Solve equations using trigonometric identities

8. Given that \(z = e ^ { \mathrm { i } \theta }\)

1. show that \(z ^ { n } + \frac { 1 } { z ^ { n } } = 2 \cos n \theta\) where \(n\) is a positive integer.
2. Show that $$\cos ^ { 6 } \theta = \frac { 1 } { 32 } ( \cos 6 \theta + 6 \cos 4 \theta + 15 \cos 2 \theta + 10 )$$
3. Hence solve the equation $$\cos 6 \theta + 6 \cos 4 \theta + 15 \cos 2 \theta = 0 \quad 0 \leqslant \theta \leqslant \pi$$ Give your answers to 3 significant figures.
4. Use calculus to determine the exact value of $$\int _ { 0 } ^ { \frac { \pi } { 3 } } \left( 32 \cos ^ { 6 } \theta - 4 \cos ^ { 2 } \theta \right) d \theta$$ Solutions relying entirely on calculator technology are not acceptable.

1. In this question you must show all stages of your working.

\section*{Solutions relying entirely on calculator technology are not acceptable.}

1. Use de Moivre's theorem to show that $$\cos 5 x \equiv \cos x \left( a \sin ^ { 4 } x + b \sin ^ { 2 } x + c \right)$$ where \(a\), \(b\) and \(c\) are integers to be determined.
2. Hence solve, for \(0 < \theta < \frac { \pi } { 2 }\) $$\cos 5 \theta = \sin 2 \theta \sin \theta - \cos \theta$$ giving your answers to 3 decimal places.

1. (a) Use de Moivre's theorem to show that

$$\tan 4 \theta \equiv \frac { 4 \tan \theta - 4 \tan ^ { 3 } \theta } { 1 - 6 \tan ^ { 2 } \theta + \tan ^ { 4 } \theta }$$ (b) Use the identity given in part (a) to find the 2 positive roots of $$x ^ { 4 } + 2 x ^ { 3 } - 6 x ^ { 2 } - 2 x + 1 = 0$$ giving your answers to 3 significant figures. \includegraphics[max width=\textwidth, alt={}, center]{0d44aec7-a6e8-47fc-a215-7c8c4790e93f-29_2255_50_314_35}

1. In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

1. Use De Moivre's theorem to show that $$\cos 6 \theta \equiv 32 \cos ^ { 6 } \theta - 48 \cos ^ { 4 } \theta + 18 \cos ^ { 2 } \theta - 1$$
2. Hence determine the smallest positive root of the equation $$48 x ^ { 6 } - 72 x ^ { 4 } + 27 x ^ { 2 } - 1 = 0$$ giving your answer to 3 decimal places.

2. (a) Use de Moivre's theorem to show that $$\cos 5 \theta = 16 \cos ^ { 5 } \theta - 20 \cos ^ { 3 } \theta + 5 \cos \theta$$ (b) Hence find 3 distinct solutions of the equation \(16 x ^ { 5 } - 20 x ^ { 3 } + 5 x + 1 = 0\), giving your answers to 3 decimal places where appropriate.

10. (a) Given that \(z = e ^ { \mathrm { i } \theta }\), show that $$z ^ { n } - \frac { 1 } { z ^ { n } } = 2 \mathrm { i } \sin n \theta$$ where \(n\) is a positive integer.
(b) Show that $$\sin ^ { 5 } \theta = \frac { 1 } { 16 } ( \sin 5 \theta - 5 \sin 3 \theta + 10 \sin \theta )$$ (c) Hence solve, in the interval \(0 \leq \theta < 2 \pi\), $$\sin 5 \theta - 5 \sin 3 \theta + 6 \sin \theta = 0$$ (5)(Total 12 marks)

6. (a) Use de Moivre's theorem to show that \(\boldsymbol { \operatorname { s i n } } 5 \boldsymbol { \theta } = \boldsymbol { \operatorname { s i n } } \boldsymbol { \theta } \left( \mathbf { 1 6 } \mathbf { c o s } ^ { 4 } \boldsymbol { \theta } - \mathbf { 1 2 } \boldsymbol { \operatorname { c o s } } ^ { 2 } \boldsymbol { \theta } + \mathbf { 1 } \right)\).
(b) Hence, or otherwise, solve, for \(0 \leq \theta < \pi\) $$\sin 5 \theta + \cos \theta \sin 2 \theta = 0$$ (6)(Total 11 marks)

