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.

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.

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. Hence show that the equation \(x ^ { 2 } - 10 x + 5 = 0\) has roots \(\tan ^ { 2 } \left( \frac { 1 } { 5 } \pi \right)\) and \(\tan ^ { 2 } \left( \frac { 2 } { 5 } \pi \right)\).

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 $$\sin \frac { 1 } { 5 } \pi \sin \frac { 2 } { 5 } \pi \sin \frac { 3 } { 5 } \pi \sin \frac { 4 } { 5 } \pi \quad \text { and } \quad \sin ^ { 2 } \left( \frac { 1 } { 5 } \pi \right) + \sin ^ { 2 } \left( \frac { 2 } { 5 } \pi \right) .$$

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)

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 \(z ^ { n }\).

10 Using de Moivre's theorem, show that $$\tan 5 \theta = \frac { 5 \tan \theta - 10 \tan ^ { 3 } \theta + \tan ^ { 5 } \theta } { 1 - 10 \tan ^ { 2 } \theta + 5 \tan ^ { 4 } \theta }$$ Hence show that the equation \(x ^ { 2 } - 10 x + 5 = 0\) has roots \(\tan ^ { 2 } \left( \frac { 1 } { 5 } \pi \right)\) and \(\tan ^ { 2 } \left( \frac { 2 } { 5 } \pi \right)\). Deduce a quadratic equation, with integer coefficients, having roots \(\sec ^ { 2 } \left( \frac { 1 } { 5 } \pi \right)\) and \(\sec ^ { 2 } \left( \frac { 2 } { 5 } \pi \right)\).
[0pt] [Question 11 is printed on the next page.]

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.

10

1. Use de Moivre's theorem to show that \(2 \cos 6 \theta \equiv 64 \cos ^ { 6 } \theta - 96 \cos ^ { 4 } \theta + 36 \cos ^ { 2 } \theta - 2\).
2. Hence find, in exact trigonometric form, the six roots of the equation $$x ^ { 6 } - 6 x ^ { 4 } + 9 x ^ { 2 } - 3 = 0$$
3. By considering the product of these six roots, determine the exact value of $$\cos \left( \frac { 1 } { 18 } \pi \right) \cos \left( \frac { 5 } { 18 } \pi \right) \cos \left( \frac { 7 } { 18 } \pi \right) .$$

11

1. Use de Moivre's theorem to prove that \(\sin 5 \theta \equiv s \left( 16 s ^ { 4 } - 20 s ^ { 2 } + 5 \right)\), where \(s = \sin \theta\), and deduce that $$\sin \frac { 2 \pi } { 5 } = \sqrt { \frac { 5 + \sqrt { 5 } } { 8 } }$$ The complex number \(\omega = 16 ( - 1 + \mathrm { i } \sqrt { 3 } )\).
2. State the value of \(| \omega |\) and find \(\arg \omega\) as a rational multiple of \(\pi\).
3. (a) Determine the five roots of the equation \(z ^ { 5 } = \omega\), giving your answers in the form ( \(\mathrm { r } , \theta\) ), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\).
   (b) These five roots are represented in the complex plane by the points \(A , B , C , D\) and \(E\). Show these points on an Argand diagram, and find the area of the pentagon \(A B C D E\) in an exact surd form.

11

1. Use de Moivre's theorem to prove that \(\sin 5 \theta \equiv s \left( 16 s ^ { 4 } - 20 s ^ { 2 } + 5 \right)\), where \(s = \sin \theta\), and deduce that $$\sin \frac { 2 \pi } { 5 } = \sqrt { \frac { 5 + \sqrt { 5 } } { 8 } } .$$ The complex number \(\omega = 16 ( - 1 + \mathrm { i } \sqrt { 3 } )\).
2. State the value of \(| \omega |\) and find \(\arg \omega\) as a rational multiple of \(\pi\).
3. 1. Determine the five roots of the equation \(z ^ { 5 } = \omega\), giving your answers in the form ( \(\mathrm { r } , \theta\) ), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\).
   2. These five roots are represented in the complex plane by the points \(A , B , C , D\) and \(E\). Show these points on an Argand diagram, and find the area of the pentagon \(A B C D E\) in an exact surd form.[3]

11

1. Use de Moivre's theorem to prove that \(\sin 5 \theta \equiv s \left( 16 s ^ { 4 } - 20 s ^ { 2 } + 5 \right)\), where \(s = \sin \theta\), and deduce that \(\sin \frac { 2 \pi } { 5 } = \sqrt { \frac { 5 + \sqrt { 5 } } { 8 } }\). The complex number \(\omega = 16 ( - 1 + \mathrm { i } \sqrt { 3 } )\).
2. State the value of \(| \omega |\) and find \(\arg \omega\) as a rational multiple of \(\pi\).
3. 1. Determine the five roots of the equation \(z ^ { 5 } = \omega\), giving your answers in the form \(( \mathrm { r } , \theta )\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\).
   2. These five roots are represented in the complex plane by the points \(A , B , C , D\) and \(E\). Show these points on an Argand diagram, and find the area of the pentagon \(A B C D E\) in an exact surd form.

10

1. Use de Moivre's theorem to prove that \(\sin 5 \theta \equiv s \left( 16 s ^ { 4 } - 20 s ^ { 2 } + 5 \right)\), where \(s = \sin \theta\), and deduce that \(\sin \frac { 2 \pi } { 5 } = \sqrt { \frac { 5 + \sqrt { 5 } } { 8 } }\). The complex number \(\omega = 16 ( - 1 + \mathrm { i } \sqrt { 3 } )\).
2. State the value of \(| \omega |\) and find \(\arg \omega\) as a rational multiple of \(\pi\).
3. (a) Determine the five roots of the equation \(z ^ { 5 } = \omega\), giving your answers in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta \leq \pi\).
   (b) These five roots are represented in the complex plane by the points \(A , B , C , D\) and \(E\). Show these points on an Argand diagram, and find the area of the pentagon \(A B C D E\) in an exact surd form.