4.02q De Moivre's theorem: multiple angle formulae

195 questions

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CAIE FP1 2018 November Q7
10 marks Challenging +1.2
  1. Use de Moivre's theorem to show that $$\sin 8\theta = 8\sin \theta \cos \theta(1 - 10\sin^2 \theta + 24\sin^4 \theta - 16\sin^6 \theta).$$ [6]
  2. Use the equation \(\frac{\sin 8\theta}{\sin 2\theta} = 0\) to find the roots of $$16x^6 - 24x^4 + 10x^2 - 1 = 0$$ in the form \(\sin k\pi\), where \(k\) is rational. [4]
CAIE FP1 2018 November Q8
10 marks Challenging +1.3
  1. By considering the binomial expansion of \(\left(z + \frac{1}{z}\right)^6\), where \(z = \cos \theta + \mathrm{i} \sin \theta\), express \(\cos^6 \theta\) in the form $$\frac{1}{32}(p + q \cos 2\theta + r \cos 4\theta + s \cos 6\theta),$$ where \(p, q, r\) and \(s\) are integers to be determined. [6]
  2. Hence find the exact value of $$\int_{-\frac{1}{4}\pi}^{\frac{1}{4}\pi} \cos^6\left(\frac{1}{2}x\right) \mathrm{d}x.$$ [4]
CAIE FP1 2019 November Q9
11 marks Challenging +1.8
  1. Use de Moivre's theorem to show that $$\sec 6\theta = \frac{\sec^6 \theta}{32 - 48 \sec^2 \theta + 18 \sec^4 \theta - \sec^6 \theta}.$$ [6]
  2. Hence obtain the roots of the equation $$3x^6 - 36x^4 + 96x^2 - 64 = 0$$ in the form \(\sec q\pi\), where \(q\) is rational. [5]
CAIE FP1 2019 November Q9
11 marks Challenging +1.8
  1. Use de Moivre's theorem to show that $$\sec 6\theta = \frac{\sec^6 \theta}{32 - 48\sec^2 \theta + 18\sec^4 \theta - \sec^6 \theta}.$$ [6]
  2. Hence obtain the roots of the equation $$3t^6 - 36t^4 + 96t^2 - 64 = 0$$ in the form \(\sec q\pi\), where \(q\) is rational. [5]
CAIE Further Paper 2 2020 June Q8
15 marks Challenging +1.8
  1. Use de Moivre's theorem to show that \(\sin^6 \theta = -\frac{1}{32}(\cos 6\theta - 6\cos 4\theta + 15\cos 2\theta - 10)\). [6]
It is given that \(\cos^6 \theta = \frac{1}{32}(\cos 6\theta + 6\cos 4\theta + 15\cos 2\theta + 10)\).
  1. Find the exact value of \(\int_0^{\frac{1}{4}\pi}\left(\cos^6\left(\frac{1}{4}x\right) + \sin^6\left(\frac{1}{4}x\right)\right)dx\). [4]
  2. Express each root of the equation \(16c^6 + 16\left(1-c^2\right)^3 - 13 = 0\) in the form \(\cos k\pi\), where \(k\) is a rational number. [5]
CAIE Further Paper 2 2021 November Q4
10 marks Challenging +1.8
  1. Write down all the roots of the equation \(x^5 - 1 = 0\). [2]
  2. Use de Moivre's theorem to show that \(\cos 4\theta = 8\cos^4 \theta - 8\cos^2 \theta + 1\). [4]
  3. Use the results of parts (a) and (b) to express each real root of the equation $$8x^9 - 8x^7 + x^5 - 8x^4 + 8x^2 - 1 = 0$$ in the form \(\cos k\pi\), where \(k\) is a rational number. [4]
CAIE Further Paper 2 2023 November Q2
