Trigonometry: Edexcel International GCSE Further Maths questions

Edexcel International GCSE Further Maths past-paper questions on 10 Trigonometry, 2019–2025, with mark schemes and examiner’s reports where available.

All 96 questions on Trigonometry, as a list

June 2025 (R), Paper 1, Q4 (7 marks): Triangle ABC is such that AB = 4 3. cm BC = 5 9. cm angle BCA 29 (a) Find in degrees, to one decimal place, the two possible values for angle CAB (4) (b) Hence,
June 2025 (R), Paper 1, Q10 (15 marks): (a) Using a formula from page 2, show that A 1 cos 2 (i) sin 2 A 2 A cos 2 1 (ii) cos 2 A 2 (5) Diagram NOT y accurately drawn C R A O x D Figure 2 Figure 2 sh
June 2025 (R), Paper 2, Q3 (6 marks): B Diagram NOT A C accurately drawn 6 cm θ rad O Figure 1 Figure 1 shows sector OABC of a circle with centre O and radius 6 cm The size of angle AOC is θ radians
June 2025 (R), Paper 2, Q6 (6 marks): V Diagram NOT accurately drawn B C A D Figure 2 Figure 2 shows a right pyramid VABCD with vertex V and rectangular base ABCD AD = VC = 3 AB Find in degrees, to
June 2025, Paper 1, Q3 (6 marks): Diagram NOT A accurately drawn 7 cm R θ rad O 7 cm B Figure 1 Figure 1 shows circle C1 with radius 7 cm and circle C2 with centre O and radius 7 cm The circles
June 2025, Paper 1, Q8 (13 marks): V Diagram NOT accurately drawn B C A D 12 cm Figure 3 Figure 3 shows a right pyramid with vertex V and square base, ABCD, of side 12 cm The size of the angle be
June 2025, Paper 2, Q1 (5 marks): Solve the equation o o 3 tan A = 2 sin A for 0 A 360 (5)
June 2025, Paper 2, Q5 (8 marks): In triangle ABC, AC =12 cm , BC =14 cm , AB = x cm and angle ABC 30 (a) Show that x P 95 where P is a prime number. 3 (5) (b) Hence or otherwise, find in cm2, t
June 2025, Paper 2, Q11 (15 marks): (a) Show that cos 4 4 sin cos 2 (4) (b) Hence, or otherwise, solve the equation 0 2 4 4 for 8 cos 2 3 2 cos 2 sin 4 8 8 Give your solutions to 2 decimal places.
November 2024, Paper 1, Q3 (5 marks): Triangle ABC is such that AC =10cm BC = 7cm angle CAB = 25 ° Given that angle ABC is obtuse, find, in cm to one decimal place, the length of AB (5)
November 2024, Paper 1, Q4 (6 marks): F 20 cm Diagram NOT E accurately drawn 15 cm B A C 40° D Figure 1 Figure 1 shows a right prism ABCDEF Triangle CDE is a cross section of the prism. 90° DCE ABF
November 2024, Paper 1, Q8 (11 marks): (i) (a) Using a formula given on page 2, show that A 2 tan tan 2 A 2 A 1 tan (2) (b) Hence, solve the equation A A 0 A tan tan 2 for 0 180 (5) (ii) Using a form
November 2024, Paper 2, Q1 (5 marks): S R Diagram NOT accurately drawn 2 cm θ rad O Figure 1 Figure 1 shows sector ROS of a circle with centre O and radius 2 cm The size of angle ROS is θ radians. T
November 2024, Paper 2, Q9 (14 marks): (a) Using a formula given on page 2, show that cos 2 2 cos 2 1 (2) (b) Hence show that 3 a b 2 2 cos 1 d 4 c 3 where a, b and c are integers to be found. (4) Di
June 2024 (R), Paper 1, Q3 (8 marks): B Diagram NOT accurately drawn x cm 30° A C (x + 7) cm Figure 1 Figure 1 shows triangle ABC where AB x AC x BAC cm 7 cm 30 o The area of triangle ABC = 36 cm2 (
