4.02q De Moivre's theorem: multiple angle formulae

8

1. Use de Moivre's theorem to find an expression for \(\tan 4 \theta\) in terms of \(\tan \theta\).
2. Deduce that \(\cot 4 \theta = \frac { \cot ^ { 4 } \theta - 6 \cot ^ { 2 } \theta + 1 } { 4 \cot ^ { 3 } \theta - 4 \cot \theta }\).
3. Hence show that one of the roots of the equation \(x ^ { 2 } - 6 x + 1 = 0\) is \(\cot ^ { 2 } \left( \frac { 1 } { 8 } \pi \right)\).
4. Hence find the value of \(\operatorname { cosec } ^ { 2 } \left( \frac { 1 } { 8 } \pi \right) + \operatorname { cosec } ^ { 2 } \left( \frac { 3 } { 8 } \pi \right)\), justifying your answer.

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.

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.

2

1. Use de Moivre's theorem to find the constants \(a , b , c\) in the identity $$\cos 5 \theta \equiv a \cos ^ { 5 } \theta + b \cos ^ { 3 } \theta + c \cos \theta$$
2. Let $$\begin{aligned} C & = \cos \theta + \cos \left( \theta + \frac { 2 \pi } { n } \right) + \cos \left( \theta + \frac { 4 \pi } { n } \right) + \ldots + \cos \left( \theta + \frac { ( 2 n - 2 ) \pi } { n } \right) \\ \text { and } S & = \sin \theta + \sin \left( \theta + \frac { 2 \pi } { n } \right) + \sin \left( \theta + \frac { 4 \pi } { n } \right) + \ldots + \sin \left( \theta + \frac { ( 2 n - 2 ) \pi } { n } \right) \end{aligned}$$ where \(n\) is an integer greater than 1 .
   By considering \(C + \mathrm { j } S\), show that \(C = 0\) and \(S = 0\).
3. Write down the Maclaurin series for \(\mathrm { e } ^ { t }\) as far as the term in \(t ^ { 2 }\). Hence show that, for \(t\) close to zero, $$\frac { t } { \mathrm { e } ^ { t } - 1 } \approx 1 - \frac { 1 } { 2 } t$$

8

1. Use de Moivre's theorem to prove that $$\tan 5 \theta \equiv \frac { 5 \tan \theta - 10 \tan ^ { 3 } \theta + \tan ^ { 5 } \theta } { 1 - 10 \tan ^ { 2 } \theta + 5 \tan ^ { 4 } \theta } .$$
2. Solve the equation \(\tan 5 \theta = 1\), for \(0 \leqslant \theta < \pi\).
3. Show that the roots of the equation $$t ^ { 4 } - 4 t ^ { 3 } - 14 t ^ { 2 } - 4 t + 1 = 0$$ may be expressed in the form \(\tan \alpha\), stating the exact values of \(\alpha\), where \(0 \leqslant \alpha < \pi\). \section*{THERE ARE NO QUESTIONS WRITTEN ON THIS PAGE}

7 Let \(S = \mathrm { e } ^ { \mathrm { i } \theta } + \mathrm { e } ^ { 2 \mathrm { i } \theta } + \mathrm { e } ^ { 3 \mathrm { i } \theta } + \ldots + \mathrm { e } ^ { 10 \mathrm { i } \theta }\).

1. (a) Show that, for \(\theta \neq 2 n \pi\), where \(n\) is an integer, $$S = \frac { \mathrm { e } ^ { \frac { 1 } { 2 } \mathrm { i } \theta } \left( \mathrm { e } ^ { 10 \mathrm { i } \theta } - 1 \right) } { 2 \mathrm { i } \sin \left( \frac { 1 } { 2 } \theta \right) }$$ (b) State the value of \(S\) for \(\theta = 2 n \pi\), where \(n\) is an integer.
2. Hence show that, for \(\theta \neq 2 n \pi\), where \(n\) is an integer, $$\cos \theta + \cos 2 \theta + \cos 3 \theta + \ldots + \cos 10 \theta = \frac { \sin \left( \frac { 21 } { 2 } \theta \right) } { 2 \sin \left( \frac { 1 } { 2 } \theta \right) } - \frac { 1 } { 2 }$$
3. Hence show that \(\theta = \frac { 1 } { 11 } \pi\) is a root of \(\cos \theta + \cos 2 \theta + \cos 3 \theta + \ldots + \cos 10 \theta = 0\) and find another root in the interval \(0 < \theta < \frac { 1 } { 4 } \pi\).

