De Moivre to derive trigonometric identities

A question is this type if and only if it asks to use De Moivre's theorem to prove an identity expressing cos(nθ) or sin(nθ) as a polynomial in cos θ and/or sin θ.

29 questions · Challenging +1.1

4.02q De Moivre's theorem: multiple angle formulae
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WJEC Further Unit 4 2023 June Q3
9 marks Standard +0.8
  1. Given that \(z = \cos\theta + i\sin\theta\), use de Moivre's theorem to show that $$z^n + \frac{1}{z^n} = 2\cos n\theta .$$ [3]
  2. Express \(32\cos^6\theta\) in the form \(a\cos 6\theta + b\cos 4\theta + c\cos 2\theta + d\), where \(a\), \(b\), \(c\), \(d\) are integers whose values are to be determined. [6]
WJEC Further Unit 4 Specimen Q9
14 marks Challenging +1.2
  1. Use mathematical induction to prove de Moivre's Theorem, namely that $$(\cos \theta + i \sin \theta)^n = \cos n\theta + i \sin n\theta,$$ where \(n\) is a positive integer. [7]
    1. Use this result to show that $$\sin 5\theta = a \sin^5 \theta - b \sin^3 \theta + c \sin \theta,$$ where \(a\), \(b\) and \(c\) are positive integers to be found.
    2. Hence determine the value of \(\lim_{\theta \to 0} \frac{\sin 5\theta}{\sin \theta}\). [7]
Pre-U Pre-U 9795/1 2011 June Q10
10 marks Challenging +1.2
  1. Use de Moivre's theorem to show that \(\cos 3\theta = 4\cos^3 \theta - 3\cos \theta\). [2]
  2. The sequence \(\{u_n\}\) is such that \(u_0 = 1\), \(u_1 = \cos \theta\) and, for \(n \geqslant 1\), $$u_{n+1} = (2\cos \theta)u_n - u_{n-1}.$$
    1. Determine \(u_2\) and \(u_3\) in terms of powers of \(\cos \theta\) only. [2]
    2. Suggest a simple expression for \(u_n\), the \(n\)th term of the sequence, and prove it for all positive integers \(n\) using induction. [6]
Pre-U Pre-U 9795/1 2013 November Q2
5 marks Standard +0.3
Use de Moivre's theorem to express \(\cos 3\theta\) in terms of powers of \(\cos \theta\) only, and deduce the identity \(\cos 6x \equiv \cos 2x(2\cos 4x - 1)\). [5]