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4PM1/01 Jan 2020, Question 8: Solving Trig Equations

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 1 · Calculator · 11 marks

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The question, in short

Solve three trigonometry equations: one sine equation to the nearest integer, one mixed tan-sine equation to one decimal place, and explain why a cos²-sin² equation has no solutions.

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

Part (a) · 3 marks

Find x to the nearest integer satisfying sin(2x − 75)° = −0.515 for 0 ≤ x < 180.

  1. 1sin⁻¹(−0.515) = −31.0°
  2. 22x − 75 = −31 or 2x − 75 = 180 + 31 = 211
  3. 3x = 44/2 = 22 or x = 286/2 = 143

Answerx = 22, 143

Part (b) · 4 marks

Solve 2 tan y° + 5 sin y° = 0 for 0 ≤ y ≤ 180, giving answers to one decimal place where needed.

  1. 12 sin y/cos y + 5 sin y = 0
  2. 2sin y (2/cos y + 5) = 0
  3. 3sin y = 0 gives y = 0, 180
  4. 4cos y = −2/5 gives y = 113.6

Answery = 0, 113.6, 180

Part (c) · 4 marks

Give a mathematical reason why 3 cos²θ° − 3 sin²θ° + sin θ° + 12 = 0 has no solutions.

  1. 1cos²θ = 1 − sin²θ: 3(1 − sin²θ) − 3 sin²θ + sin θ + 12 = 0
  2. 26 sin²θ − sin θ − 15 = 0
  3. 3(2 sin θ + 3)(3 sin θ − 5) = 0
  4. 4sin θ = −3/2 or sin θ = 5/3
  5. 5−1 ≤ sin θ ≤ 1, so neither value is possible: no values of θ

Answersin θ = −3/2 or 5/3; both lie outside −1 ≤ sin θ ≤ 1, so there are no solutions

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