5 marks Standard +0.3
Find the roots of the equation \((z + 5i)^3 = 4 + 4\sqrt{3}i\), giving your answers in the form \(r\cos\theta + ir\sin\theta - 5)\), where \(r > 0\) and \(0 \leq \theta < 2\pi\). [5]
CAIE Further Paper 2 2024 November Q4
10 marks Challenging +1.8
  1. Use de Moivre's theorem to show that $$\cot 6\theta = \frac{\cot^4 \theta - 15\cot^4 \theta + 15\cot^2 \theta - 1}{6\cot^5 \theta - 20\cot^3 \theta + 6\cot \theta}.$$ [6]
  2. Hence obtain the roots of the equation $$x^6 - 6x^5 - 15x^4 + 20x^3 + 15x^2 - 6x - 1 = 0$$ in the form \(\cot(q\pi)\), where \(q\) is a rational number. [4]
Edexcel FP2 Q2
6 marks Standard +0.3
Solve the equation $$z^2 = 4\sqrt{2} - 4\sqrt{2}i,$$ giving your answers in the form \(r(\cos \theta + i \sin \theta)\), where \(-\pi < \theta \leq \pi\). [6]
Edexcel FP2 Q4
10 marks Standard +0.3
\(z = -8 + (8\sqrt{3})i\)
  1. Find the modulus of \(z\) and the argument of \(z\). [3]
Using de Moivre's theorem,
  1. find \(z^3\). [2]
  2. find the values of \(w\) such that \(w^4 = z\), giving your answers in the form \(a + ib\), where \(a, b \in \mathbb{R}\). [5]
Edexcel FP2 Q7
11 marks Standard +0.8
  1. Use de Moivre's theorem to show that $$\sin 5\theta = 16 \sin^5 \theta - 20 \sin^3 \theta + 5 \sin \theta$$ [5]
Hence, given also that \(\sin 3\theta = 3 \sin \theta - 4 \sin^3 \theta\),
  1. find all the solutions of $$\sin 5\theta = 5 \sin 3\theta$$ in the interval \(0 \leq \theta < 2\pi\). Give your answers to 3 decimal places. [6]
Edexcel FP2 Q3
8 marks Moderate -0.3
  1. Express the complex number \(-2 + (2\sqrt{3})i\) in the form \(r(\cos \theta + i \sin \theta)\), \(-\pi < \theta \leq \pi\) [3]
  2. Solve the equation $$z^3 = -2 + (2\sqrt{3})i$$ giving the roots in the form \(r(\cos \theta + i \sin \theta)\), \(-\pi < \theta \leq \pi\). [5]
Edexcel FP2 Q6
11 marks Standard +0.8
The complex number \(z = e^{i\theta}\), where \(\theta\) is real.
  1. Use de Moivre's theorem to show that $$z^n + \frac{1}{z^n} = 2\cos n\theta$$ where \(n\) is a positive integer. [2]
  2. Show that $$\cos^n \theta = \frac{1}{16}(\cos 5\theta + 5\cos 3\theta + 10\cos \theta)$$ [5]
  3. Hence find all the solutions of $$\cos 5\theta + 5\cos 3\theta + 12\cos \theta = 0$$ in the interval \(0 \leq \theta < 2\pi\). [4]
Edexcel FP2 Q4
7 marks Standard +0.8
  1. Given that $$z = r(\cos n\theta + i \sin n\theta), \quad r \in \mathbf{R}$$ prove, by induction, that \(z^n = r^n(\cos n\theta + i \sin n\theta)\), \(n \in \mathbf{Z}^+\). [5]
  2. Find the exact value of \(w^2\), giving your answer in the form \(a + ib\), where \(a, b \in \mathbf{R}\). [2]
Edexcel FP2 2008 June Q11
Challenging +1.2
De Moivre's theorem states that \((\cos\theta + i\sin\theta)^n = \cos n\theta + i\sin n\theta\) for \(n \in \mathbb{R}\)
