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4PM1/02R Jan 2022, Question 9: Trigonometry in 3D: Pyramids and Cones

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 2R · Calculator · 10 marks

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The question, in short

Use tan α = 2/3 to find cos α, then work with a right pyramid (base 24 cm by 18 cm, slant edges 17 cm) to find height h, show tan θ° = 2/3, and find an angle between two planes.

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

Part (a) · 1 mark

Given tan α = 2/3 for acute α, find the exact value of cos α.

  1. 1Right-angled triangle with opposite 2, adjacent 3
  2. 2Hypotenuse = √(4 + 9) = √13
  3. 3cos α = 3/√13

Answercos α = 3/√13

Part (b) · 3 marks

For the pyramid with AB = 24 cm, BC = 18 cm and EC = 17 cm, find the height h.

  1. 1OC = ½ × √(24² + 18²) = 15 (= √(9² + 12²))
  2. 2h² = 17² − 15² = 64
  3. 3h = 8

Answerh = 8

Part (c) · 2 marks

Show that the angle θ between planes EBC and ABCD satisfies tan θ° = 2/3.

  1. 1Let M be the midpoint of BC; OM = 24/2 = 12
  2. 2tan θ = h/OM = 8/12 = 2/3

Answertan θ° = 2/3 shown

Part (d) · 4 marks

With P and Q midpoints of EB and EC, find the angle between planes OPQ and BCQP in degrees to one decimal place.

  1. 1Let N be the midpoint of PQ, which is the midpoint of EM
  2. 2EM = √(8² + 12²) = 4√13, so NM = 2√13
  3. 3NO = 2√13 (by the cosine rule, or as O is the right angle of triangle EOM)
  4. 4Triangle ONM is isosceles with base angles θ = 33.7°
  5. 5Angle ONM = 180° − 2 × 33.7° = 112.6°

Answer112.6°

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