June 2024 (R), Paper 1, Q4 (9 marks): (21 − r) cm Diagram NOT P Q accurately drawn r cm θ rad O Figure 2 Figure 2 shows the sector OPQ of a circle with centre O and radius r cm. OP = OQ = r cm arc P
June 2024 (R), Paper 1, Q11 (11 marks): f θ 2 cos θ sin θ 2 sin θ cos θ 3 (a) Show that f θ sin 2 θ 2 cos 2 θ 2 (3) Diagram NOT y accurately drawn S O a θ Figure 4 Figure 4 shows part of the curve S w
June 2024 (R), Paper 2, Q10 (13 marks): V Diagram NOT accurately drawn D C x cm A B 2x cm Figure 4 Figure 4 shows a right pyramid ABCDV The base of the pyramid is a rectangle where, AB = DC = x AD = B
June 2024 (R), Paper 2, Q11 (12 marks): (a) Using a formula on page 2 show that cos A A 2 2 cos 2 1 (2) cos 4 A 1 2 (b) Hence show that 2 cos 2 A 1 2 (3) 2 2 cos x 1 2 sin 2 x 1 The curve with equatio
June 2024, Paper 1, Q1 (4 marks): In triangle ABC, AB = 2 cm, BC x = 3 cm and AC x = 4 cm x The area of triangle ABC is 50 cm2 Find, to 2 decimal places, the value of x (4)
June 2024, Paper 2, Q3 (6 marks): Diagram NOT A accurately drawn 3r cm r cm θ rad O B Figure 1 Figure 1 shows the sector AOB of a circle with centre O and radius 3r cm A circle with radius r cm
June 2024, Paper 2, Q9 (15 marks): E F Diagram NOT accurately drawn 60° D A 45° 24 cm C B Figure 5 Figure 5 shows a right triangular prism ABCDEF where ABCD is a rectangle. AF = DE BF = CE AD = F
June 2024, Paper 2, Q11 (16 marks): Using formulae from page 2, show that (a) (i) cos A A 2 2 2 cos 1 (3) (ii) sin A = A A 2 2 sin cos (1) A A cos 3 3 cos (b) Show that cos A 3 4 (4) Hence, or oth
November 2023, Paper 1, Q6 (11 marks): V Diagram NOT D C accurately drawn x cm x cm O A B x cm Figure 3 Figure 3 shows a right pyramid with a horizontal square base. AB = BC = CD = DA = x cm AV = BV
November 2023, Paper 1, Q10 (17 marks): (a) Using formulae on page 2, show that (i) sin 2 A = 2 sin A cos A (ii) cos 2 A 2 cos 2 A 1 (3) 2 tan f 2 1 tan sin 2 (b) Show that f (4) (c) Solve, in radians
November 2023, Paper 2, Q2 (7 marks): In triangle ABC, AB x BC x ABC 110° 3 cm, 5 cm and (a) Find, in degrees to one decimal place, the size of ∠BCA (4) The area of triangle ABC is 24 cm2 (b) Find,
November 2023, Paper 2, Q9 (13 marks): O Diagram NOT accurately drawn x cm B 2x cm C A E D Figure 2 A logo, AEBCD, is shown shaded in Figure 2. The straight line ABC is the diameter of the semicircle
June 2023 (R), Paper 1, Q3 (9 marks): Diagram NOT A accurately drawn 100° 10 cm 8 cm B C x cm Figure 2 Figure 2 shows triangle ABC where AB = 10 cm, AC = 8 cm, BC = x cm and BAC 100o (a) Find, to 3
June 2023 (R), Paper 2, Q3 (10 marks): Diagram NOT accurately drawn A B r cm θ rad O Figure 1 Figure 1 shows the sector OAB of a circle with centre O. The radius of the circle is r cm and the angle A
June 2023 (R), Paper 2, Q7 (7 marks): y Diagram NOT accurately drawn S R m O n θ Figure 2 1 Figure 2 shows part of the curve S with equation y cos 3 θ 3 sin 3 θ 2 where m θ n The curve S meets the x
June 2023 (R), Paper 2, Q11 (12 marks): 1 11 (a) Use a formula on page 2 to show that sin A A 2 2 1 cos 2 (3) x 3 cos 4 (b) Show that sin 4 x cos 4 x 4 (5) (c) Hence solve, in degrees to one decimal p
June 2023, Paper 1, Q9 (13 marks): (a) Using the formulae on page 2, show that cos 2 A 1 (i) cos A 2 2 1 cos 2 A (ii) sin A 2 2 (4) (b) Show that (2 sin x – cos x)(sin x – 3 cos x) = 1 2 (cos 2x
