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0580/43 Oct/Nov 2022, Question 8: The Cosine Rule

Cambridge IGCSE Mathematics 0580 · Paper 43 (Extended) · Calculator · 14 marks

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The question, in short

Use triangle sides 9.5, 7.7 and 10 cm in a circle to work out angles, radius and area ratio.

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

Part (a) · 4 marks

Show the angle at B equals 70.2° using the three given side lengths.

  1. 110² = 9.5² + 7.7² − 2 × 9.5 × 7.7 × cos B
  2. 2cos B = (90.25 + 59.29 − 100)/146.3 = 49.54/146.3 = 0.3386…
  3. 3B = 70.207…° = 70.2° (1 d.p.)

Answer70.206 to 70.207 or 70.21 to 70.22

Part (bi) · 1 mark

Find the angle at the centre subtended by chord AC.

  1. 1Angle at centre = 2 × angle at circumference
  2. 2AOC = 2 × 70.2 = 140.4°

Answer140.4

Part (bii) · 1 mark

Find the base angle of the isosceles triangle formed with radii.

  1. 1Triangle AOC is isosceles (OA = OC)
  2. 2ACO = (180 − 140.4) ÷ 2 = 19.8°

Answer19.8

Part (biii) · 1 mark

Find the angle between chord AC and the tangent at C.

  1. 1Tangent is perpendicular to radius, so angle OCD = 90°
  2. 2ACD = 90 − 19.8 = 70.2°

Answer70.2

Part (c) · 3 marks

Calculate the circle radius from chord AC and its centre angle.

  1. 1Half of AC = 5 cm, cos 19.8 = 5/r
  2. 2r = 5 ÷ cos 19.8
  3. 3= 5.314… ≈ 5.31 cm

Answer5.31 or 5.314 to 5.315

Part (d) · 4 marks

Express triangle area as a percentage of circle area.

  1. 1Area ABC = 0.5 × 9.5 × 7.7 × sin 70.2 = 34.41… cm²
  2. 2Area of circle = π × 5.314…² = 88.7… cm²
  3. 334.41… ÷ 88.7… × 100 = 38.79… ≈ 38.8%

Answer38.8 or 38.9 or 38.78 to 38.85

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Also searched as: 0580/43/O/N/22 · 0580_w22_qp_43 · 0580 w22 43 · question 8.
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