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4PM1/02 Jun 2021, Question 10: Solving Trig Equations

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

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The question, in short

Solve a tangent equation in degrees, transform a trigonometric identity into a quadratic in tan, then use it to solve a related equation over a full rotation.

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

Part (a) · 3 marks

Find all x in 0 to 360 satisfying tan x° = −3, rounded to whole degrees.

  1. 1tan⁻¹(−3) = −71.6°
  2. 2tan has period 180°: x = −71.6 + 180 = 108.4, x = −71.6 + 360 = 288.4
  3. 3x = 108, 288

Answerx = 108, 288

Part (b) · 3 marks

Starting from 7sin²θ + sinθcosθ = 6, show it rearranges to tan²θ + tanθ − 6 = 0.

  1. 17sin²θ + sin θ cos θ = 6(sin²θ + cos²θ)
  2. 2sin²θ + sin θ cos θ − 6cos²θ = 0
  3. 3Divide by cos²θ: tan²θ + tan θ − 6 = 0

Answertan²θ + tan θ − 6 = 0 shown

Part (c) · 4 marks

Use the quadratic from part b to solve 7sin²y° + siny°cosy° = 6 for 0 ≤ y < 360, to whole degrees.

  1. 1tan²y + tan y − 6 = 0 ⇒ (tan y + 3)(tan y − 2) = 0
  2. 2tan y = −3: y = 108, 288 (from part (a))
  3. 3tan y = 2: y = 63.4, 243.4
  4. 4y = 63, 108, 243, 288

Answery = 63, 108, 243, 288

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