AQA GCSE Mathematics Paper 3 (Higher), June 2025: Question 17

2 marks · Medium difficulty · Short Answer

Use the sine rule to calculate the length of an unknown side in a triangle.

Practise this question

Question

A triangle with an angle of 119 degrees at the top vertex and an angle of 31 degrees at the bottom-left vertex. The side opposite the 31-degree angle has a length of 6.3 cm, and the base side opposite the 119-degree angle is labeled x. The prompt states 'Use the sine rule to work out x.' followed by workspace and an answer line for x in cm.

Mark scheme

Show the mark scheme Mark scheme table for question 17: M1 awarded for x / sin(119) = 6.3 / sin(31) or x = (6.3 * sin(119)) / sin(31). A1 awarded for an answer in the range [10.698, 10.7], allowing 11 if the correct equation is seen.

How to answer it

Calculating a Missing Side Using the Sine Rule

📌 What this question tests

This question assesses your ability to apply the Sine Rule to find an unknown side in a non-right-angled triangle. You need to identify matching "opposite" pairs of angles and sides, substitute the values accurately into the formula, rearrange the equation, and correctly evaluate the trigonometric expression using a calculator.

Question 17 Breakdown

Higher Tier • Geometry & Trigonometry • 2 Marks

💡 Key Knowledge: The Sine Rule

  • Formula: a / sin A = b / sin B
  • Use this rule when you know an angle-side opposite pair and want to find another missing side or angle.
  • Top Tip: When finding an unknown side, always write the formula with sides on top:
    x / sin(opposite angle) = known side / sin(opposite angle)

🧠 Exam Technique

  • Identify Opposite Pairs: Draw arrows across the triangle:
    • Side x is opposite angle 119°
    • Side 6.3 cm is opposite angle 31°
  • Check Calculator Mode: Always confirm a small letter D (degrees) is shown at the top of your screen, not R (radians).
  • Show Full Working: Always write the rearranged fraction before typing it into your calculator to secure the method mark ( M1 ).

📐 Step-by-Step Calculation

  1. Set up the equation: Pair side x with 119° and side 6.3 with 31° .
    x / sin(119°) = 6.3 / sin(31°)
  2. Rearrange to isolate x: Multiply both sides by sin(119°) .
    x = (6.3 × sin(119°)) / sin(31°)
    (Alternative intermediate value: 6.3 / sin(31°) ≈ 12.2316...)
  3. Calculate the final value:
    x = 10.6983... cm
  4. Round appropriately:
    Give your answer as 10.7 cm (or any value in the acceptable range 10.698 to 10.7 ).

✅ Mark Scheme Breakdown

  • M1 (Method Mark): A fully correct equation or rearranged form:
    x / sin 119 = 6.3 / sin 31
    or x = (6.3 × sin 119) / sin 31
  • A1 (Accuracy Mark): Correct numerical answer in the range [10.698, 10.7] .
    Note: 11 is accepted only if the correct equation is clearly seen in the working.
Examiner Note: If a student writes sin x = (6.3 × sin 119) / sin 31 but recovers to give 10.7 on the answer line, full marks ( M1A1 ) are awarded.

❌ Common Errors to Avoid

  • Mismatched Pairs: Pairing x with 31° or 6.3 with 119° . Always trace straight across the triangle!
  • Inverted Ratios: Mixing up fractions, e.g. x / sin 119 = sin 31 / 6.3 (one inverted, one upright). Both sides must have length on top and sine on bottom.
  • Premature Rounding: Rounding sin(31°) to 0.5 too early, which leads to 11.02 and loses the A1 mark.
  • Calculator in Radians: Forgetting to set the calculator to degrees gives an answer of 13.7 , scoring 0 marks for calculation.

Topics

Geometry and measures · 3.4.2 Mensuration and calculation

Question and mark scheme from the AQA GCSE Mathematics examination, Paper 3 (Higher), June 2025. QuestionVault is an independent revision resource; questions remain the copyright of the awarding body.