Question Simulator

IGCSE Additional Mathematics (0606)

Trigonometry Banker

25 questions · about 10% of the marks we indexed · examined in 9 of 9 sessions.

Questions by year

2026

  • 2026 May/June · Paper 12 · Q1 (a, b, c, d)5 marksTrigonometric graphs
  • 2026 May/June · Paper 12 · Q73 marksIdentities and proofs
  • 2026 May/June · Paper 22 · Q106 marksTrigonometric equations

2025

  • 2025 Oct/Nov · Paper 12 · Q134 marksIdentities and proofs
  • 2025 Oct/Nov · Paper 22 · Q11 (a, b)10 marksTrigonometric equations
  • 2025 May/June · Paper 12 · Q4 (b)3 marksIdentities and proofs
  • 2025 May/June · Paper 22 · Q9 (a, b)10 marksTrigonometric equations

2024

  • 2024 Oct/Nov · Paper 12 · Q1 (a, b)7 marksTrigonometric graphs
  • 2024 Oct/Nov · Paper 12 · Q5 (a)1 markIdentities and proofs
  • 2024 Oct/Nov · Paper 22 · Q86 marksTrigonometric equations
  • 2024 May/June · Paper 12 · Q13 marksTrigonometric graphs
  • 2024 May/June · Paper 22 · Q4 (b(iii))2 marksTrigonometric equations
  • 2024 May/June · Paper 22 · Q6 (a, b)8 marksIdentities and proofs · Trigonometric equations

2023

  • 2023 Oct/Nov · Paper 12 · Q2 (a, b, c)5 marksTrigonometric graphs
  • 2023 Oct/Nov · Paper 12 · Q55 marksTrigonometric equations
  • 2023 Oct/Nov · Paper 22 · Q10 (a, b)10 marksIdentities and proofs · Trigonometric equations
  • 2023 May/June · Paper 12 · Q13 marksTrigonometric graphs
  • 2023 May/June · Paper 12 · Q6 (a, b)7 marksIdentities and proofs · Trigonometric equations

2022

  • 2022 Oct/Nov · Paper 12 · Q13 marksTrigonometric graphs
  • 2022 Oct/Nov · Paper 12 · Q45 marksTrigonometric equations
  • 2022 Oct/Nov · Paper 22 · Q7 (a, b)8 marksIdentities and proofs · Trigonometric equations
  • 2022 Oct/Nov · Paper 22 · Q9 (a, b, c)11 marksSine and cosine rules (exact values)
  • 2022 May/June · Paper 12 · Q13 marksTrigonometric graphs
  • 2022 May/June · Paper 12 · Q24 marksIdentities and proofs
  • 2022 May/June · Paper 22 · Q8 (b)5 marksTrigonometric equations

References only, not question text. Open Official paper ↗ where Cambridge publishes the paper, or see the official Cambridge past-papers page; for other papers, ask your school. “Paper 42” means Paper 4, variant 2.

Practice questions

Original questions written by HomiLabs in the style of the exam, with answers and mark schemes. They are not Cambridge past paper questions and have not been reviewed by Cambridge. Try each one before opening the answer.

