Lemmara logoLemmaraWorked solution

4PM1/01 Nov 2023, Question 10: Solving Trig Equations

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 1 · Calculator · 17 marks

اقرأ هذا الحل مشروحاً بالعربي ←

The question, in short

Use given formulae to prove double-angle identities, simplify a tangent expression to sin 2θ, solve a trigonometric equation in radians, and evaluate a definite integral using calculus.

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

Part (a) · 3 marks

Using the formulae on page 2, show the two double-angle identities for sin 2A and cos 2A.

  1. 1(i) sin(A + A) = sin A cos A + cos A sin A
  2. 2sin 2A = 2sin A cos A
  3. 3(ii) cos(A + A) = cos A cos A − sin A sin A = cos²A − sin²A
  4. 4= cos²A − (1 − cos²A)
  5. 5= 2cos²A − 1

Answer(i) sin 2A = 2sin A cos A shown; (ii) cos 2A = 2cos²A − 1 shown

Part (b) · 4 marks

Show that f(θ) = 2tan θ/(1 + tan²θ) simplifies to sin 2θ.

  1. 1f(θ) = (2sin θ/cos θ)/(1 + sin²θ/cos²θ)
  2. 2Multiply top and bottom by cos²θ: 2sin θ cos θ/(cos²θ + sin²θ)
  3. 3= 2sin θ cos θ
  4. 4= sin 2θ

Answerf(θ) = sin 2θ shown

Part (c) · 6 marks

Solve 5tan(x + π/6) = [1 + tan²(x + π/6)][1 − 2cos²(x + π/6)] for −π/2 ≤ x ≤ π/2, radians to 3 significant figures.

  1. 1Divide by 1 + tan²(x + π/6): 5tan(x + π/6)/(1 + tan²(x + π/6)) = 1 − 2cos²(x + π/6)
  2. 2Left side = (5/2)sin(2x + π/3) by part (b); right side = −cos(2x + π/3)
  3. 3(5/2)sin(2x + π/3) = −cos(2x + π/3), so tan(2x + π/3) = −2/5
  4. 42x + π/3 = −0.38050… or 2.76108…
  5. 5x = −0.714 or x = 0.857

Answerx = −0.714, x = 0.857

Part (d) · 4 marks

Find the exact value of the integral from 0 to π/2 of 4tan θ/(1 + tan²θ) − cos 5θ + 2 using calculus.

  1. 14tan θ/(1 + tan²θ) = 2sin 2θ
  2. 2I = ∫ from 0 to π/2 of (2sin 2θ − cos 5θ + 2) dθ
  3. 3= [−cos 2θ − (sin 5θ)/5 + 2θ] from 0 to π/2
  4. 4= (1 − 1/5 + π) − (−1)
  5. 5= 9/5 + π

Answer9/5 + π

Stuck on a step? Read the free lesson: Solving Trig Equations

Practise more solving trig equations questions

Real 4PM1 past-paper questions, marked on your working like the exam, with a worked solution for every part.

Practise free on Lemmara

More past-paper questions on solving trig equations

Also searched as: 4PM1 1 November 2023 · 4PM1/01 · 4pm1 nov23 1 · question 10.
Lemmara is an independent study tool, not affiliated with or endorsed by Pearson Edexcel. The worked solutions are Lemmara’s own.