AQA GCSE Mathematics Paper 2 (Foundation), June 2025: Question 26

3 marks · Medium difficulty · Short Answer

Use right-angled trigonometry to find the length of the opposite side in a 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 side is labeled x, and the angle between the horizontal base and the hypotenuse is 36 degrees. The question instructs to use trigonometry to work out the value of x, showing all working.

Mark scheme

Show the mark scheme Mark scheme for Question 26 detailing two methods: Method 1 awards M1 for stating or using tan, M1dep for writing tan(36) = x/4 or 4 × tan(36), and A1 for an answer in the range [2.9, 2.91]. Method 2 calculates the hypotenuse first using sine or cosine before finding x. Additional guidance notes that [2.9, 2.91] with no working scores 0 marks.

How to answer it

Trigonometry: Finding an Unknown Side Using Tan

📋 What this question tests

This question assesses your ability to apply right-angled trigonometry (SOH CAH TOA). Specifically, you need to correctly identify triangle side labels relative to a given angle, choose the correct trigonometric ratio (tangent), set up an algebraic equation, and solve for an unknown length while clearly presenting all method steps.

Question 26 (3 Marks)

Calculating Side Length x

Right-angled triangle with given base = 4 cm and angle = 36°

💡 Key Knowledge (SOH CAH TOA)

  • Hypotenuse (H): The longest side, directly opposite the 90° right angle.
  • Opposite (O): The side directly opposite the given angle (here, side x is opposite 36°).
  • Adjacent (A): The side next to the given angle between the angle and the right angle (here, 4 cm ).
  • Since we have O and A, select TOA:
    tan(θ) = Opposite / Adjacent

🧠 Exam Technique & Mark Breakdown

  • M1: State or use the tangent ratio (e.g. writing tan or circling TOA in SOHCAHTOA).
  • M1 (dep): Set up the correct equation and rearrange:
    tan(36°) = x / 4 → x = 4 × tan(36°) .
  • A1: Provide a final calculated answer in the accepted range [2.9, 2.91] cm (or rounded to 3 , provided method marks M2 were earned).

📐 Step-by-Step Calculation

  1. Label the sides relative to 36°:
    Opposite = x
    Adjacent = 4 cm
  2. Select the correct trigonometric formula:
    tan(θ) = Opposite / Adjacent
  3. Substitute the given values:
    tan(36°) = x / 4
    (Note: tan(36°) ≈ 0.7265...)
  4. Rearrange to solve for x:
    x = 4 × tan(36°)
  5. Evaluate using a calculator:
    x = 4 × 0.72654... = 2.90617... cm
  6. Round appropriately:
    x = 2.91 cm (to 2 d.p.) or 2.9 cm (to 1 d.p.)

✅ Acceptable Final Answers

Any value in the range:

2.9 cm to 2.91 cm

3 is also accepted only if full method marks (M2) are clearly shown.

❌ Common Traps & Mistakes

  • Zero working shown: The prompt explicitly states "You must show your working." Writing 2.9 with no working scores 0 / 3 marks.
  • Calculator in Radian (R) or Gradient (G) mode: Ensure your calculator is set to Degree (D) mode. In radian mode, 4 × tan(36) ≈ -1.56 , which loses accuracy marks.
  • Inverting the division: Rearranging incorrectly to x = 4 / tan(36°) gives 5.51 , scoring 0 marks for rearrangement.
  • Using ruler measurement: The diagram says "Not drawn accurately". Scale drawings earn 0 / 3 marks.
Examiner Insight: Many students write the correct final value directly from their calculator without writing down tan 36 = x / 4 or 4 × tan 36 . In questions containing the instruction "You must show your working", unsupported answers automatically receive M0 M0 A0. Always write down your equation before punching it into your calculator!

Topics

Geometry and measures · 3.4.2 Mensuration and calculation

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