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4PM1/02 Jun 2024, Question 9: The Sine Rule

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 2 · Calculator · 15 marks

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The question, in short

Right triangular prism with AB = 24 cm, angles 60° and 45°; use a given formula to prove a sine value, find BF exactly, then EF and an angle with the base.

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

Part (a) · 3 marks

Show that sin AFB equals (√2 + √6) ÷ 4 using a formula from page 2.

  1. 1∠AFB = 180° − 60° − 45° = 75°
  2. 2sin 75° = sin(45° + 30°) = sin 45° cos 30° + cos 45° sin 30°
  3. 3= (√2/2)(√3/2) + (√2/2)(1/2)
  4. 4= (√6 + √2)/4

Answersin AFB = (√2 + √6)/4 shown

Part (b) · 5 marks

Without a calculator, show that BF = 12(3√2 − √6) cm.

  1. 1Sine rule: 24/sin 75° = BF/sin 60°
  2. 2BF = 24 × (√3/2) ÷ ((√6 + √2)/4) = 48√3/(√6 + √2)
  3. 3Multiply top and bottom by (√6 − √2)
  4. 4BF = 48√3(√6 − √2)/(6 − 2) = 12(√18 − √6)
  5. 5= 12(3√2 − √6)

AnswerBF = 12(3√2 − √6) cm shown

Part (c) · 3 marks

Given the angle between planes AEB and ABCD is 65°, find EF in cm to 2 significant figures.

  1. 1Let X be the foot of the perpendicular from F to AB
  2. 2FX = BF sin 45° = 36 − 12√3 = 15.215…
  3. 3The angle between plane AEB and ABCD is the angle at X, so tan 65° = FX/EF
  4. 4EF = 15.215…/tan 65° = 7.095… ≈ 7.1

AnswerEF = 7.1 cm

Part (d) · 4 marks

Find the angle between line CF and plane ABCD, in degrees to one decimal place.

  1. 1The required angle is FCX (F is vertically above X)
  2. 2FC = √(BF² + EF²) = √(21.517…² + 7.095…²) = 22.657…
  3. 3sin FCX = FX/FC = 15.215…/22.657…
  4. 4∠FCX = 42.2°

Answer42.2°

Stuck on a step? Read the free lesson: The Sine Rule

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