plumiTrigonometry: AQA A‑Level Maths questions
Every AQA A-Level Maths exam question on E Trigonometry, 2018–2025, with its mark scheme and examiner’s report.
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![June 2025 Paper 2, question 2: One of the diagrams below shows the graph of y = cosec x° for 0 ≤ x ≤ 360 Identify the correct graph. Tick (✓) one box. [1 mark] x y x y 1 –1 1 –1 180 360 90 27](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/73572-jun25-02-qp.webp)
![June 2025 Paper 2, question 5: The diagram shows a sector AOB of a circle with centre O and radius 8 cm. B A O 8 cm 1.4 The angle AOB is 1.4 radians. Find the area of the sector. [2 marks]](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/73572-jun25-05-qp.webp)
![June 2025 Paper 2, question 8: (a) Given that cos x ≠ 0 state the value of sec2 x – tan2 x [1 mark] 8 (b) Show that (3 sec x + 5 tan x)(5 sec x – 3 tan x) – 16 tan x cos x = N where N is an i](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/73572-jun25-08-qp.webp)
![June 2025 Paper 3, question 10: (a) Express 2 sin x + 5 cos x in the form R sin (x + α) where R > 0 and 0° ≤ α ≤ 90° [3 marks] 10 (b) In 2010, the water temperature at a beach could be modelle](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/73573-jun25-10-qp.webp)
![June 2024 Paper 1, question 4: One of the diagrams below shows the graph of y = arccos x Identify the graph of y = arccos x Tick (✓) one box. [1 mark] y x O π 2 –1 y x O 1 π 2](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/73571-jun24-04-qp.webp)
![June 2024 Paper 1, question 5: Solve the equation sin2 x = 1 for 0° < x < 360° [3 marks]](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/73571-jun24-05-qp.webp)
![June 2024 Paper 1, question 9: (a) Show that, for small values of θ measured in radians cos 4θ + 2 sin 3θ – tan 2θ ≈ A + Bθ + Cθ2 where A, B and C are constants to be found. [3 marks] 9 (b) U](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/73571-jun24-09-qp.webp)
![June 2024 Paper 1, question 15: (a) Show that the expression sin 2θ cosec θ + cos 2θ sec θ can be written as 4 cos θ – sec θ where sin θ ≠ 0 and cos θ ≠ 0 [4 marks] 15 (b) A student is attempt](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/73571-jun24-15-qp.webp)
![June 2024 Paper 2, question 6: It is given that (2 sin θ + 3 cos θ)2 + (6 sin θ – cos θ)2 = 30 and that θ is obtuse. Find the exact value of sin θ. Fully justify your answer. [6 marks]](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/73572-jun24-06-qp.webp)
![June 2023 Paper 1, question 4: Given that y is a small angle, find an approximation for cos 2y Circle your answer. [1 mark] 1 − y2 2 2 − 2y2 1 − 2y2 1 − y2 for the next question](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/73571-jun23-04-qp.webp)
![June 2023 Paper 1, question 13: The function f is defined by f (x) = arccosx for 0 × x × a The curve with equation y = f (x) is shown below. O a x 2 2 y 13 (a) State the value of a [1 mark] 13](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/73571-jun23-13-qp.webp)
All 58 questions on Trigonometry, as a list
- June 2025, Paper 1, Q8 (5 marks): Show that, for small values of x, the graph with equation y = 4 + sin 5x – 3 cos 2x 1 + 2 tan x can be approximated by the straight line with equation of the fo
- June 2025, Paper 1, Q16 (7 marks): The triangle ABC is shown in the diagram below. The angle ABC is π 3 radians. The angle BCA is x radians. A B C x 3 16 (a) Explain why angle BAC = 2π 3 – x [1 m