Q1Trigonometric equations · Core · Paper 1 style · no calculator · 3 marks
  1. (a) Solve the equation 2 cos(x − 30°) = 1 for 0° ≤ x ≤ 360°. [3]
Show answer
  1. (a) x = 90°, 330°
    cos(x − 30°) = ½ M1; x − 30° = 60°, 300° (range −30° to 330°) M1; x = 90°, 330° A1
Q2Trigonometric equations · Core · Paper 2 style · calculator · 3 marks
  1. (a) Solve tan 2x = 3 for 0° ≤ x ≤ 180°. Give your answers correct to 1 decimal place. [3]
Show answer
  1. (a) x = 35.8°, 125.8°
    2x = tan⁻¹3 = 71.57° M1; 2x = 71.57°, 251.57° M1; x = 35.8°, 125.8° A1
Q3Trigonometric equations · Core · Paper 2 style · calculator · 3 marks
  1. (a) Solve 5 sin y = 2 for 0 ≤ y ≤ 2π radians. Give your answers correct to 3 significant figures. [3]
Show answer
  1. (a) y = 0.412, 2.73
    sin y = 0.4 B1; y = 0.4115 M1; y = π − 0.4115 = 2.7301; y = 0.412, 2.73 A1
Q4Trigonometric equations · Standard · Paper 2 style · calculator · 6 marks
  1. (a) Show that the equation 3 sec²θ − 5 tan θ − 5 = 0 can be written as 3 tan²θ − 5 tan θ − 2 = 0. [2]
  2. (b) Hence solve 3 sec²θ − 5 tan θ − 5 = 0 for 0° ≤ θ ≤ 360°. Give your answers correct to 1 decimal place. [4]
Show answer
  1. (a) Shown
    sec²θ = 1 + tan²θ M1; 3 + 3tan²θ − 5tanθ − 5 = 0 → 3tan²θ − 5tanθ − 2 = 0 A1
  2. (b) θ = 63.4°, 161.6°, 243.4°, 341.6°
    (3tanθ + 1)(tanθ − 2) = 0 M1; tanθ = 2 → 63.4°, 243.4° A1; tanθ = −⅓ → 161.6°, 341.6° M1 A1
Q5Trigonometric equations · Standard · Paper 1 style · no calculator · 5 marks
  1. (a) Do not use a calculator in this question.Solve the equation 2 cos²x + 3 sin x = 3 for 0 ≤ x ≤ 2π radians, giving your answers as exact multiples of π. [5]
Show answer
  1. (a) x = π/6, π/2, 5π/6
    Use cos²x = 1 − sin²x M1; 2sin²x − 3sin x + 1 = 0 A1; (2sin x − 1)(sin x − 1) = 0 M1; sin x = ½ → π/6, 5π/6 A1; sin x = 1 → π/2 A1
Q6Trigonometric equations · Standard · Paper 2 style · calculator · 5 marks
  1. (a) Solve cot(2y + π/4) = 0.5 for −π ≤ y ≤ π radians. Give your answers correct to 3 significant figures. [5]
Show answer
  1. (a) y = −2.98, −1.41, 0.161, 1.73
    tan(2y + π/4) = 2 M1; 2y + π/4 = 1.1071 + nπ M1; range for 2y + π/4 is −7π/4 to 9π/4 M1; y = ((1.1071 + nπ) − π/4)/2 for n = −2, −1, 0, 1 A2 (A1 for any two correct)
Q7Trigonometric equations · Standard · Paper 2 style · calculator · 6 marks
  1. (a) Solve 4 cosec²x + cot x − 7 = 0 for 0° < x < 360°. Give your answers correct to 1 decimal place where necessary. [6]
Show answer
  1. (a) x = 53.1°, 135°, 233.1°, 315°
    cosec²x = 1 + cot²x M1; 4cot²x + cot x − 3 = 0 A1; (4cot x − 3)(cot x + 1) = 0 M1; cot x = ¾ → tan x = 4/3 → 53.1°, 233.1° A1 A1; cot x = −1 → 135°, 315° A1
Q8Trigonometric equations · Standard · Paper 2 style · calculator · 4 marks
  1. (a) Solve tan x + 3 cot x = 4 for 0° < x < 180°. [4]
Show answer
  1. (a) x = 45°, 71.6°
    Multiply by tan x: tan²x − 4tan x + 3 = 0 M1 A1; (tan x − 1)(tan x − 3) = 0 → x = 45° A1, x = 71.6° A1
Q9Trigonometric equations · Standard · Paper 2 style · calculator · 4 marks
