plumiTrigonometry: OCR A‑Level Maths questions
Every OCR 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 9: (a) Express cos 3 2 x 4 sin 2 x in the form R cos ( 2 x ) , where R 2 0 and ° 0 1 1 90 ° . [3] + - a a (b) In this question you must show detailed reasoning. 3](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/h24001-jun25-09-qp.webp?v=3)
![June 2025 Paper 2, question 4: / 4 (a) Prove that ( cos sin ) 1 sin 2 . [2] + + i i i cos sin + i i (b) Hence or otherwise prove that sec 2 tan 2 . [4] + i i / cos sin i - i](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/h24002-jun25-04-qp.webp?v=3)
![June 2025 AS Paper 1, question 4: cos 1 i 4 (a) Show that tan (where 90 n ° for any odd integer n). [3] - i i ! / 1 sin cos - i i cos 3 x 1 (b) Hence solve the equation = for 1 0 x 90 ° . [4] -](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/h23001-jun25-04-qp.webp?v=3)
![June 2025 AS Paper 2, question 3: In a triangle ABC, AB = 9 cm, BC = 7 cm and AC = 4 cm. 2 (a) Show that cos CAB . [2] = 3 (b) Hence find the exact value of sin CAB. [2] (c) Find the exact area](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/h23002-jun25-03-qp.webp?v=3)
![June 2024 Paper 1, question 11: A curve has equation y 5 ln ( 1 cos 2 x ) , where x is in radians. = - (a) State the values of x for which 5 ln ( 1 cos x 2 ) is not defined. [2] - (b) P is the](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/h24001-jun24-11-qp.webp?v=3)
![June 2024 Paper 3, question 4: - 4 (a) Show that the equation cot 2 x 9 cosec x 3 0 can be expressed in the form - = 2 + 5 sin x 9 sin x 2 0 . [3] - = (b) (i) In this question you must show d](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/h24003-jun24-04-qp.webp?v=3)
![June 2023 Paper 1, question 1: In the triangle ABC, the length AB = 6 cm, the length AC = 15 cm and the angle BAC = 30°. (a) Calculate the length BC. [2] D is the point on AC such that the le](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/h24001-jun23-01-qp.webp?v=3)
![June 2023 Paper 1, question 7: (a) Use the result cos ( A B ) cos A cos B sin A sin B to show that + = - cos ( A B ) cos A cos B sin A sin B . [2] - = + The function ( ) f i is defined as cos](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/h24001-jun23-07-qp.webp?v=3)
![June 2023 Paper 3, question 2: (a) Express sin 3 x 4 cos x in the form R sin ( x ) , where R 0 and ° 0 90 ° . Give the - - 2 1 1 a a value of a correct to 4 significant figures. [3] (b) Hence](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/h24003-jun23-02-qp.webp?v=3)
![June 2023 AS Paper 1, question 1: 1 1 (a) Prove that cos x sin x tan x (where x n r for any odd integer n). [3] + / ! cos x 2 2 2 (b) Solve the equation sin 2 x cos x for 0 x 180 c. [2] = c G G](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/h23001-jun23-01-qp.webp?v=3)
All 54 questions on Trigonometry, as a list
- June 2025, Paper 1, Q6 (8 marks): B NOT TO SCALE 6cm A C D The diagram shows a triangle ABC. The arc BD is part of a circle with centre A and radius 6 cm. 2. The area of the sector ABD is 14 4 .