June 2023, Paper 2, Q3 (6 marks): P Diagram NOT accurately drawn O r cm r cm B θ rad A Figure 1 Figure 1 shows a circle, centre O, with radius r cm. The points A, P and B lie on the circle. The
June 2023, Paper 2, Q9 (6 marks): V Diagram NOT accurately drawn 3x cm E A D 2x cm B C Figure 4 Figure 4 shows a right pyramid with vertex V and base ABCDE which is a regular pentagon. AB = BC =
January 2023 (R), Paper 1, Q2 (3 marks): In ΔABC, ∠BAC = 50° AB = 10 cm BC = 9 cm Given that ∠BCA = x° find the two possible values, to one decimal place, of x (3)
January 2023 (R), Paper 1, Q9 (13 marks): O Diagram NOT D accurately drawn x cm 30° C A 12 cm B Figure 2 Figure 2 shows the right pyramid OABCD with a square base ABCD of side 12 cm. OA = OB = OC = OD =
January 2023 (R), Paper 1, Q10 (16 marks): Using formulae from page 2 (a) show that cos(A – B) – cos(A + B) = 2 sin A sin B (2) (b) Hence show that cos 5θ – cos 9θ = 2 sin 7θ sin 2θ (1) (c) Solve the equ
January 2023 (R), Paper 2, Q6 (6 marks): O Diagram NOT accurately drawn r cm π rad 6 A B Figure 2 Figure 2 shows the sector OAB of a circle with centre O and radius r cm. π radians OA = OB = r cm ∠AOB
January 2023, Paper 1, Q8 (13 marks): A E D Diagram NOT accurately drawn x cm 0.5 rad F B y cm C G Figure 1 Figure 1 shows a badge, shown shaded, made from two identical rectangles, ABCD and DEFG, a
January 2023, Paper 2, Q2 (4 marks): Z Diagram NOT accurately drawn (2x + 4) cm (2x – 1) cm 60° X Y (x + 2) cm Figure 1 Figure 1 shows triangle XYZ in which XY = (x + 2) cm XZ = (2x + 4) cm YZ = (2
January 2023, Paper 2, Q5 (7 marks): E Diagram NOT accurately drawn D C O 12 cm A B 12 cm Figure 2 Figure 2 shows a right pyramid with a square base ABCD and vertex E. The base of the pyramid is ho
January 2023, Paper 2, Q10 (13 marks): Using formulae given on page 2 (a) show that (i) sin 2θ = 2sin θ cos θ (ii) cos 2θ = 2cos2 θ – 1 (5) Given that θ ≠ (90° + 180° n) where n (b) use the results f
June 2022 (R), Paper 1, Q3 (6 marks): O Diagram NOT θ rad accurately drawn T 4 cm R S Figure 1 Figure 1 shows sector ORS of a circle with centre O and radius 4 cm. The size of angle ROS is θ radians
June 2022 (R), Paper 1, Q6 (8 marks): Given that a + 5 = + 11 5 5 5 − 2 (a) without using a calculator, find the value of a Show your working clearly. (2) Triangle PQR is such that PR = (x + 3) cm Q
June 2022 (R), Paper 1, Q10 (11 marks): Using suitable results for sin (A + B) and sin (A – B) from the formulae page, (a) show that 2 sin 4 x cos x = sin 5 x + sin 3 x (3) y Diagram NOT accurately dr
June 2022 (R), Paper 2, Q6 (6 marks): V Diagram NOT accurately drawn B C A D Figure 1 Figure 1 shows a right pyramid VABCD with vertex V and square base ABCD. Each of the edges of the pyramid has th
June 2022 (R), Paper 2, Q7 (12 marks): (a) Solve the equation 3 cos (3x – 15)° = 2 for 0 x <180 (4) (b) Solve, giving your solutions to one decimal place where appropriate, 3 tan y° + 4 sin y° = 0 fo
June 2022, Paper 1, Q3 (7 marks): D C Diagram NOT 13 cm accurately drawn 15 cm E 35° A B 10 cm Figure 2 Figure 2 shows a rectangle ABCD with AB = 10 cm and AD = 15 cm. E is the point inside the
