): You are given that sin 5 9 cos and ° 0 360 ° . = 1 1 i i i Determine the possible values of i correct to 3 significant figures. [2]
- June 2024, Paper 1, Q8 (6 marks): The equation of a curve is y sin 4 x 2 cos 2 x , where x is in radians. = + 4 2 (a) Show that, for small values of x, y 2 x 2 x . [2] . + - The diagram shows th
- June 2024, Paper 2, Q4 (5 marks): (a) On the axes in the Printed Answer Booklet, sketch the graph of y sin 2i for 0 2 r. = i G G [2] 1 (b) Solve the equation sin2 i =- for 0 2 . [3] i r G G 2
- June 2024, Paper 3, Q6 (5 marks): In this question you must show detailed reasoning. Solve the equation tan x 3 cot x 2 for values of x in the interval 0 ° G x 360 ° . [5] - = G
- June 2024, Paper 3, Q7 (3 marks): Prove that sin 8 tan 4 cos 8 = . 1 [3] i i + i
- June 2024, Paper 3, Q8 (8 marks): In this question you must show detailed reasoning. 1 (a) Express cos x 3 sin x in the form R sin ( x ) , where R 2 0 and 0 1 1 . Give the + + a a r 2 values of
- June 2024, AS Paper 1, Q1 (2 marks): The triangle ABC has an obtuse angle at A. The angle at B is 15°. The length of AC is 10 cm and the length of BC is 13 cm. Calculate the size of the angle at A.
- June 2024, AS Paper 1, Q4 (4 marks): The perpendicular lines AC and BD intersect at E as shown in the diagram. The point E is the midpoint of AC. The angles BAC and BDC are each equal to x°. The le
- June 2024, AS Paper 2, Q14 (6 marks): In this question you must show detailed reasoning. 2 Solve the equation 5 cos 6 sin 0 for ° 0 1 1 360 ° . [6] i i i - - = for question 15
- June 2023, Paper 1, Q6 (5 marks): a k a k can be written in the form 1 1 6 (a) Show that the equation sin x cos x + = - r r 6 4 2 1 - tan x . [4] = 3 2 - a k a k for 1 1 (b) Hence solve the equa
- June 2023, Paper 1, Q10 (6 marks): 10 The diagram shows the graph of y 1 5 . sin x for 0 x 2 r. = + G G y x O 2r (a) Show that the equation of the graph can be written in the form y a b cos 2 x w
- June 2023, Paper 1, Q15 (8 marks): A projectile is launched from a point on level ground with an initial velocity u at an angle i above the horizontal. 2 2 u sin cos i i . (a) Show that the range
- June 2023, Paper 2, Q6 (4 marks): The parametric equations of a circle are x 2 cos 3 and y 2 sin + . 1 = i - = i 2 2 Determine the cartesian equation of the circle in the form ( x a ) ( y b ) k
- June 2023, Paper 2, Q17 (6 marks): In this question you must show detailed reasoning. Solve the equation sin 2 x sec x 4 cos x , where x . [6] + = - 1 1 r r
- June 2023, Paper 3, Q1 (3 marks): In this question you must show detailed reasoning. 2 The obtuse angle i is such that sin . i = 13 Find the exact value of cos i. [3]
- June 2023, Paper 3, Q7 (6 marks): A wire, 10 cm long, is bent to form the perimeter of a sector of a circle, as shown in the diagram. The radius is r cm and the angle at the centre is i radians.