- June 2025, Paper 2, Q2 (1 mark): One of the diagrams below shows the graph of y = cosec x° for 0 ≤ x ≤ 360 Identify the correct graph. Tick (✓) one box. [1 mark] x y x y 1 –1 1 –1 180 360 90 27
- June 2025, Paper 2, Q5 (2 marks): The diagram shows a sector AOB of a circle with centre O and radius 8 cm. B A O 8 cm 1.4 The angle AOB is 1.4 radians. Find the area of the sector. [2 marks]
- June 2025, Paper 2, Q8 (6 marks): (a) Given that cos x ≠ 0 state the value of sec2 x – tan2 x [1 mark] 8 (b) Show that (3 sec x + 5 tan x)(5 sec x – 3 tan x) – 16 tan x cos x = N where N is an i
- June 2025, Paper 2, Q9 (9 marks): A circle with centre C has equation (x – 12)2 + (y – 2)2 = 100 The graph with equation y = │3x – 36│ – 8 intersects the circle at the points A, B and D as shown
- June 2025, Paper 2, Q15 (8 marks): A particle moves under the actions of two forces, F1 and F2 F1 has magnitude 17 newtons and acts due East. F2 has magnitude 26 newtons and acts at a bearing of
- June 2025, Paper 3, Q2 (1 mark): The diagram shows the graph of y = arctan x x a – a y O The graph has horizontal asymptotes at y = –a and y = a State the value of a Circle your answer. [1 mark
- June 2025, Paper 3, Q10 (10 marks): (a) Express 2 sin x + 5 cos x in the form R sin (x + α) where R > 0 and 0° ≤ α ≤ 90° [3 marks] 10 (b) In 2010, the water temperature at a beach could be modelle
- June 2024, Paper 1, Q4 (1 mark): One of the diagrams below shows the graph of y = arccos x Identify the graph of y = arccos x Tick (✓) one box. [1 mark] y x O π 2 –1 y x O 1 π 2
- June 2024, Paper 1, Q5 (3 marks): Solve the equation sin2 x = 1 for 0° < x < 360° [3 marks]
- June 2024, Paper 1, Q9 (5 marks): (a) Show that, for small values of θ measured in radians cos 4θ + 2 sin 3θ – tan 2θ ≈ A + Bθ + Cθ2 where A, B and C are constants to be found. [3 marks] 9 (b) U
- June 2024, Paper 1, Q15 (6 marks): (a) Show that the expression sin 2θ cosec θ + cos 2θ sec θ can be written as 4 cos θ – sec θ where sin θ ≠ 0 and cos θ ≠ 0 [4 marks] 15 (b) A student is attempt
- June 2024, Paper 2, Q6 (6 marks): It is given that (2 sin θ + 3 cos θ)2 + (6 sin θ – cos θ)2 = 30 and that θ is obtuse. Find the exact value of sin θ. Fully justify your answer. [6 marks]
- June 2024, Paper 3, Q5 (3 marks): The diagram below shows a sector of a circle OAB. The chord AB divides the sector into a triangle and a shaded segment. Angle AOB is π 6 radians. The radius of
- June 2024, Paper 3, Q9 (9 marks): Figure 1 below shows a circle. Figure 1 y x O P Q The centre of the circle is P and the circle intersects the y-axis at Q as shown in Figure 1. The equation of
- June 2023, Paper 1, Q4 (1 mark): Given that y is a small angle, find an approximation for cos 2y Circle your answer. [1 mark] 1 − y2 2 2 − 2y2 1 − 2y2 1 − y2 for the next question
- June 2023, Paper 1, Q10 (8 marks): The curve with equation y = sinx° for −360 × x × 360 is shown below. A – 360 O y x B 360 10 (a) Point A on the curve has coordinates (a, 0.5) 10 (a) (i) Find th
- June 2023, Paper 1, Q12 (8 marks): One of the rides at a theme park is a room where the floor and ceiling both move up and down for 10p seconds. At time t seconds after the ride begins, the dista
- June 2023, Paper 1, Q13 (9 marks): The function f is defined by f (x) = arccosx for 0 × x × a The curve with equation y = f (x) is shown below. O a x 2 2 y 13 (a) State the value of a [1 mark] 13
- June 2023, Paper 2, Q8 (10 marks): (a) Given that cos y 6= ≤1 , prove the identity 1 1 − cos y + 1 1 + cos y ≥ 2 cosec2 y [4 marks] 8 (b) Hence, find the set of values of A for which the equation