2

1. Solve the equation \(z ^ { 4 } = 2 ( 1 + \mathrm { i } \sqrt { 3 } )\), giving the roots exactly in the form \(r ( \cos \theta + \mathrm { i } \sin \theta )\), where \(r > 0\) and \(0 \leqslant \theta < 2 \pi\).
2. Sketch an Argand diagram to show the lines from the origin to the point representing \(2 ( 1 + i \sqrt { 3 } )\) and from the origin to the points which represent the roots of the equation in part (i).

4 The complex numbers 0,3 and \(3 \mathrm { e } ^ { \frac { 1 } { 3 } \pi \mathrm { i } }\) are represented in an Argand diagram by the points \(O , A\) and \(B\) respectively.

1. Sketch the triangle \(O A B\) and show that it is equilateral.
2. Hence express \(3 - 3 e ^ { \frac { 1 } { 3 } \pi i }\) in polar form.
3. Hence find \(\left( 3 - 3 \mathrm { e } ^ { \frac { 1 } { 3 } \pi \mathrm { i } } \right) ^ { 5 }\), giving your answer in the form \(a + b \sqrt { 3 } \mathrm { i }\) where \(a\) and \(b\) are rational numbers.

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 \(\cos \frac { 1 } { 10 } \pi\).

3

1. Solve the equation \(z ^ { 6 } = 1\), giving your answers in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), and sketch an Argand diagram showing the positions of the roots.
2. Show that \(( 1 + \mathrm { i } ) ^ { 6 } = - 8 \mathrm { i }\).
3. Hence, or otherwise, solve the equation \(z ^ { 6 } + 8 \mathrm { i } = 0\), giving your answers in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\).

7

1. Use de Moivre's theorem to show that \(\tan 4 \theta \equiv \frac { 4 \tan \theta - 4 \tan ^ { 3 } \theta } { 1 - 6 \tan ^ { 2 } \theta + \tan ^ { 4 } \theta }\).
2. Hence find the exact roots of \(t ^ { 4 } + 4 \sqrt { 3 } t ^ { 3 } - 6 t ^ { 2 } - 4 \sqrt { 3 } t + 1 = 0\).

2

1. 1. Given that \(z = \cos \theta + \mathrm { j } \sin \theta\), express \(z ^ { n } + \frac { 1 } { z ^ { n } }\) and \(z ^ { n } - \frac { 1 } { z ^ { n } }\) in simplified trigonometric form.
   2. Beginning with an expression for \(\left( z + \frac { 1 } { z } \right) ^ { 4 }\), find the constants \(A , B , C\) in the identity $$\cos ^ { 4 } \theta \equiv A + B \cos 2 \theta + C \cos 4 \theta$$
   3. Use the identity in part (ii) to obtain an expression for \(\cos 4 \theta\) as a polynomial in \(\cos \theta\).
2. 1. Given that \(z = 4 \mathrm { e } ^ { \mathrm { j } \pi / 3 }\) and that \(w ^ { 2 } = z\), write down the possible values of \(w\) in the form \(r \mathrm { e } ^ { \mathrm { j } \theta }\), where \(r > 0\). Show \(z\) and the possible values of \(w\) in an Argand diagram.
   2. Find the least positive integer \(n\) for which \(z ^ { n }\) is real. Show that there is no positive integer \(n\) for which \(z ^ { n }\) is imaginary.
      For each possible value of \(w\), find the value of \(w ^ { 3 }\) in the form \(a + \mathrm { j } b\) where \(a\) and \(b\) are real.

2

1. Use de Moivre's theorem to find expressions for \(\sin 5 \theta\) and \(\cos 5 \theta\) in terms of \(\sin \theta\) and \(\cos \theta\).
   Hence show that, if \(t = \tan \theta\), then $$\tan 5 \theta = \frac { t \left( t ^ { 4 } - 10 t ^ { 2 } + 5 \right) } { 5 t ^ { 4 } - 10 t ^ { 2 } + 1 }$$
2. 1. Find the 5th roots of \(- 4 \sqrt { 2 }\) in the form \(r \mathrm { e } ^ { \mathrm { j } \theta }\), where \(r > 0\) and \(0 \leqslant \theta < 2 \pi\). These 5th roots are represented in the Argand diagram, in order of increasing \(\theta\), by the points A , \(\mathrm { B } , \mathrm { C } , \mathrm { D } , \mathrm { E }\).
   2. Draw the Argand diagram, making clear which point is which. The mid-point of AB is the point P which represents the complex number \(w\).
   3. Find, in exact form, the modulus and argument of \(w\).
   4. \(w\) is an \(n\)th root of a real number \(a\), where \(n\) is a positive integer. State the least possible value of \(n\) and find the corresponding value of \(a\).

