De Moivre to derive trigonometric identities

6

1. Use de Moivre's theorem to show that \(\sin ^ { 4 } \theta = \frac { 1 } { 8 } ( \cos 4 \theta - 4 \cos 2 \theta + 3 )\).
2. Find the solution of the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} \theta } + y \cot \theta = \sin ^ { 3 } \theta$$ for which \(y = 0\) when \(\theta = \frac { 1 } { 2 } \pi\).

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. By considering \(\left( z - \frac { 1 } { z } \right) ^ { 4 } \left( z + \frac { 1 } { z } \right) ^ { 2 }\), find \(A , B , C\) and \(D\) such that $$\sin ^ { 4 } \theta \cos ^ { 2 } \theta = A \cos 6 \theta + B \cos 4 \theta + C \cos 2 \theta + D$$
2. 1. Find the modulus and argument of \(4 + 4 \mathrm { j }\).
   2. Find the fifth roots of \(4 + 4 \mathrm { j }\) in the form \(r \mathrm { e } ^ { \mathrm { j } \theta }\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\). Illustrate these fifth roots on an Argand diagram.
   3. Find integers \(p\) and \(q\) such that \(( p + q \mathrm { j } ) ^ { 5 } = 4 + 4 \mathrm { j }\).

2

1. Use de Moivre's theorem to show that \(\sin 5 \theta = 5 \sin \theta - 20 \sin ^ { 3 } \theta + 16 \sin ^ { 5 } \theta\).
2. 1. Find the cube roots of \(- 2 + 2 \mathrm { j }\) in the form \(r \mathrm { e } ^ { \mathrm { j } \theta }\) where \(r > 0\) and \(- \pi < \theta \leqslant \pi\). These cube roots are represented by points \(\mathrm { A } , \mathrm { B }\) and C in the Argand diagram, with A in the first quadrant and ABC going anticlockwise. The midpoint of AB is M , and M represents the complex number \(w\).
   2. Draw an Argand diagram, showing the points \(\mathrm { A } , \mathrm { B } , \mathrm { C }\) and M .
   3. Find the modulus and argument of \(w\).
   4. Find \(w ^ { 6 }\) in the form \(a + b \mathrm { j }\).

2

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.
   Hence find the constants \(A , B , C\) in the identity $$\sin ^ { 5 } \theta \equiv A \sin \theta + B \sin 3 \theta + C \sin 5 \theta$$
2. 1. Find the 4th roots of - 9 j in the form \(r \mathrm { e } ^ { \mathrm { j } \theta }\), where \(r > 0\) and \(0 < \theta < 2 \pi\). Illustrate the roots on an Argand diagram.
   2. Let the points representing these roots, taken in order of increasing \(\theta\), be \(\mathrm { P } , \mathrm { Q } , \mathrm { R } , \mathrm { S }\). The mid-points of the sides of PQRS represent the 4th roots of a complex number \(w\). Find the modulus and argument of \(w\). Mark the point representing \(w\) on your Argand diagram.

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

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, $$\frac { t } { \mathrm { e } ^ { t } - 1 } \approx 1 - \frac { 1 } { 2 } t$$

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 Given that \(z = \cos \theta + \mathrm { i } \sin \theta\), show that

1. \(z - \frac { 1 } { z } = 2 \mathrm { i } \sin \theta\).
2. \(z ^ { n } + z ^ { - n } = 2 \cos n \theta\). Hence show that $$\sin ^ { 6 } \theta = \frac { 1 } { 32 } ( 10 - 15 \cos 2 \theta + 6 \cos 4 \theta - \cos 6 \theta )$$ Find a similar expression for \(\cos ^ { 6 } \theta\), and hence express \(\cos ^ { 6 } \theta - \sin ^ { 6 } \theta\) in the fom \(a \cos 2 \theta + b \cos 6 \theta\).

10 Use de Moivre's theorem to express \(\cos 8 \theta\) as a polynomial in \(\cos \theta\). Hence

1. express \(\cos 8 \theta\) as a polynomial in \(\sin \theta\),
2. find the exact value of $$4 x ^ { 4 } - 8 x ^ { 3 } + 5 x ^ { 2 } - x$$ where \(x = \cos ^ { 2 } \left( \frac { 1 } { 8 } \pi \right)\).

6 Use de Moivre's theorem to show that $$\cos 4 \theta = 8 \cos ^ { 4 } \theta - 8 \cos ^ { 2 } \theta + 1$$ Without using a calculator, verify that \(\cos 4 \theta = - \cos 3 \theta\) for each of the values \(\theta = \frac { 1 } { 7 } \pi , \frac { 3 } { 7 } \pi , \frac { 5 } { 7 } \pi , \pi\). Using the result \(\cos 3 \theta = 4 \cos ^ { 3 } \theta - 3 \cos \theta\), show that the roots of the equation $$8 c ^ { 4 } + 4 c ^ { 3 } - 8 c ^ { 2 } - 3 c + 1 = 0$$ are \(\cos \frac { 1 } { 7 } \pi , \cos \frac { 3 } { 7 } \pi , \cos \frac { 5 } { 7 } \pi , - 1\). Deduce that \(\cos \frac { 1 } { 7 } \pi + \cos \frac { 3 } { 7 } \pi + \cos \frac { 5 } { 7 } \pi = \frac { 1 } { 2 }\).

