A Level Maths: Pure 3 chapter 3: Trigonometry MCQs with answers

By Team Megalecture · Updated 4 Nov 2026

30 exam-style questions on Trigonometry, each with the answer and a short explanation. Ten of them are below to try. The chapter test draws ten fresh questions each time and is marked the moment you finish, with your score by topic.

30 questions · 1 topics · 10 per test

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What chapter 3 covers

Ten Trigonometry questions with answers

Try each one, then open the answer. These ten stay the same; the test draws new ones every time.

1. Find the exact value of cot(π/6).

  1. A
    √3
  2. B
    1/√3
  3. C
    2
  4. D
    √3/2
Show answer

Answer: A. tan(π/6) = 1/√3, so cot(π/6) = 1/tan(π/6) = √3. The value 2 is cosec(π/6).

2. Find the greatest value of 10/(7 + 3 sin x − 4 cos x).

  1. A
    5/6
  2. B
    10/7
  3. C
    2
  4. D
    5
Show answer

Answer: D. 3 sin x − 4 cos x = 5 sin(x − α), which varies between −5 and 5, so the denominator varies between 2 and 12. The fraction is greatest when the denominator is least: 10/2 = 5. The value 5/6 is the least value.

3. Which statement describes the graph of y = sec x for 0 ≤ x ≤ 2π?

  1. A
    asymptotes at x = 0, x = π and x = 2π; no y-values strictly between −1 and 1
  2. B
    asymptotes at x = π/2 and x = 3π/2; no y-values strictly between −1 and 1
  3. C
    asymptotes at x = π/2 and x = 3π/2; y-values only between −1 and 1
  4. D
    asymptotes at x = π/2 and x = 3π/2; y-values only greater than or equal to 1
Show answer

Answer: B. sec x = 1/cos x is undefined where cos x = 0, at π/2 and 3π/2. Since |cos x| ≤ 1, |sec x| ≥ 1, so the graph has branches with y ≥ 1 and branches with y ≤ −1. Asymptotes at 0, π and 2π belong to cosec x.

4. Solve the equation sin 2θ = cos θ for 0° ≤ θ ≤ 180°.

  1. A
    θ = 30°, 90° or 150°
  2. B
    θ = 30° or 150° only
  3. C
    θ = 90° only
  4. D
    θ = 60°, 90° or 120°
Show answer

Answer: A. 2 sin θ cos θ − cos θ = 0, so cos θ(2 sin θ − 1) = 0. cos θ = 0 gives 90°, and sin θ = ½ gives 30° and 150°. Dividing both sides by cos θ loses the solution 90°.

5. By writing sin 3x as sin(2x + x), express sin 3x in terms of sin x only.

  1. A
    3 sin x − 4 sin3 x
  2. B
    4 sin3 x − 3 sin x
  3. C
    2 sin x − 3 sin3 x
  4. D
    3 sin x − 3 sin3 x
Show answer

Answer: A. sin 3x = sin 2x cos x + cos 2x sin x = 2 sin x cos2 x + (1 − 2 sin2 x) sin x. Replacing cos2 x by 1 − sin2 x gives 2 sin x − 2 sin3 x + sin x − 2 sin3 x = 3 sin x − 4 sin3 x.

6. Solve the equation sec2 θ = 3 tan θ − 1 for 0° ≤ θ ≤ 180°.

  1. A
    θ = 135° or 116.6°
  2. B
    θ = 0°, 71.6° or 180°
  3. C
    θ = 29.3° or 105.7°
  4. D
    θ = 45° or 63.4°
Show answer

Answer: D. Replace sec2 θ by 1 + tan2 θ: tan2 θ − 3 tan θ + 2 = 0, so (tan θ − 1)(tan θ − 2) = 0. tan θ = 1 gives 45° and tan θ = 2 gives 63.4°; the next solutions, 225° and 243.4°, are outside the range.

7. Which expression is identical to (1 − cos 2θ)/sin 2θ?

  1. A
    cot θ
  2. B
    2 tan θ
  3. C
    ½ tan θ
  4. D
    tan θ
Show answer

Answer: D. 1 − cos 2θ = 2 sin2 θ and sin 2θ = 2 sin θ cos θ, so the fraction is sin θ/cos θ = tan θ. Using 1 − cos 2θ = 2 cos2 θ by mistake gives cot θ.

8. Find the exact value of sin 75°.

  1. A
    (√6 + √2)/4
  2. B
    (√6 − √2)/4
  3. C
    (√2 + 1)/2
  4. D
    (√3 + 1)/4
Show answer

Answer: A. sin(45° + 30°) = sin 45° cos 30° + cos 45° sin 30° = (√2/2)(√3/2) + (√2/2)(½) = (√6 + √2)/4. sin(A + B) is not sin A + sin B, which would give (√2 + 1)/2.

9. Simplify cos(x + 60°) + cos(x − 60°).

  1. A
    cos x
  2. B
    2 cos x
  3. C
    cos 2x
  4. D
    −√3 sin x
Show answer

Answer: A. Expanding both, the sin x sin 60° terms cancel, leaving 2 cos x cos 60° = 2 cos x × ½ = cos x. Forgetting that cos 60° = ½ gives 2 cos x.

10. Solve the equation sec x = 3 for 0 < x < 2π, giving answers in radians correct to 3 significant figures.

  1. A
    x = 1.23 or x = 5.05
  2. B
    x = 1.23 or x = 1.91
  3. C
    x = 1.23 only
  4. D
    x = 0.340 or x = 2.80
Show answer

Answer: A. cos x = 1/3, so x = cos−1(1/3) = 1.23 or x = 2π − 1.23 = 5.05, as cosine is positive in the first and fourth quadrants. The values 0.340 and 2.80 solve cosec x = 3 instead.

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Ways to practise this chapter

Trigonometry: questions students ask

How many Trigonometry MCQs are there?

30, each with the answer and a short explanation, across 1 topics: Trigonometry. Every chapter test draws ten of them, so you can take it many times.

Is the A Level Maths: Pure 3 chapter 3 test free?

Yes. The test, the answers and the explanations are free and need no sign-up. Sign in only if you want your scores kept across devices.

How is the chapter test marked?

The moment you press Finish: your score, the right answer and an explanation for every question, and the ones you got wrong saved in My mistakes to retry.

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