4.02n Euler's formula: e^(i*theta) = cos(theta) + i*sin(theta)

2

1. The infinite series \(C\) and \(S\) are defined as follows. $$\begin{gathered} C = a \cos \theta + a ^ { 2 } \cos 2 \theta + a ^ { 3 } \cos 3 \theta + \ldots \\ S = a \sin \theta + a ^ { 2 } \sin 2 \theta + a ^ { 3 } \sin 3 \theta + \ldots \end{gathered}$$ where \(a\) is a real number and \(| a | < 1\).
   By considering \(C + \mathrm { j } S\), show that $$S = \frac { a \sin \theta } { 1 - 2 a \cos \theta + a ^ { 2 } }$$ Find a corresponding expression for \(C\).
2. P is one vertex of a regular hexagon in an Argand diagram. The centre of the hexagon is at the origin. P corresponds to the complex number \(\sqrt { 3 } + \mathrm { j }\).
   1. Find, in the form \(x + \mathrm { j } y\), the complex numbers corresponding to the other vertices of the hexagon.
   2. The six complex numbers corresponding to the vertices of the hexagon are squared to form the vertices of a new figure. Find, in the form \(x + \mathrm { j } y\), the vertices of the new figure. Find the area of the new figure.

2

1. Express \(\frac { \sqrt { 3 } + \mathrm { i } } { \sqrt { 3 } - \mathrm { i } }\) in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(0 \leqslant \theta < 2 \pi\).
2. Hence find the smallest positive value of \(n\) for which \(\left( \frac { \sqrt { 3 } + \mathrm { i } } { \sqrt { 3 } - \mathrm { i } } \right) ^ { n }\) is real and positive.

8

1. By expressing \(\sin \theta\) in terms of \(\mathrm { e } ^ { \mathrm { i } \theta }\) and \(\mathrm { e } ^ { - \mathrm { i } \theta }\), show that $$\sin ^ { 6 } \theta \equiv - \frac { 1 } { 32 } ( \cos 6 \theta - 6 \cos 4 \theta + 15 \cos 2 \theta - 10 )$$
2. Replace \(\theta\) by ( \(\frac { 1 } { 2 } \pi - \theta\) ) in the identity in part (i) to obtain a similar identity for \(\cos ^ { 6 } \theta\).
3. Hence find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \left( \sin ^ { 6 } \theta - \cos ^ { 6 } \theta \right) \mathrm { d } \theta\).

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 \(0 < \theta < \frac { 1 } { 4 } \pi\).

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 \(\left( \begin{array} { c c } 0 & - 1 \\ 1 & 0 \end{array} \right)\).

4 In an Argand diagram, the complex numbers \(0 , z\) and \(z \mathrm { e } ^ { \frac { 1 } { 6 } \mathrm { i } \pi }\) are represented by the points \(O , A\) and \(B\) respectively.

1. Sketch a possible Argand diagram showing the triangle \(O A B\). Show that the triangle is isosceles and state the size of angle \(A O B\). The complex numbers \(1 + \mathrm { i }\) and \(5 + 2 \mathrm { i }\) are represented by the points \(C\) and \(D\) respectively. The complex number \(w\) is represented by the point \(E\), such that \(C D = C E\) and angle \(D C E = \frac { 1 } { 6 } \pi\).
2. Calculate the possible values of \(w\), giving your answers exactly in the form \(a + b \mathrm { i }\).

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 It is given that the set of complex numbers of the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\) for \(- \pi < \theta \leqslant \pi\) and \(r > 0\), under multiplication, forms a group.

1. Write down the inverse of \(5 \mathrm { e } ^ { \frac { 1 } { 3 } \pi \mathrm { i } }\).
2. Prove the closure property for the group.
3. \(Z\) denotes the element \(\mathrm { e } ^ { \mathrm { i } \gamma }\), where \(\frac { 1 } { 2 } \pi < \gamma < \pi\). Express \(Z ^ { 2 }\) in the form \(\mathrm { e } ^ { \mathrm { i } \theta }\), where \(- \pi < \theta < 0\).

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 $$\sum _ { k = 0 } ^ { n - 1 } \frac { \cos \left( k \cos ^ { - 1 } x \right) } { x ^ { k } } = \frac { \sin \left( n \cos ^ { - 1 } x \right) } { x ^ { n - 1 } \sqrt { } \left( 1 - x ^ { 2 } \right) }$$

6 Write down the values of \(\theta\), in the interval \(0 \leqslant \theta < 2 \pi\), for which \(\cos \theta + \mathrm { i } \sin \theta\) is a fifth root of unity. By writing the equation \(( z + 1 ) ^ { 5 } = z ^ { 5 }\) in the form $$\left( \frac { z + 1 } { z } \right) ^ { 5 } = 1$$ show that its roots are $$- \frac { 1 } { 2 } \left\{ 1 + \mathrm { i } \cot \left( \frac { k \pi } { 5 } \right) \right\} , \quad k = 1,2,3,4$$

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 \(\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \sin ^ { 6 } \theta \mathrm {~d} \theta\).