- June 2023, Paper 3, Q7 (14 marks): A new design for a company logo is to be made from two sectors of a circle, ORP and OQS, and a rhombus OSTR, as shown in the diagram below. θ O P R S T Q The po
- June 2022, Paper 1, Q6 (6 marks): (a) Find the first two terms, in ascending powers of x, of the binomial expansion of 1 × x 2 1 2 [2 marks] 6 (b) Hence, for small values of x, show that sin 4x
- June 2022, Paper 1, Q7 (3 marks): Sketch the graph of y = cot x × p 2 for 0 − x − 2p [3 marks] x y O 2
- June 2022, Paper 1, Q10 (12 marks): The diagram shows a sector of a circle OAB. O θ C A B The point C lies on OB such that AC is perpendicular to OB. Angle AOB is y radians. 10 (a) Given the area
- June 2022, Paper 1, Q12 (8 marks): (a) A geometric sequence has first term 1 and common ratio 1 2 12 (a) (i) Find the sum to infinity of the sequence. [2 marks] 12 (a) (ii) Hence, or otherwise, e
- June 2022, Paper 1, Q15 (16 marks): (a) Given that y = cosecy 15 (a) (i) Express y in terms of siny. [1 mark] 15 (a) (ii) Hence, prove that dy dy = × cosecy cot y [3 marks] 15 (a) (iii) Show that
- June 2022, Paper 2, Q4 (3 marks): 8.7 cm 6.1 cm A C B 38° The diagram shows a triangle ABC. AB is the shortest side. The lengths of AC and BC are 6.1 cm and 8.7 cm respectively. The size of angl
- June 2022, Paper 3, Q5 (3 marks): (a) Sketch the graph of y = sin 2x for 0° × x ×360° x O y 90° 180° 270° 360° [2 marks] 5 (b) The equation sin 2x = A has exactly two solutions for 0° × x ×360°
- November 2021, Paper 1, Q8 (9 marks): (a) Given that 9 sin2 y + sin 2y = 8 show that 8 cot2 y − 2 cot y − 1 = 0 [4 marks] 8 (b) Hence, solve 9 sin2 y + sin 2y = 8 in the interval 0 < y < 2p Give you
- November 2021, Paper 1, Q10 (8 marks): (a) Given that y = tanx use the quotient rule to show that dy dx = sec2 x [3 marks] 10 (b) The region enclosed by the curve y = tan2 x and the horizontal line,
- November 2021, Paper 1, Q15 (10 marks): (a) Show that sin x − sinx cos 2x ≤ 2x3 for small values of x. [3 marks] 15 (b) Hence, show that the area between the graph with equation y = 8( sinx − sin x co
- November 2021, Paper 2, Q9 (9 marks): A robotic arm which is attached to a flat surface at the origin O, is used to draw a graphic design. The arm is made from two rods OP and PQ, each of length d,
- November 2021, Paper 2, Q18 (11 marks): Two particles, P and Q, are projected at the same time from a fixed point X, on the ground, so that they travel in the same vertical plane. P is projected at an
- November 2021, Paper 3, Q1 (1 mark): The graph of y = arccosx is shown. P x y State the coordinates of the end point P. Circle your answer. [1 mark] (−p, 1) (−1, p) − p 2 , 1 −1, p 2
- November 2021, Paper 3, Q5 (13 marks): A gardener is creating flowerbeds in the shape of sectors of circles. The gardener uses an edging strip around the perimeter of each of the flowerbeds. The cost
- November 2020, Paper 1, Q3 (1 mark): The diagram shows a sector OAB of a circle with centre O and radius 2 B 2 A O θ The angle AOB is y radians and the perimeter of the sector is 6 Find the value o
- November 2020, Paper 1, Q8 (7 marks): Mike, an amateur astronomer who lives in the South of England, wants to know how the number of hours of darkness changes through the year. On various days betwe
- November 2020, Paper 2, Q2 (1 mark): Which one of the following equations has no real solutions? Tick (3) one box. [1 mark] cot x = 0 ln x = 0 jx + 1j = 0 sec x = 0
