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0580/41 May/Jun 2024, Question 6: Bearings with Sine and Cosine Rules

Cambridge IGCSE Mathematics 0580 · Paper 41 (Extended) · Calculator · 17 marks

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The question, in short

Using the given diagram with AB = 11 km, AH = 14 km, CH = 32 km, angle BCH = 25° and angle BHC = 55°, work out bearings, a length, a travel time and a closest-approach distance.

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

Part (a) · 1 mark

State the bearing of H as measured from C.

  1. 1Bearing of B from C = 270°
  2. 2H is 25° anticlockwise from CB
  3. 3Bearing = 270 − 25 = 245°

Answer245

Part (bi) · 1 mark

Prove that the angle at B in triangle BCH equals 100°.

  1. 1Angles in a triangle add to 180°
  2. 2Angle CBH = 180 − (55 + 25) = 100°

Answer180 − (55 + 25) = 100

Part (bii) · 3 marks

Prove that the length BH rounds to 13.7 km.

  1. 1Sine rule: BH/sin 25 = 32/sin 100
  2. 2BH = 32 × sin 25 / sin 100
  3. 3= 13.73… = 13.7 km (1 d.p.)

Answer32 × sin 25 / sin 100 = 13.73…

Part (c) · 5 marks

Work out the bearing of A as seen from B.

  1. 1cos ABH = (11² + 13.73…² − 14²)/(2 × 11 × 13.73…)
  2. 2= 0.3759…
  3. 3Angle ABH = 67.9°
  4. 4Bearing of H from B = 90 + 100 = 190°
  5. 5Bearing of A from B = 190 + 67.9 = 257.9 ≈ 258°

Answer258 or 257.9 to 258.0…

Part (di) · 4 marks

Boat C travels 32 km straight to H at 10 knots; find the arrival time to the nearest minute using 1 knot = 1.852 km/h.

  1. 1Speed = 10 × 1.852 = 18.52 km/h
  2. 2Time = 32 ÷ 18.52 = 1.7278… h
  3. 30.7278… × 60 = 43.67… min ≈ 1 h 44 min
  4. 41 pm + 1 h 44 min = 2 44 pm

Answer2:44 pm or 14:44

Part (dii) · 3 marks

Find how far C is from H at the moment C is nearest to lighthouse B.

  1. 1Shortest distance from B is the perpendicular from B to CH, meeting CH at N
  2. 2NH = BH × cos 55
  3. 3= 13.73… × cos 55 = 7.876… km

Answer7.857 to 7.88

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