June 2022, Paper 2, Q4 (7 marks): A Diagram NOT C1 C2 accurately drawn O1 O2 R 6 cm 6 cm B Figure 1 Figure 1 shows two circles, C1 and C2 , each with a radius of 6 cm. The centre of C1 is O1 suc
June 2022, Paper 2, Q6 (8 marks): (a) Using a formula from page 2, show that cos 2 A = 1 – 2 sin2A (2) The finite region R is bounded by the curve with equation y = 3 + 2 sin x , the x‑axis, π t
June 2022, Paper 2, Q7 (11 marks): (i) (a) Using a formula from page 2, show that 2tan θ tan 2θ = 1 − tan θ 2 (2) Given that tan 2α = 1 (b) show that tan α = a ± b where a and b are integers whos
January 2022 (R), Paper 1, Q4 (8 marks): A D Diagram NOT accurately drawn R π rad 3 O r cm B Figure 1 Figure 1 shows sector OAB of a circle with centre O and radius r cm. π radians. The angle AOB = 3 T
January 2022 (R), Paper 1, Q10 (15 marks): (a) Use the factor theorem to show that (4x − 3) is a factor of 16x3 + 11x − 15 (2) (b) Using formulae given on page 2, show that (i) sin 2θ = 2 sin θ cos θ (ii
January 2022 (R), Paper 2, Q7 (8 marks): Given that 3 + sin θ 2 = 3 cos θ cos θ − 2 (a) show that cos θ = − 1 2 (4) (b) Hence solve the equation 3 + sin 3 x 2 = 3 cos 3x for 0° x < 180° cos 3 x − 2 (4)
January 2022 (R), Paper 2, Q9 (10 marks): Given that α is the acute angle such that tan α = 2 3 (a) find the exact value of cos α (1) E Diagram NOT accurately drawn 17 cm h cm D C 18 cm O A B 24 cm Figu
January 2022 (R), Paper 2, Q10 (12 marks): y Diagram NOT accurately drawn C2 A R1 B C1 R2 O π 3 π x 2 Figure 2 Figure 2 shows part of the curve C1 with equation y = sin x + 1 and part of the curve C2 wit
January 2022, Paper 1, Q6 (10 marks): In triangle ABC, AC = x cm, AB = (x + 3) cm and ∠ABC = 30° Given that ∠ACB = θ° where 0 < θ < 90 (a) show that (i) sin θ° = x + 3 2 x x 2 − x − 3 6 9 (ii) cos θ
January 2022, Paper 2, Q1 (4 marks): A Diagram NOT accurately drawn 7 cm 1.2 rad O B Figure 1 Figure 1 shows sector AOB of a circle with centre O and radius 7 cm. The angle AOB is 1.2 radians. Calc
January 2022, Paper 2, Q2 (5 marks): Solve the equation sin(2θ – 20)° – 3 cos(2θ – 20)° = 0 for 0 θ 180 (5)
January 2022, Paper 2, Q4 (9 marks): E Diagram NOT accurately drawn H F G 10 cm D C 10 cm A 10 cm B Figure 2 Figure 2 shows a solid cube ABCDEFGH with sides of length 10 cm. (a) Find, in cm, the ex
January 2022, Paper 2, Q8 (11 marks): (a) Given that x is measured in radians, complete the table of values for y = sin x – 3 cos 2x – 0.5 Give your answers to one decimal place. x 0 0.5 0.8 1 1.6 2
November 2021, Paper 1, Q3 (8 marks): A Diagram NOT accurately drawn r cm θ rad O B Figure 1 Figure 1 shows the sector AOB of a circle with centre O and radius r cm, where r is an integer. The size
November 2021, Paper 1, Q7 (9 marks): B Diagram NOT accurately drawn θ° 6 cm 8 cm A C k cm Figure 2 Figure 2 shows triangle ABC AB = 6 cm BC = 8 cm AC = k cm ∠ABC = θ° − k (a) Show that cos θ° = 100
November 2021, Paper 1, Q10 (15 marks): y Diagram NOT accurately drawn A C O M N x Figure 4 2 + sin 3x where 0 x < 2 π Figure 4 shows the curve C with equation y = 1 3 The curve C crosses the x‑axis a