- June 2025, Paper 1, Q9 (10 marks): (a) Express cos 3 2 x 4 sin 2 x in the form R cos ( 2 x ) , where R 2 0 and ° 0 1 1 90 ° . [3] + - a a (b) In this question you must show detailed reasoning. 3
- June 2025, Paper 2, Q4 (6 marks): / 4 (a) Prove that ( cos sin ) 1 sin 2 . [2] + + i i i cos sin + i i (b) Hence or otherwise prove that sec 2 tan 2 . [4] + i i / cos sin i - i
- June 2025, Paper 3, Q6 (12 marks): The compound angle formulae for sin A ( B ) and sin A ( B ) are + - sin ( A B ) sin A cos B cos A sin B and + = + sin ( A B ) sin A cos B cos A sin B . - = - (a
- June 2025, AS Paper 1, Q4 (7 marks): cos 1 i 4 (a) Show that tan (where 90 n ° for any odd integer n). [3] - i i ! / 1 sin cos - i i cos 3 x 1 (b) Hence solve the equation = for 1 0 x 90 ° . [4] -
- June 2025, AS Paper 2, Q3 (6 marks): In a triangle ABC, AB = 9 cm, BC = 7 cm and AC = 4 cm. 2 (a) Show that cos CAB . [2] = 3 (b) Hence find the exact value of sin CAB. [2] (c) Find the exact area
- June 2024, Paper 1, Q9 (9 marks): The depth of the water, d metres, in a tidal river during a given day is modelled by the equation d 1 9 . 1 1 . cos ( 30 t 60 )° = + - where t is the number of
- June 2024, Paper 1, Q11 (12 marks): A curve has equation y 5 ln ( 1 cos 2 x ) , where x is in radians. = - (a) State the values of x for which 5 ln ( 1 cos x 2 ) is not defined. [2] - (b) P is the
- June 2024, Paper 2, Q6 (10 marks): In this question you must show detailed reasoning. (a) (i) Use the formula for cos A ( B ) , and the double angle formulae, to show that + 3i cos 3 4 cos 3 cos
- June 2024, Paper 3, Q2 (5 marks): In this question you must show detailed reasoning. A O 2rad (3x + 1) cm B The diagram shows a sector AOB of a circle with centre O and radius ( x . The angle AO
- June 2024, Paper 3, Q4 (9 marks): - 4 (a) Show that the equation cot 2 x 9 cosec x 3 0 can be expressed in the form - = 2 + 5 sin x 9 sin x 2 0 . [3] - = (b) (i) In this question you must show d
- June 2024, AS Paper 1, Q3 (5 marks): In the triangle ABC, AB = 3, AC = 5 and angle BAC = a. 5. The area of the triangle is 2 Determine the two possible values of cos a. Give your answers in exact f
- June 2024, AS Paper 2, Q2 (5 marks): D C 5 cm A B 14cm The diagram shows a parallelogram ABCD where AB 14 cm , AD 5 cm and angle DAB is = = 2. obtuse. The area of ABCD is 50 cm (a) Determine, in de
- June 2024, AS Paper 2, Q7 (9 marks): y x O 12 The diagram shows the curve with equation y 5 3 cos ( 30 x 60 )° , for 0 x 12 . = + + G G (a) (i) State the greatest value of y for points on the curve
- June 2023, Paper 1, Q1 (5 marks): In the triangle ABC, the length AB = 6 cm, the length AC = 15 cm and the angle BAC = 30°. (a) Calculate the length BC. [2] D is the point on AC such that the le
- June 2023, Paper 1, Q7 (9 marks): (a) Use the result cos ( A B ) cos A cos B sin A sin B to show that + = - cos ( A B ) cos A cos B sin A sin B . [2] - = + The function ( ) f i is defined as cos
- June 2023, Paper 2, Q5 (12 marks): In this question you must show detailed reasoning. The function f is defined by f( ) x cos x 3 sin x with domain 0 x 2 r. = + G G (a) Solve the following equati
- June 2023, Paper 3, Q2 (5 marks): (a) Express sin 3 x 4 cos x in the form R sin ( x ) , where R 0 and ° 0 90 ° . Give the - - 2 1 1 a a value of a correct to 4 significant figures. [3] (b) Hence