7

1. Use de Moivre's theorem to prove that $$\tan 3 \theta \equiv \frac { \tan \theta \left( 3 - \tan ^ { 2 } \theta \right) } { 1 - 3 \tan ^ { 2 } \theta } .$$
2. (a) By putting \(\theta = \frac { 1 } { 12 } \pi\) in the identity in part (i), show that \(\tan \frac { 1 } { 12 } \pi\) is a solution of the equation $$t ^ { 3 } - 3 t ^ { 2 } - 3 t + 1 = 0 .$$ (b) Hence show that \(\tan \frac { 1 } { 12 } \pi = 2 - \sqrt { 3 }\).
3. Use the substitution \(t = \tan \theta\) to show that $$\int _ { 0 } ^ { 2 - \sqrt { 3 } } \frac { t \left( 3 - t ^ { 2 } \right) } { \left( 1 - 3 t ^ { 2 } \right) \left( 1 + t ^ { 2 } \right) } \mathrm { d } t = a \ln b$$ where \(a\) and \(b\) are positive constants to be determined.

10 By considering \(\sum _ { n = 1 } ^ { N } z ^ { 2 n - 1 }\), where \(z = \mathrm { e } ^ { \mathrm { i } \theta }\), show that $$\sum _ { n = 1 } ^ { N } \cos ( 2 n - 1 ) \theta = \frac { \sin ( 2 N \theta ) } { 2 \sin \theta }$$ where \(\sin \theta \neq 0\). Deduce that $$\sum _ { n = 1 } ^ { N } ( 2 n - 1 ) \sin \left[ \frac { ( 2 n - 1 ) \pi } { N } \right] = - N \operatorname { cosec } \frac { \pi } { N }$$

By considering \(\sum _ { k = 0 } ^ { n - 1 } ( 1 + \mathrm { i } \tan \theta ) ^ { k }\), show that $$\sum _ { k = 0 } ^ { n - 1 } \cos k \theta \sec ^ { k } \theta = \cot \theta \sin n \theta \sec ^ { n } \theta$$ provided \(\theta\) is not an integer multiple of \(\frac { 1 } { 2 } \pi\). Hence or otherwise show that $$\sum _ { k = 0 } ^ { n - 1 } 2 ^ { k } \cos \left( \frac { 1 } { 3 } k \pi \right) = \frac { 2 ^ { n } } { \sqrt { 3 } } \sin \left( \frac { 1 } { 3 } n \pi \right)$$ Given that \(0 < x < 1\), show that $$\sum _ { k = 0 } ^ { n - 1 } \frac { \cos \left( k \cos ^ { - 1 } x \right) } { x ^ { k } } = \frac { \sin \left( n \cos ^ { - 1 } x \right) } { x ^ { n - 1 } \sqrt { } \left( 1 - x ^ { 2 } \right) }$$

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.

Use de Moivre's theorem to prove that $$\tan 3 \theta = \frac { 3 \tan \theta - \tan ^ { 3 } \theta } { 1 - 3 \tan ^ { 2 } \theta }$$ State the exact values of \(\theta\), between 0 and \(\pi\), that satisfy \(\tan 3 \theta = 1\). Express each root of the equation \(t ^ { 3 } - 3 t ^ { 2 } - 3 t + 1 = 0\) in the form \(\tan ( k \pi )\), where \(k\) is a positive rational number. For each of these values of \(k\), find the exact value of \(\tan ( k \pi )\).