  1. (a) Solve 3 sin x cos x = 2 cos x for 0° ≤ x ≤ 360°. Give your answers correct to 1 decimal place where necessary. [4]
Show answer
  1. (a) x = 41.8°, 90°, 138.2°, 270°
    cos x(3 sin x − 2) = 0 M1 (do not divide by cos x); cos x = 0 → 90°, 270° A1; sin x = ⅔ → 41.8° A1, 138.2° A1
Q10Trigonometric equations · Standard · Paper 2 style · calculator · 4 marks
  1. (a) Solve cosec(x + 1) = −3 for 0 ≤ x ≤ 2π, where x is in radians. Give your answers correct to 3 significant figures. [4]
Show answer
  1. (a) x = 2.48, 4.94
    sin(x + 1) = −⅓ M1; x + 1 lies between 1 and 2π + 1; x + 1 = π + 0.3398 = 3.4814 or 2π − 0.3398 = 5.9434 M1 A1; x = 2.48, 4.94 A1
Q11Trigonometric equations · Challenge · Paper 2 style · calculator · 5 marks
  1. (a) Solve sec(3x − 20°) = −2.5 for 0° ≤ x ≤ 180°. Give your answers correct to 1 decimal place. [5]
Show answer
  1. (a) x = 44.5°, 88.8°, 164.5°
    cos(3x − 20°) = −0.4 M1; 3x − 20° = 113.58°, 246.42°, 473.58° (range −20° to 520°) M1 A1; x = values ÷ 3 after adding 20° M1; all three correct, no extras A1
Q12Trigonometric equations · Challenge · Paper 2 style · calculator · 6 marks
  1. (a) Show that sin²A/(1 − cos A) ≡ 1 + cos A. [2]
  2. (b) Hence solve 5 sin²(2x) = 3(1 − cos 2x) for 0° < x < 180°. Give your answers correct to 1 decimal place. [4]
Show answer
  1. (a) Shown
    sin²A = 1 − cos²A = (1 − cos A)(1 + cos A) M1; cancel (1 − cos A) A1
  2. (b) x = 56.8°, 123.2°
    sin²2x/(1 − cos 2x) = 3/5 so 1 + cos 2x = 0.6 M1; cos 2x = −0.4 A1; 2x = 113.58°, 246.42° M1; x = 56.8°, 123.2° A1 (cos 2x = 1 gives x = 0° or 180°, outside the range)
Q13Trigonometric equations · Challenge · Paper 1 style · no calculator · 6 marks
  1. (a) Do not use a calculator in this question.Factorise 2s³ − s² − 2s + 1 completely. [2]
  2. (b) Hence solve 2 sin³x − sin²x − 2 sin x + 1 = 0 for 0° ≤ x ≤ 360°. [4]
Show answer
  1. (a) (2s − 1)(s − 1)(s + 1)
    s²(2s − 1) − (2s − 1) M1; (2s − 1)(s² − 1) = (2s − 1)(s − 1)(s + 1) A1
  2. (b) x = 30°, 90°, 150°, 270°
    sin x = ½ → 30°, 150° B1 B1; sin x = 1 → 90° B1; sin x = −1 → 270° B1
Q14Trigonometric equations · Challenge · Paper 2 style · calculator · 6 marks
  1. (a) Solve 3 cos²y − 2 = sin y for 0 ≤ y ≤ 2π radians. Give your answers correct to 3 significant figures. [6]
Show answer
  1. (a) y = 0.449, 2.69, 4.02, 5.41
    3(1 − sin²y) − 2 − sin y = 0 M1; 3sin²y + sin y − 1 = 0 A1; sin y = (−1 ± √13)/6 = 0.4343 or −0.7676 M1; y = 0.449, 2.69 A1; y = 4.02, 5.41 A1 A1
Q15Trigonometric graphs · Core · Paper 1 style · no calculator · 4 marks
  1. (a) For the curve y = 3 sin 2x + 1, state the amplitude. [1]
  2. (b) State the period of y = 3 sin 2x + 1 in degrees. [1]
  3. (c) State the greatest and least values of y. [2]
Show answer
  1. (a) 3
    B1
  2. (b) 180°
    360° ÷ 2 B1
  3. (c) greatest 4, least −2
    3 + 1 = 4 B1; −3 + 1 = −2 B1
Q16Trigonometric graphs · Core · Paper 1 style · no calculator · 4 marks
