Express roots in trigonometric form

A question is this type if and only if it asks to express roots of a polynomial equation in the form cos(kπ), sin(kπ), tan(kπ), cot(kπ), or similar trigonometric expressions.

32 questions · Challenging +1.3

4.02q De Moivre's theorem: multiple angle formulae

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CAIE FP1 2019 November Q9

11 marks Challenging +1.8

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]
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 2021 November Q4

10 marks Challenging +1.8

Write down all the roots of the equation $x^5 - 1 = 0$. [2]
Use de Moivre's theorem to show that $\cos 4\theta = 8\cos^4 \theta - 8\cos^2 \theta + 1$. [4]
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]

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 $n \in \mathbb{R}$

Use induction to prove de Moivre's theorem for $n \in \mathbb{Z}^+$. (5)
Show that $\cos 5\theta = 16\cos^5\theta - 20\cos^3\theta + 5\cos\theta$ (5)
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

Use de Moivre's theorem to show that $$\cos 5\theta = 16\cos^5 \theta - 20\cos^3 \theta + 5\cos \theta.$$ [6]
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]

OCR FP3 Q5

8 marks Standard +0.8

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]
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 2010 January Q7

13 marks Challenging +1.3

Solve the equation $\cos 6\theta = 0$, for $0 < \theta < \pi$. [3]
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]
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]

Pre-U Pre-U 9795/1 2018 June Q9

8 marks Standard +0.3

Use de Moivre's theorem to prove that $\cos 3\theta = 4c^3 - 3c$, where $c = \cos\theta$. [3]
Solve the equation $2\cos 3\theta - \sqrt{3} = 0$ for $0 < \theta < \pi$, giving each answer in an exact form. [2]
Deduce, in trigonometric form, the three roots of the equation $x^3 - 3x - \sqrt{3} = 0$. [3]