AQA GCSE Mathematics Paper 2 (Higher), June 2025: Question 10

3 marks · Easy difficulty · Short Answer

Use trigonometry to calculate the unknown side length x of a right-angled triangle given an adjacent side of 4 cm and an angle of 36°.

Practise this question

Question

A right-angled triangle labeled 'Not drawn accurately'. The horizontal base has length 4 cm, the vertical height is labeled x, and the angle between the base and the hypotenuse is 36 degrees. The text below asks: 'Use trigonometry to work out the value of x. You must show your working.' with an answer line 'x = _____ cm'.

Mark scheme

Show the mark scheme Mark scheme for question 10 showing two alternative methods. Alternative method 1: M1 for tan stated or used; M1dep for tan 36 = x/4 or 4 * tan 36 (or using tan 54 = 4/x); A1 for answer in range [2.9, 2.91] with M2 awarded. Alternative method 2: M1 for calculating hypotenuse as 4/cos 36 or 4/sin 54; M1dep for using Pythagoras or sine/cosine to find x; A1 for [2.9, 2.91]. Additional guidance specifies 0 marks for answer with no working or from a scale drawing.

How to answer it

Calculating a Side Using Trigonometry (SOH CAH TOA)

WHAT THIS QUESTION TESTS

This question assesses your ability to identify the correct trigonometric ratio (sine, cosine, or tangent) for a given right-angled triangle, set up an algebraic equation relating an angle and two side lengths, and rearrange the equation to find an unknown side. It also assesses precision and the requirement to show complete mathematical working.

Question 10 • 3 Marks

Finding Missing Side x

Right-angled triangle with angle 36° and adjacent side 4 cm

📐 Step-by-Step Calculation

  1. 1 Label the sides:
    • Opposite to 36° = x
    • Adjacent to 36° = 4 cm
    • Hypotenuse = longest side (not involved)
  2. 2 Choose the correct ratio:
    Since we know the Adjacent and need the Opposite, use TOA:
    tan(θ) = Opposite ÷ Adjacent
  3. 3 Form the equation:
    tan(36°) = x / 4
  4. 4 Rearrange and solve:
    Multiply both sides by 4:
    x = 4 × tan(36°)
    x = 4 × 0.72654... = 2.90617... cm
  5. 5 Round appropriately:
    x = 2.91 cm (3 s.f.) or 2.9 cm (1 d.p.)

✅ Correct Answer & Mark Breakdown

Final Answer: 2.91 cm (any value in range [2.9, 2.91] , or 3 if full working is shown).

Mark Scheme Breakdown:
  • M1: Stating or using tan (e.g. circling "TOA" from SOH CAH TOA, or calculating tan(36°) ≈ 0.726 ).
  • M1 (dep): Correct equation or rearranged form: tan(36) = x / 4 or 4 × tan(36) .
  • A1: Accurate answer in the range [2.9, 2.91] with both method marks awarded.

💡 Key Knowledge: SOH CAH TOA

  • SOH: sin(θ) = Opposite / Hypotenuse
  • CAH: cos(θ) = Adjacent / Hypotenuse
  • TOA: tan(θ) = Opposite / Adjacent
  • Opposite: Side directly across from the reference angle.
  • Adjacent: Side between the reference angle and the 90° right angle.
  • Hypotenuse: The longest side, directly opposite the 90° right angle.

❌ Common Errors to Avoid

  • No working shown (0/3 marks): The question states "You must show your working". Writing just 2.9 without working gets M0 M0 A0 .
  • Wrong calculator angle mode: Ensure your calculator is set to DEG (Degrees), not RAD (Radians) or GRA (Gradians). tan(36 rad) gives -7.50 .
  • Inverting the ratio: Incorrectly writing tan(36) = 4 / x , giving x = 4 / tan(36) = 5.51 cm .
  • Measuring by ruler: The diagram is labelled "Not drawn accurately"; scale drawings get 0 marks.

🧠 Exam Technique & Examiner Insight

  • Alternative Valid Method: You could find the missing angle first ( 90° - 36° = 54° ). From 54°, the adjacent side is x and the opposite is 4 cm , giving tan(54°) = 4 / x , so x = 4 / tan(54°) = 2.91 cm . This earns full marks but requires more steps.
  • Sense-Check Your Answer: In this triangle, 36° is smaller than 45°. Since the angle is less than 45°, the opposite side ( x ) must be shorter than the adjacent side ( 4 cm ). An answer of 2.91 cm makes physical sense, whereas 5.51 cm would indicate an inverted fraction!

Topics

Geometry and measures · 3.4.2 Mensuration and calculation

Question and mark scheme from the AQA GCSE Mathematics examination, Paper 2 (Higher), June 2025. QuestionVault is an independent revision resource; questions remain the copyright of the awarding body.