- June 2023, Paper 3, Q8 (7 marks): A circle with centre A and radius 8 cm and a circle with centre C and radius 12 cm intersect at points B and D. 2. Quadrilateral ABCD has area 60 cm Determine t
- June 2023, Paper 3, Q10 (6 marks): 2 3 6 4 2 2 4 6 10 (a) You are given that ( x y ) x 3 x y 3 x y y . + = + + + 6 6 3 2 Hence, or otherwise, prove that sin cos 1 sin 2 for all values of i. [4] +
- June 2023, AS Paper 1, Q2 (3 marks): The height of the first part of a rollercoaster track is h m at a horizontal distance of x m from the start. A student models this using the equation h 17 15 co
- June 2023, AS Paper 1, Q4 (5 marks): In this question you must show detailed reasoning. 2 + Solve the equation cos 6 x sin x = , giving all the roots in the interval 5 180 ° x 180 ° . [5] - G G
- June 2023, AS Paper 2, Q10 (5 marks): In this question you must show detailed reasoning. The diagram shows triangle ABC, where AB = .3 9 cm, BC = .4 5 cm and AC = .3 5 cm. A 3.5cm 3.9cm C 4.5 cm B D
- June 2022, Paper 1, Q3 (8 marks): (a) Sketch the graph of y arctan x where x is in radians. [2] = (b) In this question you must show detailed reasoning. 2 Find all points of intersection of the
- June 2022, Paper 1, Q10 (8 marks): A triangle ABC is made from two thin rods hinged together at A and a piece of elastic which joins B and C. AB is a 30 cm rod and AC is a 15 cm rod. The angle BA
- June 2022, Paper 2, Q1 (4 marks): Express cos 3 sin in the form Rcos(i – a), where R and a are exact values to be determined. i + i [4]
- June 2022, Paper 3, Q4 (5 marks): In this question you must show detailed reasoning. 2 = Determine the exact solutions of the equation cos 2 x 3 sin x for 0 x 2 r [5] . G G
- June 2022, Paper 3, Q9 (2 marks): Show that y = has the same gradient as x y sin x when x 0 , as stated in line 5. [2] = =
- June 2022, Paper 3, Q11 (3 marks): ( 180 ) - i i 11 Show that, for the angle 45°, the formula sin given in line 28 gives the i . 40500 ( 180 ) - - i i 16 x ( x ) r - same approximation for the si
- June 2022, Paper 3, Q12 (5 marks): a k. r 12 (a) Show that cos x sin x [2] = + 2 16 x ( x ) 2 2 r - 4 x r - (b) Hence show that sin x gives the approximation cos x , as stated . . 2 2 2 x x x 5 4
- June 2022, AS Paper 1, Q5 (6 marks): Part of the graph of y f ( ) x is shown below. The graph is the image of y tan x ° after a stretch in = = the x-direction. y 15 10 5 x -60 0 60 -5 -10 -15 (a) F
- June 2022, AS Paper 1, Q10 (9 marks): A triangle has vertices A ( , ), 1 4 B ( , ) 7 0 and C - ( 4 , 1 ). - (a) Show that the equation of the line AC is y x 3 . [2] = + M is the midpoint of AB. The
- November 2021, Paper 1, Q6 (7 marks): (a) The diagram shows part of the graph of y cosec x , where x is in radians. = State the equations of the three vertical asymptotes that can be seen. [1] y x O
- November 2021, Paper 2, Q2 (3 marks): (a) Write 65° in radians, giving your answer in the form kr, where k is a fraction in its lowest terms. [2] (b) Write 0.211 radians in degrees, giving your answ
- November 2021, Paper 2, Q7 (4 marks): The parametric equations of a circle are x 7 5 cos , y 5 sin - , for 0 3 2 r. = + i = i G i G (a) Find a cartesian equation of the circle. [3] (b) State the coo
- November 2021, Paper 3, Q2 (2 marks): Solve the equation sin x 2 0 3 . for ° 0 x 180 ° . Give your answer(s) correct to 1 decimal place. = G G [2]
- November 2021, Paper 3, Q11 (5 marks): In this question you must show detailed reasoning. The diagram shows triangle ABC, with BC = 8 cm and angle BAC = 45°. The point D on AC is such that DC = 5 cm
- November 2021, Paper 3, Q12 (3 marks): c m, as given in line 15. 1 12 Show that arctan 3 [3] b =
- November 2021, Paper 3, Q13 (3 marks): c m 1 r 13 (a) Use triangle ABE in Fig. C2 to show that arctan x arctan , as given in line 29. [1] + = 2 x (b) Sketch the graph of y arctan x . [1] = c m, as gi
- November 2021, Paper 3, Q14 (5 marks): (a) Show that c m c m c m c m 1 c 1 m 1 1 1 & arctan arctan arctan arctan arctan arctan 1 . [1] + = + = 2 n 1 n n n 2 3 1 + + + (b) Use the arctan addition form
- November 2021, Paper 3, Q15 (4 marks): Prove that arctan 1 arctan 2 arctan 3 , as given in line 41. [4] + + = r
- November 2021, AS Paper 1, Q5 (5 marks): The diagram shows the triangle ABC in which AC 13 cm and AB is the shortest side. The = 4 perimeter of the triangle is 32 cm. The area is 24cm2 and sinB . = 5 A
- November 2021, AS Paper 2, Q3 (3 marks): In this question you must show detailed reasoning. 1 You are given that tan 30 ° . = 3 1 Explain why tan 690 ° . [3] =- 3
