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Trigonometry practice problems A level Maths, Topic 3

Written by STEM-Tutor's tutors · Updated October 2026 · 8 questions and 1 exam-style question

Radians, arc length and sector area, exact values, solving trigonometric equations, identities and the R cos(x − α) form. Try each question before you look: use the hint if you are stuck, then reveal the worked solution one step at a time.

On this page
  1. Key equations
  2. Warm-up questions
  3. Exam-style questions
  4. Exam question practice
  5. Common mistakes
  6. Keep going

Key equations for trigonometry

s = rθ, A = ½r2θarc length and sector area (radians)
sin2x + cos2x = 1Pythagorean identity
tan x = sin x / cos xquotient identity
a cos x + b sin x = R cos(x − α)R = √(a2 + b2), tan α = b/a

A formula booklet is provided in the exam. Answers within 2.5% are marked correct, and you can type fractions such as 1/6. What the booklet gives you, and what it leaves out, is in our A level Maths formula booklet guide.

Warm-up questions

Quick checks on the core ideas. If these feel easy, go straight to the exam-style questions.

Q1Warm-up
Convert 135° to radians, to 3 significant figures.
rad

Hint

Multiply by π/180.

Worked solution
  1. 135 × π/180 = 3π/4.
  2. = 2.36 rad.
Q2Warm-up
Find the length of an arc of a circle of radius 6 cm subtending an angle of 1.2 radians.
cm

Hint

s = rθ, with θ in radians.

Worked solution
  1. s = 6 × 1.2 = 7.2 cm.
Q3Warm-up
What is the exact value of sin 60°?

Hint

Use the equilateral triangle of side 2, cut in half.

Worked solution
  1. The height is √3, the hypotenuse 2.
  2. sin 60° = √3/2: C.

Exam-style questions

Questions at the level of the A level Maths exam. The ones marked Challenge combine more than one idea.

Q4Exam-style
Find the area of a sector of radius 5 cm and angle 0.8 radians.
cm²

Hint

A = ½r2θ.

Worked solution
  1. A = ½ × 25 × 0.8 = 10 cm².
Q5Exam-style
Solve sin x = 0.5 for 0° ≤ x ≤ 360°. Give the larger solution.
°

Hint

sin is positive in the first and second quadrants.

Worked solution
  1. Principal value 30°; the other is 180° − 30°.
  2. Larger solution: 150°.
Q6Exam-style
sin θ = 0.6 and θ is acute. Find tan θ.

Hint

Use sin2θ + cos2θ = 1 to find cos θ.

Worked solution
  1. cos θ = √(1 − 0.36) = 0.8.
  2. tan θ = 0.6/0.8 = 0.75.
Q7Exam-styleYear 2
Write 3 cos x + 4 sin x in the form R cos(x − α). Find R.

Hint

R = √(a2 + b2).

Worked solution
  1. R = √(9 + 16) = 5.
Q8Challenge
Solve 2cos2x − cos x − 1 = 0 for 0° < x < 360°. Give the smallest solution.
°

Hint

Treat it as a quadratic in cos x.

Worked solution
  1. (2cos x + 1)(cos x − 1) = 0, so cos x = −½ or cos x = 1.
  2. cos x = 1 gives x = 0° or 360°, both outside the range. cos x = −½ gives 120° and 240°.
  3. Smallest: 120°.

Exam question practice

One original exam-style question in the style of an A level paper. Answer it in full, with working, before you open the mark scheme: M marks are for method, A marks for accurate answers that follow from it, and B marks for correct statements on their own.

Exam-style question style · 9 points · about 9 minutes

f(x) = 3 cos x + 4 sin x.

  1. (a) Express f(x) in the form R cos(x − α), where R > 0 and 0° < α < 90°.
  2. (b) Write down the maximum value of f(x) and the smallest positive value of x where it occurs.
  3. (c) Solve f(x) = 2.5 for 0° ≤ x < 360°.

Write your full answer on paper first, then open the scoring guide and tick each point you earned.

Scoring guide and model answer

Part (a)

R = 5, tan α = 4/3, so α = 53.13°. f(x) = 5 cos(x − 53.13°).

Part (b)

Maximum 5, when x − 53.13° = 0, so x = 53.13°.

Part (c)

cos(x − 53.13°) = 0.5, so x − 53.13° = −60° or 60°. x = 113.13° or 353.13°.

Your mark: 0 of 9

Common mistakes in trigonometry

Keep going

Revise the ideas behind these questions:

Then try real exam questions: A level Maths past papers, with official scoring guidelines. · All A level Maths units