  1. (a) State the amplitude and the period, in degrees, of y = 6 cos(x/3). [2]
  2. (b) State the period of y = tan 4x in degrees. [1]
  3. (c) The curve y = 2 sin x + k has least value −5. Find k. [1]
Show answer
  1. (a) amplitude 6, period 1080°
    B1 B1
  2. (b) 45°
    180° ÷ 4 B1
  3. (c) k = −3
    −2 + k = −5 B1
Q17Trigonometric graphs · Standard · Paper 1 style · no calculator · 7 marks
  1. (a) The curve y = 2 cos 3x − 1 is drawn for 0° ≤ x ≤ 180°. State the period of the curve. [1]
  2. (b) State the coordinates of the maximum and minimum points of the curve in this interval. [3]
  3. (c) Find the x-coordinates of the points where the curve crosses the x-axis. [3]
Show answer
  1. (a) 120°
    360° ÷ 3 B1
  2. (b) maxima (0°, 1) and (120°, 1); minima (60°, −3) and (180°, −3)
    B1 maxima, B1 minima values, B1 all four x-coordinates
  3. (c) x = 20°, 100°, 140°
    cos 3x = ½ M1; 3x = 60°, 300°, 420° M1; x = 20°, 100°, 140° A1
Q18Trigonometric graphs · Standard · Paper 1 style · no calculator · 3 marks
  1. (a) The curve y = a sin bx + c, where a, b and c are positive constants, has a maximum point at (30°, 7). The next minimum point to the right is at (90°, −1). Find a, b and c. [3]
Show answer
  1. (a) a = 4, b = 3, c = 3
    a = (7 − (−1))/2 = 4 B1; c = (7 + (−1))/2 = 3 B1; half period 60° so period 120°, b = 3 B1 (check: 3 × 30° = 90° gives sin = 1)
Q19Trigonometric graphs · Standard · Paper 2 style · calculator · 6 marks
  1. (a) For the curve y = 4 tan(x/2), where x is in radians and −π < x < 3π, state the period and the equations of the asymptotes. [3]
  2. (b) Find the value of y when x = 2π/3, giving your answer correct to 3 significant figures. [1]
  3. (c) Find the values of x in the interval for which y = −4. [2]
Show answer
  1. (a) period 2π; asymptotes x = π (and only that one in this interval)
    period π ÷ ½ = 2π B1; asymptotes where x/2 = π/2 + nπ, i.e. x = π + 2nπ M1; in −π < x < 3π only x = π A1
  2. (b) 6.93
    4 tan(π/3) = 4√3 = 6.93 B1
  3. (c) x = −π/2, 3π/2
    tan(x/2) = −1 M1; x/2 = −π/4, 3π/4 → x = −π/2, 3π/2 A1
Q20Trigonometric graphs · Challenge · Paper 1 style · no calculator · 5 marks
  1. (a) The curve y = 5 cos 2x − 2 is drawn for 0 ≤ x ≤ π. State the coordinates of its end points and of its minimum point. [2]
  2. (b) Find the set of values of k for which the equation 5 cos 2x − 2 = k has exactly two roots for 0 ≤ x ≤ π. [3]
Show answer
  1. (a) (0, 3), (π, 3); minimum (π/2, −7)
    B1 end points; B1 minimum
  2. (b) −7 < k ≤ 3
    The line y = k cuts the curve twice when it lies between the minimum and the maximum M1; k = −7 gives one root (x = π/2) A1; k = 3 gives x = 0 and x = π, two roots, so −7 < k ≤ 3 A1
Q21Trigonometric graphs · Challenge · Paper 1 style · no calculator · 7 marks
  1. (a) Describe the curve y = |2 sin x − 1| for 0° ≤ x ≤ 360°: state the points where it meets the x-axis, the y-intercept, and the greatest value of y and where it occurs. [4]