November 2021, Paper 2, Q1 (5 marks): (a) Use the formula for cos(A + B) to show that cos 2A = 1 – 2 sin2 A (2) Given that sin A = x + 1 and cos 2A = 4 - y 2 3 (b) show that y = 1 2 (3x2 + 6x + 5) (
November 2021, Paper 2, Q9 (10 marks): Diagram NOT D accurately drawn 8 cm 8 cm C A 10 cm 10 cm B Figure 2 Figure 2 shows a triangular pyramid ABCD with base ABC AB = BC = 10 cm AD = CD = 8 cm ∠ABC =
June 2021, Paper 1, Q2 (6 marks): Angle α is acute such that cos α = 3 5 Angle β is obtuse such that sin β = 1 2 (a) Find the exact value of (i) tan α (ii) tan β (3) (b) Hence show that 3 tan(α
June 2021, Paper 1, Q11 (15 marks): (a) Using a formula from page 2, show that cos 2x = 1 – 2 sin2 x (3) y Diagram NOT accurately drawn y = sin x + 2 R2 A B R1 y = cos 2x + 2 C O x Figure 2 Figure
June 2021, Paper 2, Q3 (9 marks): Diagram NOT accurately drawn P Q 2 π rad 18 cm 18 cm 3 O Figure 1 Figure 1 shows a sector OPQ of a circle with centre O. The radius of the circle is 18 cm and t
June 2021, Paper 2, Q10 (10 marks): (a) Solve the equation tan x° = –3 for 0 x < 360 Give your solutions to the nearest whole number. (3) Given that 7 sin2 θ + sin θ cos θ = 6 (b) show that tan2θ
November 2020 (R), Paper 1, Q1 (7 marks): Here is a formula P = 3 + 2 sin 3 πt 0 t 12 8 (a) Find the exact value of P when t = 10 3 (2) (b) Find (i) the largest value of P (ii) the smallest value of P (
November 2020 (R), Paper 1, Q6 (7 marks): (a) Show that sin (A + B) + sin(A − B) = 2 sin A cos B (2) (b) Hence express 2 sin 7x cos x in the form sin mx + sin nx where m and n are integers, giving the v
November 2020 (R), Paper 1, Q10 (11 marks): (a) Solve the equation π 3 2 for 0 x 2π sin x + = 3 Give your solutions in terms of π , where appropriate. (3) (b) Solve the equation 3 sin θ + 5 cos θ = 0 for
November 2020 (R), Paper 1, Q12 (7 marks): y S C B Diagram NOT accurately drawn R O A x Figure 2 The region R, shown shaded in Figure 2, is bounded by the x-axis, the curve S with equation y = 2 sin x an
November 2020 (R), Paper 2, Q3 (8 marks): A Diagram NOT accurately drawn 10 cm 12 cm x cm B 6 cm D 2 cm C Figure 1 Figure 1 shows the triangle ABC in which AB = 10 cm and AC = 12 cm. The point D lies on
November 2020 (R), Paper 2, Q5 (8 marks): B Diagram NOT r cm accurately drawn 2 π radians O A 3 C Figure 2 In Figure 2, AB and AC are tangents to a circle with centre O and radius r cm. The points B and
November 2020, Paper 1, Q3 (9 marks): Diagram NOT E accurately drawn 14 cm B C 16 cm M A D 12 cm Figure 1 Figure 1 shows the right pyramid ABCDE. The base, ABCD, of the pyramid is a horizontal recta
November 2020, Paper 1, Q9 (12 marks): B Diagram NOT accurately drawn θ° ϕ° 12 cm 6 cm 6 cm A D C 2x cm Figure 3 Figure 3 shows triangle ABC with AB = 12 cm, BC = 6 cm and AC = 2x cm. The point D is
November 2020, Paper 2, Q10 (16 marks): (a) Show that cos (A + B) + cos (A – B) = 2 cos A cos B (2) (b) Hence show that P Q P Q + − P + Q = cos cos 2 cos cos 2 2 (3) π (c) Solve, for 0 θ , the equatio
January 2020 (R), Paper 2, Q1 (3 marks): Diagram NOT accurately drawn A 2 rad B 13 cm O Figure 1 Figure 1 shows the sector AOB of a circle with centre O. The radius of the circle is 13 cm and angle AOB
