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4024/22 May/Jun 2023, Question 9: The Cosine Rule

Cambridge O Level Mathematics 4024 · Paper 22 (Calculator) · Calculator · 12 marks

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The question, in short

Bearings and trigonometry: B and C from A on 072° and 205°, 170 m and 95 m away; find BC, a back bearing, a distance due south and an angle of elevation.

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

Part (a) · 4 marks

Find the angle at A, then BC by the cosine rule.

  1. 1Angle BAC = 205 − 72 = 133°
  2. 2BC² = 170² + 95² − 2 × 170 × 95 × cos 133°
  3. 3BC² = 37 925 + 22 028.5… = 59 953.5…
  4. 4BC = 244.8… ≈ 245 m

Answer245 or 244.8 to 244.9

Part (b) · 2 marks

Bearing of A from C.

  1. 1A from C is the opposite direction to C from A.
  2. 2Back bearing: 205 − 180 = 25
  3. 3Bearing of A from C = 025°

Answer025

Part (ci) · 2 marks

D due south of A and east of C: show AD is 86.1 m.

  1. 1Angle CAD = 205 − 180 = 25°
  2. 2AD = 95 × cos 25°
  3. 3AD = 86.099… = 86.1 m (1 d.p.)

Answer86.099… or 86.10

Part (cii) · 4 marks

Mast at A seen at 7° from B: find its height, then the elevation from D.

  1. 1AX = 170 × tan 7° = 20.87…
  2. 2tan(elevation from D) = 20.87…/86.1
  3. 3Angle = tan⁻¹(0.2424…) = 13.62… ≈ 13.6°

Answer13.6 or 13.62 to 13.63

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