  2. (b) Hence state the number of solutions of |2 sin x − 1| = 1 for 0° ≤ x ≤ 360°, and find them. [3]
Show answer
  1. (a) meets x-axis at (30°, 0) and (150°, 0); y-intercept (0°, 1); greatest value 3 at x = 270°
    B1 zeros (2 sin x = 1); B1 y-intercept |−1| = 1; B1 greatest value 3; B1 at 270° (sin x = −1)
  2. (b) 4 solutions: 0°, 90°, 180°, 360°
    2 sin x − 1 = 1 → sin x = 1 → 90° B1; 2 sin x − 1 = −1 → sin x = 0 → 0°, 180°, 360° B1; 4 solutions B1
Q22Identities and proofs · Core · Paper 1 style · no calculator · 3 marks
  1. (a) Prove that (1 + cos x)(cosec x − cot x) ≡ sin x. [3]
Show answer
  1. (a) Proof
    cosec x − cot x = (1 − cos x)/sin x M1; (1 + cos x)(1 − cos x)/sin x = (1 − cos²x)/sin x M1; = sin²x/sin x = sin x A1
Q23Identities and proofs · Core · Paper 1 style · no calculator · 3 marks
  1. (a) Prove that tan x + cot x ≡ sec x cosec x. [3]
Show answer
  1. (a) Proof
    sin x/cos x + cos x/sin x M1; = (sin²x + cos²x)/(sin x cos x) M1; = 1/(sin x cos x) = sec x cosec x A1
Q24Identities and proofs · Standard · Paper 2 style · calculator · 5 marks
  1. (a) Show that (sec x − cos x)/tan x ≡ sin x. [3]
  2. (b) Hence solve (sec x − cos x)/tan x = −0.6 for 0° < x < 360°. Give your answers correct to 1 decimal place. [2]
Show answer
  1. (a) Shown
    sec x − cos x = (1 − cos²x)/cos x = sin²x/cos x M1 A1; ÷ (sin x/cos x) gives sin x A1
  2. (b) x = 216.9°, 323.1°
    sin x = −0.6 M1; 216.9°, 323.1° A1
Q25Identities and proofs · Standard · Paper 2 style · calculator · 5 marks
  1. (a) Show that cosec²θ/cot θ − cot θ ≡ tan θ. [3]
  2. (b) Hence solve cosec²θ/cot θ − cot θ = 3 for 0° < θ < 360°. Give your answers correct to 1 decimal place. [2]
Show answer
  1. (a) Shown
    cosec²θ = 1 + cot²θ M1; (1 + cot²θ)/cot θ − cot θ = 1/cot θ M1; = tan θ A1
  2. (b) θ = 71.6°, 251.6°
    tan θ = 3 M1; 71.6°, 251.6° A1
Q26Identities and proofs · Challenge · Paper 2 style · calculator · 7 marks
  1. (a) Prove that (1 − sin x)/(1 + sin x) ≡ (sec x − tan x)². [4]
  2. (b) Hence, or otherwise, solve (sec x − tan x)² = 1/9 for 0° < x < 180°, x ≠ 90°. Give your answers correct to 1 decimal place. [3]
Show answer
  1. (a) Proof
    (sec x − tan x)² = (1 − sin x)²/cos²x M1 A1; cos²x = 1 − sin²x = (1 − sin x)(1 + sin x) M1; cancel (1 − sin x) to get (1 − sin x)/(1 + sin x) A1
  2. (b) x = 53.1°, 126.9°
    (1 − sin x)/(1 + sin x) = 1/9 M1; 9 − 9 sin x = 1 + sin x → sin x = 0.8 A1; x = 53.1°, 126.9° A1
Q27Identities and proofs · Challenge · Paper 1 style · no calculator · 6 marks
  1. (a) Show that sin⁴x − cos⁴x ≡ 1 − 2cos²x. [3]
  2. (b) Do not use a calculator in this part.Hence solve sin⁴x − cos⁴x = ½ for 0 ≤ x ≤ π radians. [3]
Show answer
  1. (a) Shown
    (sin²x − cos²x)(sin²x + cos²x) M1; = sin²x − cos²x A1; = 1 − cos²x − cos²x = 1 − 2cos²x A1
  2. (b) x = π/3, 2π/3
    1 − 2cos²x = ½ → cos²x = ¼ M1; cos x = ±½ M1; x = π/3, 2π/3 A1
Q28Sine and cosine rules (exact values) · Core · Paper 1 style · no calculator · 5 marks