7 Expand \(\left( z + \frac { 1 } { z } \right) ^ { 4 } \left( z - \frac { 1 } { z } \right) ^ { 2 }\) and, by substituting \(z = \cos \theta + \mathrm { i } \sin \theta\), find integers \(p , q , r , s\) such that $$64 \sin ^ { 2 } \theta \cos ^ { 4 } \theta = p + q \cos 2 \theta + r \cos 4 \theta + s \cos 6 \theta$$ Using the substitution \(x = 2 \cos \theta\), show that $$\int _ { 1 } ^ { 2 } x ^ { 4 } \sqrt { } \left( 4 - x ^ { 2 } \right) \mathrm { d } x = \frac { 4 } { 3 } \pi + \sqrt { } 3$$

7 By considering the binomial expansion of \(\left( z - \frac { 1 } { z } \right) ^ { 6 }\), where \(z = \cos \theta + \mathrm { i } \sin \theta\), express \(\sin ^ { 6 } \theta\) 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. Hence find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \sin ^ { 6 } \theta \mathrm {~d} \theta\).

Show the cube roots of 1 on an Argand diagram. Show that the two non-real cube roots can be expressed in the form \(\omega\) and \(\omega ^ { 2 }\), and find these cube roots in exact cartesian form \(x + i y\). Evaluate the determinant $$\left| \begin{array} { c c c } 1 & 3 \omega & 2 \omega ^ { 2 } \\ 3 \omega ^ { 2 } & 2 & \omega \\ 2 \omega & \omega ^ { 2 } & 3 \end{array} \right|$$ It is given that \(z = ( 4 \sqrt { } 3 ) \left( \cos \frac { 4 } { 3 } \pi + i \sin \frac { 4 } { 3 } \pi \right) - 4 \left( \cos \frac { 11 } { 6 } \pi + i \sin \frac { 11 } { 6 } \pi \right)\). Express \(z\) in the form \(r ( \cos \theta + \mathrm { i } \sin \theta )\), giving exact values for \(r\) and \(\theta\). Hence find the cube roots of \(z\) in the form \(r ( \cos \theta + \mathrm { i } \sin \theta )\).

7 Use de Moivre's theorem to show that $$\tan 5 \theta = \frac { 5 t - 10 t ^ { 3 } + t ^ { 5 } } { 1 - 10 t ^ { 2 } + 5 t ^ { 4 } }$$ where \(t = \tan \theta\). Deduce that the roots of the equation \(t ^ { 4 } - 10 t ^ { 2 } + 5 = 0\) are \(\pm \tan \frac { 1 } { 5 } \pi\) and \(\pm \tan \frac { 2 } { 5 } \pi\). Hence show that \(\tan \frac { 1 } { 5 } \pi \tan \frac { 2 } { 5 } \pi = \sqrt { } 5\).

8 By considering \(\sum _ { r = 1 } ^ { n } z ^ { 2 r - 1 }\), where \(z = \cos \theta + \mathrm { i } \sin \theta\), show that, if \(\sin \theta \neq 0\), $$\sum _ { r = 1 } ^ { n } \sin ( 2 r - 1 ) \theta = \frac { \sin ^ { 2 } n \theta } { \sin \theta }$$ Deduce that $$\sum _ { r = 1 } ^ { n } ( 2 r - 1 ) \cos \left[ \frac { ( 2 r - 1 ) \pi } { 2 n } \right] = - \operatorname { cosec } \left( \frac { \pi } { 2 n } \right) \cot \left( \frac { \pi } { 2 n } \right)$$

6 Use de Moivre's theorem to express \(\cot 7 \theta\) in terms of \(\cot \theta\). Use the equation \(\cot 7 \theta = 0\) to show that the roots of the equation $$x ^ { 6 } - 21 x ^ { 4 } + 35 x ^ { 2 } - 7 = 0$$ are \(\cot \left( \frac { 1 } { 14 } k \pi \right)\) for \(k = 1,3,5,9,11,13\), and deduce that $$\cot ^ { 2 } \left( \frac { 1 } { 14 } \pi \right) \cot ^ { 2 } \left( \frac { 3 } { 14 } \pi \right) \cot ^ { 2 } \left( \frac { 5 } { 14 } \pi \right) = 7$$

8

1. Let \(z = \cos \theta + \mathrm { i } \sin \theta\). Show that \(z - \frac { 1 } { z } = 2 \mathrm { i } \sin \theta\) and hence express \(16 \sin ^ { 5 } \theta\) in the form \(\sin 5 \theta + p \sin 3 \theta + q \sin \theta\), where \(p\) and \(q\) are integers to be determined.
2. Hence find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 3 } \pi } 16 \sin ^ { 5 } \theta \mathrm {~d} \theta\).