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0580/41 May/Jun 2025, Question 17: The Sine Rule

Cambridge IGCSE Mathematics 0580 · Paper 41 (Extended) · Calculator · 10 marks

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The question, in short

Using the given side lengths and angles in quadrilateral ABCD with diagonal BD, work out side AD, an obtuse angle at D, and the total area.

Summarised in our own words. Use the official paper for the exact wording and the diagram.

Part (a) · 3 marks

Find the length of side AD from triangle ABD using AB = 9.6 cm, BD = 12.1 cm and angle ABD = 56°.

  1. 1AD² = 9.6² + 12.1² − 2 × 9.6 × 12.1 × cos 56°
  2. 2AD² = 108.65…
  3. 3AD = 10.42… ≈ 10.4 cm

Answer10.4 or 10.42…

Part (b) · 4 marks

In triangle BCD, use BC = 15.8 cm, BD = 12.1 cm and angle BCD = 47° to find the obtuse angle BDC.

  1. 1sin BDC/15.8 = sin 47°/12.1
  2. 2sin BDC = 15.8 × sin 47° ÷ 12.1 = 0.9549…
  3. 3sin⁻¹(0.9549…) = 72.74…°
  4. 4Obtuse angle BDC = 180 − 72.74… = 107.25… ≈ 107.3°

Answer107.2 to 107.3

Part (c) · 3 marks

Work out the combined area of triangles ABD and BCD to give the quadrilateral's area.

  1. 1Triangle ABD = 1/2 × 9.6 × 12.1 × sin 56° = 48.15…
  2. 2Angle DBC = 180 − 47 − 107.25… = 25.74…°
  3. 3Triangle BCD = 1/2 × 15.8 × 12.1 × sin 25.74…° = 41.52…
  4. 4Area = 48.15… + 41.52… = 89.67… cm²

Answer89.59 to 89.80…

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