- June 2023, Paper 3, Q6 (6 marks): The first, third and fourth terms of an arithmetic progression are u1, u3 and u4 respectively, where 7 u 2 sin , u 3 cos , u sin , = i =- i = i 1 3 4 2 1 and 2
- June 2023, AS Paper 1, Q1 (5 marks): 1 1 (a) Prove that cos x sin x tan x (where x n r for any odd integer n). [3] + / ! cos x 2 2 2 (b) Solve the equation sin 2 x cos x for 0 x 180 c. [2] = c G G
- June 2023, AS Paper 1, Q8 (7 marks): In this question you must show detailed reasoning. 2 2 A circle has equation x y 6 x 4 y 12 0 . Two tangents to this circle pass through the + - - + = point (0,
- June 2023, AS Paper 2, Q3 (8 marks): A Ferris wheel at a fairground rotates in a vertical plane. The height above the ground of a seat on the wheel is h metres at time t seconds after the seat is a
- June 2022, Paper 1, Q4 (8 marks): + 2 4 (a) Write x x p x q + , where p, q and r are constants. r [3] 2 6 + in the form 7 ( ) + 2 (b) State the coordinates of the minimum point on the graph of y
- June 2022, Paper 1, Q10 (12 marks): B irad O A M The diagram shows a sector OAB of a circle with centre O and radius OA. The angle AOB is i radians. M is the mid-point of OA. The ratio of areas OM
- June 2022, Paper 3, Q7 (8 marks): In this question you must show detailed reasoning. (a) Show that the equation m sec 3 cos 4 sin can be expressed in the form i + i = i 2i m tan 4 tan ( m 3 ) 0
- June 2022, AS Paper 1, Q4 (8 marks): In this question you must show detailed reasoning. Solve the following equations, for ° 0 x 360 ° . G G (a) 2 tan x 1 4 [3] + = 2 (b) 5 sin x 1 2 cos x [5] - =
- June 2022, AS Paper 2, Q7 (9 marks): A 2i + 4j C O 4i – 3j B The diagram shows the parallelogram OACB where OA 2 i 4 j and OB 4 i 3 j . = + = - 2 5 (a) Show that cos AOB . [5] = - 25 (b) Hence find
- November 2021, Paper 1, Q8 (10 marks): Functions f and g are defined for 0 x 2 by ( ) f x 2 tan x and ( ) g x sec x . G G = = r (a) (i) State the range of f. [1] (ii) State the range of g. [1] (b) (i
- November 2021, Paper 1, Q10 (11 marks): (a) C x y b a A D B The diagram shows triangle ABC. The perpendicular from C to AB meets AB at D. Angle ACD = , angle x D B = C y , length BC = a and length AC
- November 2021, Paper 2, Q4 (10 marks): The size, P, of a population of a certain species of insect at time t months is modelled by the following formula. P = 5000 - 1000cos (30t)° (a) Write down the
- November 2021, Paper 2, Q7 (4 marks): Differentiate cos x with respect to x, from first principles. [4]
- November 2021, Paper 3, Q2 (6 marks): B A 60° C The diagram shows triangle ABC in which angle A is 60° and the lengths of AB and AC are ( 4 h ) cm and ( 4 h ) cm respectively. + - (a) Show that the
- November 2021, Paper 3, Q5 (6 marks): - , A particle P moves along a straight line in such a way that at time t seconds P has velocity v m s 1 5 where v 12 cos t 5 sin t . = + 1 (a) Express v in the
- November 2021, Paper 3, Q6 (6 marks): 6 The equation 6 arcsin ( 2 x 1 ) x 0 has exactly one real root. - - = (a) Show by calculation that the root lies between 0.5 and 0.6. [2] In order to find the
- November 2021, AS Paper 1, Q1 (7 marks): In the triangle ABC, AB = 3, BC = 4 and angle ABC = 30°. Find the following. (a) The area of the triangle. [2] (b) The length AC. [2] (c) The angle ACB. [3]
- November 2021, AS Paper 2, Q5 (9 marks): = 5 (a) Show that the equation 2 cos tan x x 3 1 ( cos x ) can be expressed in the form + 2 + 5 cos x 3 cos x 2 0 . [3] - = (b) In this question you must show d
