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4PM1/01 Jun 2021, Question 2: Trig Ratios of Any Angle

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 1 · Calculator · 6 marks

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The question, in short

Given acute α with cos α = 3/5 and obtuse β with sin β = 1/2, find exact tangents and use them to show a form for tan(α + β), finding integers m and n.

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

Part (a) · 3 marks

Find the exact value of tan α and of tan β using the given cosine and sine.

  1. 1α acute with cos α = 3/5: sides 3, 4, 5, so sin α = 4/5
  2. 2tan α = (4/5)/(3/5) = 4/3
  3. 3β obtuse with sin β = 1/2: cos β = −√3/2
  4. 4tan β = (1/2)/(−√3/2) = −1/√3 = −√3/3

Answertan α = 4/3, tan β = −1/√3 (= −√3/3)

Part (b) · 3 marks

Using your tangents, show tan(α + β) equals (m√3 − n)/(n√3 + m) and find positive integers m and n.

  1. 1tan(α + β) = (tan α + tan β)/(1 − tan α tan β)
  2. 2= (4/3 − 1/√3)/(1 + (4/3)(1/√3))
  3. 3Multiply top and bottom by 3√3: = (4√3 − 3)/(3√3 + 4)
  4. 4So m = 4, n = 3

Answertan(α + β) = (4√3 − 3)/(3√3 + 4), so m = 4 and n = 3

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