- November 2020, Paper 1, Q14 (10 marks): Douglas wants to construct a model for the height of the tide in Liverpool during the day, using a cosine graph to represent the way the height changes. He know
- November 2020, Paper 2, Q1 (2 marks): Fig. 1 shows triangle ABC. C 22.1 m 133° A B 18.0 m Fig. 1 Calculate the area of triangle ABC, giving your answer correct to 3 significant figures. [2]
- November 2020, Paper 2, Q2 (3 marks): Fig. 2 shows a sector of a circle of radius 8 cm. The angle of the sector is 2.1 radians. 8 cm 2.1 L radians Fig. 2 (a) Calculate the length of the arc L. [1] (
- November 2020, Paper 2, Q3 (4 marks): You are given that y 4 x sin 8 x . = + d . y (a) Find x [2] d d = y (b) Find the smallest positive value of x for which x 0 , giving your answer in an exact for
- November 2020, Paper 2, Q10 (9 marks): In this question you must show detailed reasoning. The equation of a curve is sin 2 x - . x y = x sin x (a) Use the small angle approximation given in the list
- November 2020, Paper 2, Q14 (8 marks): In this question you must show detailed reasoning. 1 - sin2x cosx. Fig. 14 shows the graphs of y = sin x cos 2x and y = 2 y 1 1 y= –sin2x cosx 2 0.5 x 0 0.5 1 y
- November 2020, Paper 3, Q5 (11 marks): Fig. 5 shows part of the curve y = cosec x together with the x- and y-axes. y x O Fig. 5 (a) For the section of the curve which is shown in Fig. 5, write down (
- November 2020, AS Paper 1, Q2 (4 marks): Fig. 2 shows a quadrilateral ABCD. The lengths AB and BC are 5 cm and 6 cm respectively. The angles ABC, ACD and DAC are 60°, 60° and 75° respectively. D A 75°
- November 2020, AS Paper 2, Q8 (6 marks): In this question you must show detailed reasoning. Solve the equation 3 cos 8 tan 0 for ° 0 1 1 360 ° , giving your answers correct to the + = i i i nearest deg
- June 2019, Paper 1, Q6 (7 marks): sin 1 i 6 (a) Prove that cot . [4] - = i 1 cos sin - i i sin 1 i (b) Hence find the exact roots of the equation 3 tan in the interval 0 G r. - = G i i 1 cos sin
- June 2019, Paper 1, Q10 (7 marks): 10 (a) Express cos 7 x 2 sin x in the form R cos ( x ) where R 2 0 and 0 1 1 r, giving the - + a a 2 exact value of R and the value of a correct to 3 significan
- June 2019, Paper 1, Q14 (8 marks): Fig. 14 shows a circle with centre O and radius r cm. The chord AB is such that angle AOB x radians . The area of the shaded segment formed by AB is 5% of the a
- June 2019, Paper 2, Q7 (5 marks): 288cm2. The angle of the sector is i radians and the radius of 7 The area of a sector of a circle is . the circle is r cm. (a) Find an expression for i in terms
- June 2019, AS Paper 1, Q2 (3 marks): In this question you must show detailed reasoning. Fig. 2 shows the graphs of y 4 sin x ° and y 3 cos x ° for 0 x 360 . = = G G y 4 2 x 0 0 90 180 270 360 -2 -4
- June 2019, AS Paper 1, Q4 (5 marks): A triangle ABC has sides AB = 5 cm, AC = 9 cm and BC = 10 cm. (a) Find the cosine of angle BAC, giving your answer as a fraction in its lowest terms. [2] (b) Fi
- June 2019, AS Paper 2, Q2 (2 marks): Fig. 2 shows a triangle with one angle of 117° given. The lengths are given in centimetres. 117° 14 8 19 Fig. 2 [2] Calculate the area of the triangle, giving y
- June 2018, Paper 1, Q3 (4 marks): In this question you must show detailed reasoning. 2i [4] Solve the equation sec 2 tan 4 for ° 0 1 360 ° . + i = G i
- June 2018, Paper 1, Q8 (6 marks): 2 can be written as 8 (i) Show that sin 8 x cos x 1 cos x 4 . [3] - y 2 2 (ii) Hence find sin x cos x x d . [3]
- June 2018, Paper 1, Q12 (14 marks): 2 12 Fig. 12 shows the circle ( x 1 ) ( y 1 ) 25 , the line y 4 3 x 32 and the tangent to the circle at the - + + = = - point A (5, 2). D is the point of inters
- June 2018, Paper 2, Q6 (5 marks): r . 6 (i) Express 7cosx ‒ 24sinx in the form Rcos(x + α), where 0 < α < 2 [3] (ii) Write down the range of the function f(x) = 12 + 7cosx ‒ 24sin x , x 0 G r. 2
- June 2018, Paper 3, Q1 (3 marks): Triangle ABC is shown in Fig. 1. A 14cm 85° C 32cm B Fig. 1 Find the perimeter of triangle ABC. [3]
- June 2018, Paper 3, Q3 (2 marks): Fig. 3 shows a circle with centre O and radius 1 unit. Points A and B lie on the circle with angle AOB = i radians. C lies on AO, and BC is perpendicular to AO.
- June 2018, AS Paper 1, Q5 (7 marks): (i) Sketch the graphs of y 4 cos x and y 2 sin x for ° 0 x 180 ° on the same axes. [2] = = G G (ii) Find the exact coordinates of the point of intersection of t
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