- November 2020, Paper 2, Q4 (4 marks): Using small angle approximations, show that for small, non-zero, values of x x tan 5x cos 4x − 1 × A where A is a constant to be determined. [4 marks]
- November 2020, Paper 2, Q8 (10 marks): The curve defined by the parametric equations x = t2 and y = 2t − 2 p × t × 2 p is shown in Figure 1 below. Figure 1 x y O 8 (a) Find a Cartesian equation of th
- November 2020, Paper 2, Q17 (6 marks): A ball is projected forward from a fixed point, P, on a horizontal surface with an initial speed u m s−1, at an acute angle y above the horizontal. The ball nee
- November 2020, Paper 3, Q2 (1 mark): Given that 6 cos y + 8 sin y × R cos (y + a) find the value of R. Circle your answer. [1 mark] 6 8 10 14
- November 2020, Paper 3, Q9 (5 marks): (a) For cos y 6= 0 , prove that cosec 2y + cot 2y = coty [4 marks] 9 (b) Explain why cot y 6= cosec 2y + cot 2y when cos y = 0 [1 mark]
- June 2019, Paper 1, Q3 (1 mark): The diagram below shows a sector of a circle. θ The radius of the circle is 4 cm and y = 0:8 radians. Find the area of the sector. Circle your answer. [1 mark]
- June 2019, Paper 1, Q7 (11 marks): (a) By sketching the graphs of y = 1 x and y = sec 2x on the axes below, show that the equation 1 x = sec 2x has exactly one solution for x > 0 [3 marks] x ϖ 2ϖ
- June 2019, Paper 1, Q11 (4 marks): Jodie is attempting to use differentiation from first principles to prove that the gradient of y = sinx is zero when x = p 2 Jodie’s teacher tells her that she
- June 2019, Paper 1, Q12 (7 marks): (a) Show that the equation 2 cot2 x + 2 cosec2 x = 1 + 4 cosecx can be written in the form a cosec2 x + b cosecx + c = 0 [2 marks] 12 (b) Hence, given x is obtu
- June 2019, Paper 2, Q6 (6 marks): A curve has equation y = a sinx + b cos x where a and b are constants. The maximum value of y is 4 and the curve passes through the point p 3 , 2 3 p as shown i
- June 2019, Paper 3, Q1 (1 mark): f (x) = arcsinx State the maximum possible domain of f Tick (3) one box. [1 mark] fx 2 R : −1 × x × 1g n x 2 R : − p 2 × x × p 2 o fx 2 R : −p × x × pg fx 2 R :
- June 2019, Paper 3, Q5 (5 marks): A circle has equation x2 + y2 − 6x − 8y = 264 AB is a chord of the circle. The angle at the centre of the circle, subtended by AB, is 0.9 radians, as shown in t
- June 2018, Paper 1, Q3 (1 mark): A periodic sequence is defined by Un = sin np 2 State the period of this sequence. Circle your answer. [1 mark] 8 2p 4 p
- June 2018, Paper 1, Q8 (8 marks): The diagram shows a sector of a circle OAB. C is the midpoint of OB. Angle AOB is y radians. C O θ B A 8 (a) Given that the area of the triangle OAC is equal to
- June 2018, Paper 1, Q12 (10 marks): p(x) = 30x3 − 7x2 − 7x + 2 12 (a) Prove that (2x + 1) is a factor of p(x) [2 marks] 12 (b) Factorise p(x) completely. [3 marks] 12 (c) Prove that there are no r
- June 2018, Paper 1, Q14 (7 marks): Some students are trying to prove an identity for sin (A + B) . They start by drawing two right-angled triangles ODE and OEF, as shown. The students’ incomplete
- June 2018, Paper 2, Q8 (10 marks): (a) Determine a sequence of transformations which maps the graph of y = sinx onto the graph of y = 3 p sin x − 3 cos x + 4 Fully justify your answer. [7 marks]
- June 2018, Paper 3, Q5 (3 marks): Show that, for small values of x, the graph of y = 5 + 4 sin x 2 + 12 tan x 3 can be approximated by a straight line. [3 marks]
- June 2018, Paper 3, Q8 (9 marks): (a) Prove the identity sin 2x 1 + tan2 x × 2 sin x cos3 x [3 marks] 8 (b) Hence find ( 4 sin 4y 1 + tan2 2y dy [6 marks]
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