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4024/23 May/Jun 2026, Question 19: The Sine Rule

Cambridge O Level Mathematics 4024 · Paper 23 (Calculator) · Calculator · 8 marks

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The question, in short

Quadrilateral ABCD with diagonal AC = 14 cm and angles BAC = 67°, CAD = 30°, ADC = 110°; find side AD, verify AB, and find BC.

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Part (a) · 3 marks

Use triangle ACD with AC = 14 cm and angles CAD = 30°, ADC = 110° to calculate the length AD.

  1. 1Angle ACD = 180 − 110 − 30 = 40°
  2. 2AD/sin 40 = 14/sin 110
  3. 3AD = 14 sin 40 ÷ sin 110 = 9.576… ≈ 9.58 cm

Answer9.58 or 9.576…

Part (b) · 2 marks

Given triangle ABC has area 50.9 cm², with AC = 14 cm and angle BAC = 67°, show AB = 7.9 cm to 1 decimal place.

  1. 1½ × 14 × AB × sin 67 = 50.9
  2. 2AB = 50.9 ÷ (7 sin 67)
  3. 3AB = 7.899… = 7.9 cm to 1 d.p.

Answer½ × 14 × AB × sin 67 = 50.9 leading to AB = 7.89 to 7.90

Part (c) · 3 marks

Using triangle ABC with AC = 14 cm, AB = 7.9 cm and angle BAC = 67°, calculate the length BC.

  1. 1BC² = 14² + 7.9² − 2 × 14 × 7.9 × cos 67
  2. 2BC² = 196 + 62.41 − 86.43… = 171.98…
  3. 3BC = 13.1 cm

Answer13.1 or 13.11 to 13.12

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