  1. Use induction to prove de Moivre's theorem for \(n \in \mathbb{Z}^+\). (5)
  2. Show that \(\cos 5\theta = 16\cos^5\theta - 20\cos^3\theta + 5\cos\theta\) (5)
  3. Hence show that \(2\cos\frac{\pi}{10}\) is a root of the equation $$x^4 - 5x^2 + 5 = 0$$ (3)
Edexcel FP2 Q23
10 marks Challenging +1.2
  1. Use de Moivre's theorem to show that $$\cos 5\theta = 16\cos^5 \theta - 20\cos^3 \theta + 5\cos \theta.$$ [6]
  2. Hence find \(3\) distinct solutions of the equation \(16x^5 - 20x^3 + 5x + 1 = 0\), giving your answers to \(3\) decimal places where appropriate. [4]
AQA FP2 2011 June Q7
16 marks Challenging +1.3
    1. Use de Moivre's Theorem to show that $$\cos 5\theta = \cos^5 \theta - 10 \cos^3 \theta \sin^2 \theta + 5 \cos \theta \sin^4 \theta$$ and find a similar expression for \(\sin 5\theta\). [5 marks]
    2. Deduce that $$\tan 5\theta = \frac{\tan \theta(5 - 10 \tan^2 \theta + \tan^4 \theta)}{1 - 10 \tan^2 \theta + 5 \tan^4 \theta}$$ [3 marks]
  2. Explain why \(t = \tan \frac{\pi}{5}\) is a root of the equation $$t^4 - 10t^2 + 5 = 0$$ and write down the three other roots of this equation in trigonometrical form. [3 marks]
  3. Deduce that $$\tan \frac{\pi}{5} \tan \frac{2\pi}{5} = \sqrt{5}$$ [5 marks]
AQA FP2 2016 June Q8
13 marks Challenging +1.8
  1. By applying de Moivre's theorem to \((\cos \theta + \mathrm{i} \sin \theta)^4\), where \(\cos \theta \neq 0\), show that $$(1 + \mathrm{i} \tan \theta)^4 + (1 - \mathrm{i} \tan \theta)^4 = \frac{2\cos 4\theta}{\cos^4 \theta}$$ [3 marks]
  2. Hence show that \(z = \mathrm{i} \tan \frac{\pi}{8}\) satisfies the equation \((1 + z)^4 + (1 - z)^4 = 0\), and express the three other roots of this equation in the form \(\mathrm{i} \tan \phi\), where \(0 < \phi < \pi\). [2 marks]
  3. Use the results from part (b) to find the values of:
    1. \(\tan^2 \frac{\pi}{8} \tan^2 \frac{3\pi}{8}\); [4 marks]
    2. \(\tan^2 \frac{\pi}{8} + \tan^2 \frac{3\pi}{8}\). [4 marks]
OCR MEI FP2 2011 January Q2
19 marks Standard +0.3
    1. Given that \(z = \cos \theta + j \sin \theta\), express \(z^n + z^{-n}\) and \(z^n - z^{-n}\) in simplified trigonometrical form. [2]
    2. By considering \((z + z^{-1})^6\), show that $$\cos^6 \theta = \frac{1}{32}(\cos 6\theta + 6 \cos 4\theta + 15 \cos 2\theta + 10).$$ [3]
    3. Obtain an expression for \(\cos^6 \theta - \sin^6 \theta\) in terms of \(\cos 2\theta\) and \(\cos 6\theta\). [5]
  2. The complex number \(w\) is \(8e^{i\pi/3}\). You are given that \(z_1\) is a square root of \(w\) and that \(z_2\) is a cube root of \(w\). The points representing \(z_1\) and \(z_2\) in the Argand diagram both lie in the third quadrant.