January 2020 (R), Paper 2, Q3 (5 marks): In triangle ABC, AB = 11 cm and BC = 12 cm. The area of triangle ABC = 33 cm2 Find, in cm to 3 significant figures, the two possible lengths of AC. (5)
January 2020 (R), Paper 2, Q4 (9 marks): Diagram NOT X accurately drawn h cm A 8 cm C 8 cm 8 cm B Figure 3 Figure 3 shows a triangular pyramid ABCX. The base ABC of the pyramid is an equilateral triang
January 2020 (R), Paper 2, Q8 (10 marks): (a) Write down the value of k such that sin 2A = k sin A cos A (1) g(A) = 2 + 3cos A − sin A − 3sin 2 A − 2cos2A Given that g(A) can be written in the form (p c
January 2020, Paper 1, Q4 (6 marks): Diagram NOT accurately drawn A O C 0.8 rad r cm B Figure 1 Figure 1 shows a sector AOB of a circle with centre O and radius r cm and a triangle BOC. The angle o
January 2020, Paper 1, Q8 (11 marks): (a) Solve, to the nearest integer, the equation sin (2x − 75)° = −0.515 for 0 x < 180 (3) (b) Giving your solutions to one decimal place, where appropriate, sol
January 2020, Paper 2, Q3 (9 marks): D Diagram NOT accurately drawn A B C M Figure 1 Figure 1 shows a triangular pyramid ABCD. The base, ABC, of the pyramid is a horizontal isosceles triangle with
January 2020, Paper 2, Q11 (13 marks): (a) Express the equation 3 sin(A – B) = sin(A + B) in the form tan A = k tan B , giving the value of the integer k. (4) (b) Given that θ ≠ (2 n + 1) π where n ∈
June 2019 (R), Paper 1, Q1 (3 marks): A Diagram NOT U cm accurately drawn O 1.5 rad 12 cm B Figure 1 Figure 1 shows sector AOB of a circle with centre O and radius U cm. The angle AOB is 1.5 radians
June 2019 (R), Paper 1, Q2 (6 marks): A Diagram NOT accurately drawn 3x cm 2x cm B C 4x cm Figure 2 Figure 2 shows triangle ABC in which AB = 2x cm AC = 3x cm BC = 4x cm (a) Show that sin ABC = 3 15
June 2019 (R), Paper 2, Q8 (13 marks): Diagram NOT accurately drawn E F 10 cm D A C N 60° 12 cm 8 cm B Figure 3 Figure 3 shows a right prism ABCDEF. The cross section BCF of the prism is a triangle.
June 2019 (R), Paper 2, Q10 (15 marks): (a) Use the formula for cos (A + B) to show that cos 2 A = 2 cos2௘A í (2) (b) Show that cos 4 A = 8 cos4௘A í FRV2௘A + 1 (4) í 1 θ π ⎡ θ π ⎤ ⎛ ⎞ ⎛ ⎞ 16 0 - ș 2ʌ
June 2019, Paper 1, Q3 (6 marks): In triangle ABC, AC = 7cm, BC = 10cm and angle BAC = 65° (a) Find, to the nearest 0.1°, the size of angle ABC. (3) (b) Find, in cm' to 3 significant figures, th
June 2019, Paper 1, Q4 (6 marks): A Diagram NOT accurately drawn O rad B D —15 cm Figure 1 Figure 1 shows a sector OAB of a circle where angle AOB = 0 radians. The circle has centre O and radius
June 2019, Paper 1, Q7 (12 marks): (a) Solve, in degrees to one decimal place, (3 cost + 5) (5 sin 0 - 3) = 0 for 0 ≤ 0 <180° (2) (b) Show that the equation 8 sin (x - a) = 3 sin (x + a) can be w
June 2019, Paper 2, Q4 (5 marks): In triangle ABC, AB = 5x cm, BC = (3x – 1) cm, AC = (2x + 5) cm and angle ABC = 60° Find, to 3 significant figures, the value of x. (5)
June 2019, Paper 2, Q11 (16 marks): V Diagram NOT accurately drawn B C A 16 cm D Figure 1 Figure 1 shows a right pyramid with vertex V and square base, ABCD, of side 16 cm. The size of angle AVC i

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