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0580/42 Feb/Mar 2022, Question 4: The Cosine Rule

Cambridge IGCSE Mathematics 0580 · Paper 42 (Extended) · Calculator · 12 marks

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The question, in short

Two non-right-angled triangle problems: use the sine and cosine rules and ½ab sin C to find angles, a missing side, an area and a perpendicular height.

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

Part (ai) · 3 marks

Triangle with sides 45, 29.5, 35.3 cm and a 51.6° angle at A: find the angle at B.

  1. 1sin ABC/29.5 = sin 51.6°/35.3
  2. 2sin ABC = 29.5 × sin 51.6° ÷ 35.3 = 0.6549…
  3. 3Angle ABC = 40.91… ≈ 40.9°

Answer40.9 or 40.91…

Part (aii) · 2 marks

Using the same triangle, work out its area.

  1. 1Area = 1/2 × 29.5 × 45 × sin 51.6°
  2. 2= 520.17… ≈ 520 cm²

Answer520 or 520.0 to 520.2…

Part (bi) · 4 marks

Quadrilateral made of isosceles triangle PQS (32 cm, 32 cm, 56°) and triangle QRS (QR 47 cm, 60° at Q): find SR.

  1. 1SQ = 2 × 32 × sin 28° = 30.04… cm
  2. 2SR² = 47² + 30.04…² − 2 × 47 × 30.04… × cos 60°
  3. 3SR² = 1699.6…
  4. 4SR = 41.22… ≈ 41.2 cm

Answer41.2 or 41.21 to 41.23

Part (bii) · 3 marks

Find the perpendicular distance from P down to the diagonal SQ.

  1. 1Base angle PQS = (180 − 56) ÷ 2 = 62°
  2. 2The shortest distance is perpendicular to SQ
  3. 3Distance = 32 × sin 62°
  4. 4= 28.25… ≈ 28.3 cm

Answer28.3 or 28.25 to 28.29…

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