5 State the sum of the series \(z + z ^ { 2 } + z ^ { 3 } + \ldots + z ^ { n }\), for \(z \neq 1\). By letting \(z = \cos \theta + \mathrm { i } \sin \theta\), show that $$\cos \theta + \cos 2 \theta + \cos 3 \theta + \ldots + \cos n \theta = \frac { \sin \frac { 1 } { 2 } n \theta } { \sin \frac { 1 } { 2 } \theta } \cos \frac { 1 } { 2 } ( n + 1 ) \theta$$ where \(\sin \frac { 1 } { 2 } \theta \neq 0\).

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 $$\sum _ { r = 1 } ^ { n } ( 2 r - 1 ) \cos \left[ \frac { ( 2 r - 1 ) \pi } { 2 n } \right] = - \operatorname { cosec } \left( \frac { \pi } { 2 n } \right) \cot \left( \frac { \pi } { 2 n } \right)$$

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 \(\int _ { 0 } ^ { \frac { 1 } { 3 } \pi } 16 \sin ^ { 5 } \theta \mathrm {~d} \theta\).

8

1. Prove by mathematical induction that, for \(z \neq 1\) and all positive integers \(n\), $$1 + z + z ^ { 2 } + \ldots + z ^ { n - 1 } = \frac { z ^ { n } - 1 } { z - 1 }$$
2. By letting \(z = \frac { 1 } { 2 } ( \cos \theta + \mathrm { i } \sin \theta )\), use de Moivre's theorem to deduce that $$\sum _ { m = 1 } ^ { \infty } \left( \frac { 1 } { 2 } \right) ^ { m } \sin m \theta = \frac { 2 \sin \theta } { 5 - 4 \cos \theta }$$

3

1. Write down the fifth roots of unity.
2. Find all the roots of the equation $$z ^ { 10 } + z ^ { 5 } + 1 = 0$$ giving each root in the form \(\mathrm { e } ^ { \mathrm { i } \theta }\).

9 Prove by mathematical induction that, for every positive integer \(n\), $$( \cos \theta + i \sin \theta ) ^ { n } = \cos n \theta + i \sin n \theta$$ Express \(\sin ^ { 5 } \theta\) in the form \(p \sin 5 \theta + q \sin 3 \theta + r \sin \theta\), where \(p , q\) and \(r\) are rational numbers to be determined.

6 Use de Moivre's theorem to show that $$\cos 5 \theta \equiv \cos \theta \left( 16 \sin ^ { 4 } \theta - 12 \sin ^ { 2 } \theta + 1 \right)$$ By considering the equation \(\cos 5 \theta = 0\), show that the exact value of \(\sin ^ { 2 } \left( \frac { 1 } { 10 } \pi \right)\) is \(\frac { 3 - \sqrt { 5 } } { 8 }\).

10 Let \(z = \cos \theta + \mathrm { i } \sin \theta\). Show that $$z ^ { n } + \frac { 1 } { z ^ { n } } = 2 \cos n \theta \quad \text { and } \quad z ^ { n } - \frac { 1 } { z ^ { n } } = 2 \mathrm { i } \sin n \theta$$ By considering \(\left( z - \frac { 1 } { z } \right) ^ { 4 } \left( z + \frac { 1 } { z } \right) ^ { 2 }\), show that $$\sin ^ { 4 } \theta \cos ^ { 2 } \theta = \frac { 1 } { 32 } ( \cos 6 \theta - 2 \cos 4 \theta - \cos 2 \theta + 2 ) .$$ Hence find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \sin ^ { 4 } \theta \cos ^ { 2 } \theta d \theta\).
[0pt] [Question 11 is printed on the next page.]

6 Let \(z = \cos \theta + \mathrm { i } \sin \theta\). Use the binomial expansion of \(( 1 + z ) ^ { n }\), where \(n\) is a positive integer, to show that $$\binom { n } { 1 } \cos \theta + \binom { n } { 2 } \cos 2 \theta + \ldots + \binom { n } { n } \cos n \theta = 2 ^ { n } \cos ^ { n } \left( \frac { 1 } { 2 } \theta \right) \cos \left( \frac { 1 } { 2 } n \theta \right) - 1$$ Find $$\binom { n } { 1 } \sin \theta + \binom { n } { 2 } \sin 2 \theta + \ldots + \binom { n } { n } \sin n \theta$$

8 Let \(z = \cos \theta + \mathrm { i } \sin \theta\). Show that $$1 + z = 2 \cos \frac { 1 } { 2 } \theta \left( \cos \frac { 1 } { 2 } \theta + \mathrm { i } \sin \frac { 1 } { 2 } \theta \right)$$ By considering \(( 1 + z ) ^ { n }\), where \(n\) is a positive integer, deduce the sum of the series $$\binom { n } { 1 } \sin \theta + \binom { n } { 2 } \sin 2 \theta + \ldots + \binom { n } { n } \sin n \theta$$