Express roots in trigonometric form

7

1. Use de Moivre's theorem to show that $$\operatorname { cosec } 7 \theta = \frac { \operatorname { cosec } ^ { 7 } \theta } { 7 \operatorname { cosec } ^ { 6 } \theta - 56 \operatorname { cosec } ^ { 4 } \theta + 112 \operatorname { cosec } ^ { 2 } \theta - 64 }$$
2. Hence obtain the roots of the equation $$x ^ { 7 } - 14 x ^ { 6 } + 112 x ^ { 4 } - 224 x ^ { 2 } + 128 = 0$$ in the form \(\operatorname { cosec } q \pi\), where \(q\) is rational.

6

1. Use de Moivre's theorem to show that $$\operatorname { cosec } 5 \theta = \frac { \operatorname { cosec } ^ { 5 } \theta } { 5 \operatorname { cosec } ^ { 4 } \theta - 20 \operatorname { cosec } ^ { 2 } \theta + 16 }$$
2. Hence obtain the roots of the equation $$x ^ { 5 } - 10 x ^ { 4 } + 40 x ^ { 2 } - 32 = 0$$ in the form \(\operatorname { cosec } ( q \pi )\), where \(q\) is rational.

4

1. Write down all the roots of the equation \(x ^ { 5 } - 1 = 0\).
2. Use de Moivre's theorem to show that \(\cos 4 \theta = 8 \cos ^ { 4 } \theta - 8 \cos ^ { 2 } \theta + 1\).
3. Use the results of parts (a) and (b) to express each real root of the equation $$8 x ^ { 9 } - 8 x ^ { 7 } + x ^ { 5 } - 8 x ^ { 4 } + 8 x ^ { 2 } - 1 = 0$$ in the form \(\cos k \pi\), where \(k\) is a rational number.

5

1. Write down the fourth roots of unity.
2. Use de Moivre's theorem to show that $$\cos 4 \theta = 8 \cos ^ { 4 } \theta - 8 \cos ^ { 2 } \theta + 1$$
3. Hence obtain the real roots of the equation $$16 \left( 8 x ^ { 4 } - 8 x ^ { 2 } + 1 \right) ^ { 4 } - 9 = 0$$ in the form \(\cos ( q \pi )\), where \(q\) is a rational number.

3

1. Use de Moivre's theorem to show that $$\cos 5 \theta = 16 \cos ^ { 5 } \theta - 20 \cos ^ { 3 } \theta + 5 \cos \theta$$
2. Hence obtain the roots of the equation $$32 x ^ { 5 } - 40 x ^ { 3 } + 10 x - \sqrt { 2 } = 0$$ in the form \(\cos ( q \pi )\), where \(q\) is a rational number.

4

1. Use de Moivre's theorem to show that $$\cot 6 \theta = \frac { \cot ^ { 6 } \theta - 15 \cot ^ { 4 } \theta + 15 \cot ^ { 2 } \theta - 1 } { 6 \cot ^ { 5 } \theta - 20 \cot ^ { 3 } \theta + 6 \cot \theta } .$$ \includegraphics[max width=\textwidth, alt={}, center]{bc601b16-c106-43a2-a2fc-676b5c836096-08_2718_35_107_2012}
   \includegraphics[max width=\textwidth, alt={}, center]{bc601b16-c106-43a2-a2fc-676b5c836096-09_2723_33_99_22}
2. Hence obtain the roots of the equation $$x ^ { 6 } - 6 x ^ { 5 } - 15 x ^ { 4 } + 20 x ^ { 3 } + 15 x ^ { 2 } - 6 x - 1 = 0$$ in the form \(\cot ( q \pi )\), where \(q\) is a rational number.

6

1. Using de Moivre's theorem, show that $$\tan 5 \theta = \frac { 5 \tan \theta - 10 \tan ^ { 3 } \theta + \tan ^ { 5 } \theta } { 1 - 10 \tan ^ { 2 } \theta + 5 \tan ^ { 4 } \theta }$$
2. Hen esh th the eq tion \(x ^ { 2 } - 4 x + 5 = 0\) s ro \(\operatorname { stan } ^ { 2 } \left( \frac { 1 } { 5 } \pi \right)\) ad \(\operatorname { an } ^ { 2 } \left( \frac { 2 } { 5 } \pi \right)\).

7．（a）Use de Moivre＇s theorem to show that $$\cos 7 \theta \equiv 64 \cos ^ { 7 } \theta - 112 \cos ^ { 5 } \theta + 56 \cos ^ { 3 } \theta - 7 \cos \theta$$ （b）Hence find the four distinct roots of the equation $$64 x ^ { 7 } - 112 x ^ { 5 } + 56 x ^ { 3 } - 7 x + 1 = 0$$ giving your answers to 3 decimal places where necessary．

8

1. Use de Moivre's theorem to show that \(\cos 5 \theta \equiv 16 \cos ^ { 5 } \theta - 20 \cos ^ { 3 } \theta + 5 \cos \theta\).
2. Hence find the roots of \(16 x ^ { 4 } - 20 x ^ { 2 } + 5 = 0\) in the form \(\cos \alpha\) where \(0 \leqslant \alpha \leqslant \pi\).
3. Hence find the exact value of \(\cos \frac { 1 } { 10 } \pi\).