Do not use a calculator in this question.In triangle ABC, AB = 4 cm, AC = 6 cm and angle BAC = 60°.

  1. (a) Find the exact length of BC, giving your answer in the form a√b where a and b are integers. [3]
  2. (b) Find the exact area of triangle ABC. [2]
Show answer
  1. (a) BC = 2√7 cm
    BC² = 16 + 36 − 2(4)(6)cos 60° M1; = 28 A1; BC = √28 = 2√7 A1
  2. (b) 6√3 cm²
    ½ × 4 × 6 × sin 60° M1; = 12 × √3/2 = 6√3 A1
Q29Sine and cosine rules (exact values) · Standard · Paper 1 style · no calculator · 6 marks

Do not use a calculator in this question.In triangle PQR, angle PQR = 45°, angle QPR = 30° and QR = 6 cm.You may use sin 105° = (√6 + √2)/4.

  1. (a) Find the exact length of PR. [3]
  2. (b) Find the exact area of triangle PQR, giving your answer in the form a + b√3. [3]
Show answer
  1. (a) PR = 6√2 cm
    PR/sin 45° = 6/sin 30° M1; PR = 6 × (√2/2) ÷ ½ M1; = 6√2 A1
  2. (b) 9 + 9√3 cm²
    angle PRQ = 105° B1; ½ × 6 × 6√2 × (√6 + √2)/4 M1; = (9√2/2)(√6 + √2) = 9√3 + 9 A1
Q30Sine and cosine rules (exact values) · Challenge · Paper 1 style · no calculator · 7 marks

Do not use a calculator in this question.A triangular plot of land in Murree has sides XY = 5 m, YZ = 7 m and XZ = 8 m.

  1. (a) Show that angle YXZ = 60°. [3]
  2. (b) Find the exact area of the plot. [2]
  3. (c) Find the exact value of sin XZY. [2]
Show answer
  1. (a) Shown
    cos YXZ = (5² + 8² − 7²)/(2 × 5 × 8) M1; = 40/80 = ½ A1; so angle = 60° A1
  2. (b) 10√3 m²
    ½ × 5 × 8 × sin 60° M1; = 10√3 A1
  3. (c) sin XZY = 5√3/14
    sin XZY/5 = sin 60°/7 M1; sin XZY = 5√3/14 A1

Similar questions

Questions that test the same subtopic in different years, grouped so you can work on one skill at a time.

References only: find each question in its paper (official link where available, otherwise through your school).

Trigonometric equations Banker12 questions · 9/9 sessions
Trigonometric graphs 7 questions · 7/9 sessions
Identities and proofs Banker9 questions · 9/9 sessions

Subtopics

Our analysis of how often each subtopic was examined and how many marks it carried, 2022 to May/June 2026.

SubtopicMarks indexedSessions
Trigonometric equations Banker689/9
Trigonometric graphs 297/9
Identities and proofs Banker299/9
Sine and cosine rules (exact values) Rare111/9

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