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4024/23 May/Jun 2025, Question 19: The Cosine Rule

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

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The question, in short

Using triangle ABD with sides AB = 12 cm, BD = 7 cm and AD = 6 cm, find angle BAD, then use angle DAC = 32° and angle BCA = 48° to find AC.

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Part (a) · 4 marks

Calculate angle BAD from the three given side lengths of triangle ABD.

  1. 17² = 12² + 6² − 2 × 12 × 6 × cos BAD
  2. 2cos BAD = (12² + 6² − 7²)/(2 × 12 × 6) = 131/144
  3. 3Angle BAD = 24.53… ≈ 24.5°

Answer24.5

Part (b) · 4 marks

Calculate the length AC using the given angles DAC = 32° and BCA = 48°.

  1. 1Angle BAC = 24.53… + 32 = 56.53…
  2. 2Angle ABC = 180 − 56.53… − 48 = 75.46…
  3. 3AC/sin 75.46… = 12/sin 48°
  4. 4AC = 12 × sin 75.46… ÷ sin 48° = 15.63… ≈ 15.6 cm

Answer15.6

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