1. (a) Use de Moivre's theorem to show that

$$\sin 5 \theta = 16 \sin ^ { 5 } \theta - 20 \sin ^ { 3 } \theta + 5 \sin \theta$$ Hence, given also that \(\sin 3 \theta = 3 \sin \theta - 4 \sin ^ { 3 } \theta\),
(b) find all the solutions of $$\sin 5 \theta = 5 \sin 3 \theta$$ in the interval \(0 \leqslant \theta < 2 \pi\). Give your answers to 3 decimal places.

6. The complex number \(z = \mathrm { e } ^ { \mathrm { i } \theta }\), 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. Show that $$\cos ^ { 5 } \theta = \frac { 1 } { 16 } ( \cos 5 \theta + 5 \cos 3 \theta + 10 \cos \theta )$$
3. Hence find all the solutions of $$\cos 5 \theta + 5 \cos 3 \theta + 12 \cos \theta = 0$$ in the interval \(0 \leqslant \theta < 2 \pi\)

4. (a) Use de Moivre's theorem to show that $$\cos 6 \theta = 32 \cos ^ { 6 } \theta - 48 \cos ^ { 4 } \theta + 18 \cos ^ { 2 } \theta - 1$$ (b) Hence solve for \(0 \leqslant \theta \leqslant \frac { \pi } { 2 }\) $$64 \cos ^ { 6 } \theta - 96 \cos ^ { 4 } \theta + 36 \cos ^ { 2 } \theta - 3 = 0$$ giving your answers as exact multiples of \(\pi\).

6. (a) Use de M oivre's Theorem to show that $$\sin 5 \theta = 16 \sin ^ { 5 } \theta - 20 \sin ^ { 3 } \theta + 5 \sin \theta .$$ (b) Hence or otherwise, prove that the only real solutions of the equation $$\sin 5 \theta = 5 \sin \theta ,$$ are given by \(\theta = n \tau\), where \(n\) is an integer.

5

1. By expressing \(\sin \theta\) and \(\cos \theta\) in terms of \(\mathrm { e } ^ { \mathrm { i } \theta }\) and \(\mathrm { e } ^ { - \mathrm { i } \theta }\), prove that $$\sin ^ { 3 } \theta \cos ^ { 2 } \theta \equiv - \frac { 1 } { 16 } ( \sin 5 \theta - \sin 3 \theta - 2 \sin \theta )$$
2. Hence show that all the roots of the equation $$\sin 5 \theta = \sin 3 \theta + 2 \sin \theta$$ are of the form \(\theta = \frac { n \pi } { k }\), where \(n\) is any integer and \(k\) is to be determined.

7

1. By expressing \(\sin \theta\) in terms of \(\mathrm { e } ^ { \mathrm { i } \theta }\) and \(\mathrm { e } ^ { - \mathrm { i } \theta }\), show that $$\sin ^ { 5 } \theta \equiv \frac { 1 } { 16 } ( \sin 5 \theta - 5 \sin 3 \theta + 10 \sin \theta ) .$$
2. Hence solve the equation $$\sin 5 \theta + 4 \sin \theta = 5 \sin 3 \theta$$ for \(- \frac { 1 } { 2 } \pi \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\). 8 consists of the set of matrices of the form \(\left( \begin{array} { c c } a & - b \\ b & a \end{array} \right)\), where \(a\) and \(b\) are real and \(a ^ { 2 } + b ^ { 2 } \neq 0\), combined under the operation of matrix multiplication.
3. Prove that \(G\) is a group. You may assume that matrix multiplication is associative.
4. Determine whether \(G\) is commutative.
5. Find the order of \(\left( \begin{array} { c c } 0 & - 1 \\ 1 & 0 \end{array} \right)\).

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]

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]

Find the smallest value \(\theta\) of for which \((\cos \theta + i \sin \theta)^5 = \frac{1}{\sqrt{2}}(1 - i)\) \(\{\theta \in \mathbb{R} : \theta > 0\}\) [4 marks]