- November 2020, Paper 1, Q1 (5 marks): . 1 (a) For a small angle i, where i is in radians, show that 2 cos ( 1 tan ) 3 2 . [3] i + - i - i 2 (b) Hence determine an approximate solution to 2 cos ( 1 t
- November 2020, Paper 2, Q4 (5 marks): In this question you must show detailed reasoning. 4 2 Solve the equation 3 sin sin 4 , for 0 1 2 r, where z is measured in radians. [5] + z = G z z
- November 2020, Paper 2, Q6 (3 marks): 2 6 Prove that 2 cos ( 2 45 °) cos 2 sin cos sin , where i is measured in degrees. [3] + - - i / i i i i
- November 2020, Paper 3, Q1 (2 marks): Triangle ABC has AB = 8.5 cm, BC = 6.2 cm and angle B = 35°. Calculate the area of the triangle. [2]
- November 2020, AS Paper 1, Q3 (8 marks): In this question you must show detailed reasoning. 4sin2 2 (a) Solve the equation tan for 0 ° 180 ° . [5] i = i G i G sin2 1 cos - + i i (b) Prove that cos ( co
- November 2020, AS Paper 2, Q1 (5 marks): B 25° O A 9.5 cm The diagram shows a sector AOB of a circle with centre O and radius 9.5 cm. The angle AOB is 25°. (a) Calculate the length of the straight line
- June 2019, Paper 1, Q9 (11 marks): 9 (a) Express cos 3 3 x 7 sin 3 x in the form R cos ( 3 x ) , where R 2 0 and 0 1 1 r. [3] + - a a 2 (b) Give full details of a sequence of three transformation
- June 2019, Paper 1, Q10 (11 marks): A i rad O 6cm B The diagram shows a sector AOB of a circle with centre O and radius 6 cm. The angle AOB is i radians. The area of the segment bounded by the cho
- June 2019, Paper 3, Q1 (2 marks): C 13.5cm 8.3cm 32° A B The diagram shows triangle ABC, with AC 13 5cm . , BC 8 3cm . and angle ABC 32° . = = = Find angle CAB. [2]
- June 2019, Paper 3, Q5 (9 marks): In this question you must show detailed reasoning. 1 cos + i 2 (a) Prove that ( cot cosec ) . [4] + = i i 1 cos - i 2 (b) Hence solve, for 0 1 1 2 , r ( 3 cot c
- June 2019, AS Paper 1, Q3 (10 marks): In this question you must show detailed reasoning. 3 2 (a) The polynomial ( )x f is defined by ( )x f 2 x 3 x 8 x + . 3 = + - (i) Show that ( ) f 1 0 . [1] = (i
- June 2019, AS Paper 1, Q6 (6 marks): B xcm 60° A ycm C The diagram shows triangle ABC, with AB = x cm, AC = y cm and angle BAC = 60°. It is given that ^ h 3cm2 the area of the triangle is x y . + (
- June 2019, AS Paper 2, Q6 (7 marks): In this question you must show detailed reasoning. 2i (a) Show that the equation cos 6 tan cos + 4 = i i 2i can be expressed in the form 6 sin sin 2 0 . [2] + -
- June 2018, Paper 2, Q4 (4 marks): Prove that sin2(i + 45)° - cos2(i + 45)° / sin 2i °. 4 [4]
- June 2018, Paper 3, Q6 (8 marks): a k a k. 1 1 6 It is given that the angle i satisfies the equation sin 2 3 cos 2 i + r = i + r 4 4 1 (i) Show that tan2 [3] . i = 2 (ii) Hence find, in surd for
- June 2018, AS Paper 1, Q3 (6 marks): 3 (i) Solve the equation sin 0 25 . for º 0 360 º . [3] 1 i = G i (ii) In this question you must show detailed reasoning. Solve the equation tan 3 3 for º 0 90
- June 2018, AS Paper 2, Q1 (6 marks): In triangle ABC, AB = 20 cm and angle B = 45°. (i) Given that AC = 16 cm, find the two possible values for angle C, correct to 1 decimal place. [4] (ii) Given i
- June 2018, AS Paper 2, Q7 (8 marks): (i) Show that the equation x x x 2 sin tan cos 5 = + can be expressed in the form 2 + x x [3] 3 cos 5 cos 2 0 . - = (ii) Hence solve the equation 2 sin 2 tan 2
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