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4024/22 Oct/Nov 2020, Question 9: The Sine Rule

Cambridge O Level Mathematics 4024 · Paper 22 (Calculator) · Calculator · 11 marks

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The question, in short

Triangle ABC has AC = 6.4 cm, BC = 9.5 cm and angle BAC = 79°. Find angle ABC and the triangle's area, then use ABC as the base of pyramid ABCD with BD = 9.8 cm, CD = 8.2 cm and right angles at A to find angle BDC and the elevation angle of D from C.

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Part (ai) · 3 marks

Work out the size of angle ABC using the given sides and angle BAC = 79°.

  1. 1sin ABC/6.4 = sin 79/9.5
  2. 2sin ABC = 6.4 × sin 79 ÷ 9.5 = 0.6613…
  3. 3Angle ABC = 41.39… ≈ 41.40°

Answer41.40 or 41.39...

Part (aii) · 3 marks

Work out the area of triangle ABC in cm².

  1. 1Angle ACB = 180 − 79 − 41.40 = 59.60°
  2. 2Area = 1/2 × 6.4 × 9.5 × sin 59.60°
  3. 3= 26.22… cm²

Answer26.22...

Part (bi) · 3 marks

In the pyramid, work out angle BDC using BD = 9.8 cm, CD = 8.2 cm and the base triangle.

  1. 1cos BDC = (9.8² + 8.2² − 9.5²)/(2 × 9.8 × 8.2)
  2. 2= (96.04 + 67.24 − 90.25)/160.72 = 73.03/160.72
  3. 3Angle BDC = 62.97… ≈ 63.0°

Answer63.0 or 62.97...

Part (bii) · 2 marks

Work out the angle of elevation of D as seen from C.

  1. 1DA is vertical, so triangle DAC has a right angle at A
  2. 2cos DCA = 6.4/8.2
  3. 3Angle of elevation = 38.69… ≈ 38.7°

Answer38.7 or 38.69...

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