5 Use de Moivre's theorem to show that $$\sin 5 \theta = 16 \sin ^ { 5 } \theta - 20 \sin ^ { 3 } \theta + 5 \sin \theta$$ Hence find all the roots of the equation $$32 x ^ { 5 } - 40 x ^ { 3 } + 10 x + 1 = 0$$ in the form \(\sin ( q \pi )\), where \(q\) is a positive rational number.

3

1. Use de Moivre's theorem to show that $$\cos 4 \theta = \cos ^ { 4 } \theta - 6 \cos ^ { 2 } \theta \sin ^ { 2 } \theta + \sin ^ { 4 } \theta$$
2. Hence find all the roots of the equation $$x ^ { 4 } - 6 x ^ { 2 } + 1 = 0$$ in the form \(\tan q \pi\), where \(q\) is a positive rational number.

10

1. Use de Moivre's theorem to show that $$\sin 5 \theta = 5 \sin \theta - 20 \sin ^ { 3 } \theta + 16 \sin ^ { 5 } \theta$$
2. Hence explain why the roots of the equation \(16 x ^ { 4 } - 20 x ^ { 2 } + 5 = 0\) are \(x = \pm \sin \frac { 1 } { 5 } \pi\) and \(x = \pm \sin \frac { 2 } { 5 } \pi\).
3. Without using a calculator, find the exact values of $$\sin \frac { 1 } { 5 } \pi \sin \frac { 2 } { 5 } \pi \sin \frac { 3 } { 5 } \pi \sin \frac { 4 } { 5 } \pi \quad \text { and } \quad \sin ^ { 2 } \left( \frac { 1 } { 5 } \pi \right) + \sin ^ { 2 } \left( \frac { 2 } { 5 } \pi \right) .$$

7

1. Use de Moivre's theorem to show that $$\sin 8 \theta = 8 \sin \theta \cos \theta \left( 1 - 10 \sin ^ { 2 } \theta + 24 \sin ^ { 4 } \theta - 16 \sin ^ { 6 } \theta \right) .$$
2. Use the equation \(\frac { \sin 8 \theta } { \sin 2 \theta } = 0\) to find the roots of $$16 x ^ { 6 } - 24 x ^ { 4 } + 10 x ^ { 2 } - 1 = 0$$ in the form \(\sin k \pi\), where \(k\) is rational.

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

5 Find the smallest positive integer values of \(p\) and \(q\) for which $$\frac { \left( \cos \frac { \pi } { 8 } + \mathrm { i } \sin \frac { \pi } { 8 } \right) ^ { p } } { \left( \cos \frac { \pi } { 12 } - \mathrm { i } \sin \frac { \pi } { 12 } \right) ^ { q } } = \mathrm { i }$$

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

$$\sin 7 \theta = 7 \sin \theta - 56 \sin ^ { 3 } \theta + 112 \sin ^ { 5 } \theta - 64 \sin ^ { 7 } \theta$$ (b) Hence find the distinct roots of the equation $$1 + 7 x - 56 x ^ { 3 } + 112 x ^ { 5 } - 64 x ^ { 7 } = 0$$ giving your answer to 3 decimal places where appropriate.

4 In this question you must show detailed reasoning.
You are given that \(z = \sqrt { 3 } + \mathrm { i }\).
\(n\) is the smallest positive whole number such that \(z ^ { n }\) is a positive whole number.

1. Determine the value of \(n\).
2. Find the value of \(z ^ { n }\).

3 Use De Moivre's Theorem to find the smallest positive angle \(\theta\) for which $$( \cos \theta + \mathrm { i } \sin \theta ) ^ { 15 } = - \mathrm { i }$$ (5 marks)

5

1. Use de Moivre's theorem to prove that $$\cos 6 \theta = 32 \cos ^ { 6 } \theta - 48 \cos ^ { 4 } \theta + 18 \cos ^ { 2 } \theta - 1$$
2. Hence find the largest positive root of the equation $$64 x ^ { 6 } - 96 x ^ { 4 } + 36 x ^ { 2 } - 3 = 0$$ giving your answer in trigonometrical form.

2 In this question you must show detailed reasoning.
You are given that \(z = \sqrt { 3 } + \mathrm { i }\).
\(n\) is the smallest positive whole number such that \(z ^ { n }\) is a positive whole number.

1. Determine the value of \(n\).
2. Find the value of \(z ^ { n }\).