    1. Find \(z_1\) and \(z_2\) in the form \(re^{i\theta}\). Draw an Argand diagram showing \(w\), \(z_1\) and \(z_2\). [6]
    2. Find the product \(z_1z_2\), and determine the quadrant of the Argand diagram in which it lies. [3]
OCR FP3 Q5
8 marks Standard +0.8
  1. Use de Moivre's theorem to prove that $$\cos 6\theta = 32\cos^6 \theta - 48\cos^4 \theta + 18\cos^2 \theta - 1.$$ [4]
  2. Hence find the largest positive root of the equation $$64x^6 - 96x^4 + 36x^2 - 3 = 0,$$ giving your answer in trigonometrical form. [4]
OCR FP3 Q7
11 marks Challenging +1.2
    1. Verify, without using a calculator, that \(\theta = \frac{1}{8}\pi\) is a solution of the equation \(\sin 6\theta = \sin 2\theta\). [1]
    2. By sketching the graphs of \(y = \sin 6\theta\) and \(y = \sin 2\theta\) for \(0 \leqslant \theta \leqslant \frac{1}{2}\pi\), or otherwise, find the other solution of the equation \(\sin 6\theta = \sin 2\theta\) in the interval \(0 < \frac{1}{2}\pi\). [2]
  2. Use de Moivre's theorem to prove that $$\sin 6\theta \equiv \sin 2\theta(16\cos^4 \theta - 16\cos^2 \theta + 3).$$ [5]
  3. Hence show that one of the solutions obtained in part (i) satisfies \(\cos^2 \theta = \frac{1}{4}(2 - \sqrt{2})\), and justify which solution it is. [3]
OCR FP3 2008 January Q7
11 marks Challenging +1.3
    1. Verify, without using a calculator, that \(\theta = \frac{1}{8}\pi\) is a solution of the equation \(\sin 6\theta = \sin 2\theta\). [1]
    2. By sketching the graphs of \(y = \sin 6\theta\) and \(y = \sin 2\theta\) for \(0 < \theta < \frac{1}{2}\pi\), or otherwise, find the other solution of the equation \(\sin 6\theta = \sin 2\theta\) in the interval \(0 < \theta < \frac{1}{2}\pi\). [2]
  2. Use de Moivre's theorem to prove that $$\sin 6\theta = \sin 2\theta (16 \cos^4 \theta - 16 \cos^2 \theta + 3).$$ [5]
  3. Hence show that one of the solutions obtained in part (i) satisfies \(\cos^2 \theta = \frac{1}{4}(2 - \sqrt{2})\), and justify which solution it is. [3]
OCR FP3 2010 January Q7
13 marks Challenging +1.3
  1. Solve the equation \(\cos 6\theta = 0\), for \(0 < \theta < \pi\). [3]
  2. By using de Moivre's theorem, show that $$\cos 6\theta \equiv (2\cos^2\theta - 1)(16\cos^4\theta - 16\cos^2\theta + 1).$$ [5]
  3. Hence find the exact value of $$\cos\left(\frac{1}{12}\pi\right)\cos\left(\frac{5}{12}\pi\right)\cos\left(\frac{7}{12}\pi\right)\cos\left(\frac{11}{12}\pi\right),$$ justifying your answer. [5]
OCR FP3 2010 June Q3
9 marks Standard +0.8
In this question, \(w\) denotes the complex number \(\cos \frac{2\pi}{5} + i \sin \frac{2\pi}{5}\).
  1. Express \(w^2\), \(w^3\) and \(w^4\) in polar form, with arguments in the interval \(0 \leq \theta < 2\pi\). [4]
  2. The points in an Argand diagram which represent the numbers $$1, \quad 1 + w, \quad 1 + w + w^2, \quad 1 + w + w^2 + w^3, \quad 1 + w + w^2 + w^3 + w^4$$ are denoted by \(A\), \(B\), \(C\), \(D\), \(E\) respectively. Sketch the Argand diagram to show these points and join them in the order stated. (Your diagram need not be exactly to scale, but it should show the important features.) [4]
  3. Write down a polynomial equation of degree 5 which is satisfied by \(w\). [1]
OCR FP3 2011 June Q8
11 marks Challenging +1.2
  1. Use de Moivre's theorem to express \(\cos 4\theta\) as a polynomial in \(\cos \theta\). [4]
  2. Hence prove that \(\cos 4\theta \cos 2\theta \equiv 16 \cos^6 \theta - 24 \cos^4 \theta + 10 \cos^2 \theta - 1\). [1]
  3. Use part (ii) to show that the only roots of the equation \(\cos 4\theta \cos 2\theta = 1\) are \(\theta = n\pi\), where \(n\) is an integer. [3]
  4. Show that \(\cos 4\theta \cos 2\theta = -1\) only when \(\cos \theta = 0\). [3]