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 \(\frac { 3 - \sqrt { 5 } } { 8 }\).

5 By expanding \(\left( z ^ { 2 } + \frac { 1 } { z ^ { 2 } } \right) ^ { 3 }\), where \(z = e ^ { \mathrm { i } \theta }\), show that \(4 \cos ^ { 3 } 2 \theta = \cos 6 \theta + 3 \cos 2 \theta\).

9 In this question you must show detailed reasoning.

1. Show that \(\operatorname { Re } \left( \mathrm { e } ^ { \mathrm { Ai } \theta } \left( \mathrm { e } ^ { \mathrm { i } \theta } + \mathrm { e } ^ { - \mathrm { i } \theta } \right) ^ { 4 } \right) = a \cos 4 \theta \cos ^ { 4 } \theta\), where \(a\) is an integer to be determined.
2. Hence show that \(\cos \frac { 1 } { 12 } \pi = \frac { 1 } { 2 } \sqrt [ 4 ] { \mathrm { b } + \mathrm { c } \sqrt { 3 } }\), where \(b\) and \(c\) are integers to be determined.

5

1. Prove by induction that, if \(n\) is a positive integer, $$( \cos \theta + \mathrm { i } \sin \theta ) ^ { n } = \cos n \theta + \mathrm { i } \sin n \theta$$
2. Find the value of \(\left( \cos \frac { \pi } { 6 } + \mathrm { i } \sin \frac { \pi } { 6 } \right) ^ { 6 }\).
3. Show that $$( \cos \theta + \mathrm { i } \sin \theta ) ( 1 + \cos \theta - \mathrm { i } \sin \theta ) = 1 + \cos \theta + \mathrm { i } \sin \theta$$
4. Hence show that $$\left( 1 + \cos \frac { \pi } { 6 } + i \sin \frac { \pi } { 6 } \right) ^ { 6 } + \left( 1 + \cos \frac { \pi } { 6 } - i \sin \frac { \pi } { 6 } \right) ^ { 6 } = 0$$

5

1. Prove by induction that, if \(n\) is a positive integer, $$( \cos \theta + \mathrm { i } \sin \theta ) ^ { n } = \cos n \theta + \mathrm { i } \sin n \theta$$
2. Hence, given that $$z = \cos \theta + \mathrm { i } \sin \theta$$ show that $$z ^ { n } + \frac { 1 } { z ^ { n } } = 2 \cos n \theta$$
3. Given further that \(z + \frac { 1 } { z } = \sqrt { 2 }\), find the value of $$z ^ { 10 } + \frac { 1 } { z ^ { 10 } }$$

13

1. Two of the solutions to the equation \(\cos 6 \theta = 0\) are \(\theta = \frac { \pi } { 4 }\) and \(\theta = \frac { 3 \pi } { 4 }\) Find the other solutions to the equation \(\cos 6 \theta = 0\) for \(0 \leq \theta \leq \pi\) 13
2. Use de Moivre's theorem to show that $$\cos 6 \theta = 32 \cos ^ { 6 } \theta - 48 \cos ^ { 4 } \theta + 18 \cos ^ { 2 } \theta - 1$$ 13
3. Use the fact that \(\theta = \frac { \pi } { 4 }\) and \(\theta = \frac { 3 \pi } { 4 }\) are solutions to the equation \(\cos 6 \theta = 0\) to find a factor of \(32 \cos ^ { 6 } \theta - 48 \cos ^ { 4 } \theta + 18 \cos ^ { 2 } \theta - 1\) in the form ( \(a \cos ^ { 2 } \theta + b\) ), where \(a\) and \(b\) are integers.
   [0pt] [4 marks]
4. Hence show that $$\cos \left( \frac { 11 \pi } { 12 } \right) = - \sqrt { \frac { 2 + \sqrt { 3 } } { 4 } }$$ \includegraphics[max width=\textwidth, alt={}, center]{13abb93f-2fef-465c-980c-3b412de06618-25_2492_1721_217_150}

5 Let \(z = \cos \theta + \mathrm { i } \sin \theta\).

1. Prove the result \(z ^ { n } - \frac { 1 } { z ^ { n } } = 2 \mathrm { i } \sin n \theta\).
2. Use this result to express \(\sin ^ { 5 } \theta\) in the form \(A \sin 5 \theta + B \sin 3 \theta + C \sin \theta\), for constants \(A , B\) and \(C\) to be determined.

4 Let \(z = \cos \theta + \mathrm { i } \sin \theta\).

1. Use de Moivre's theorem to prove that \(z ^ { n } + z ^ { - n } = 2 \cos n \theta\).
2. Deduce the identity \(\cos ^ { 5 } \theta = \frac { 1 } { 16 } ( \cos 5 \theta + 5 \cos 3 \theta + 10 \cos \theta )\).

The matrix \(\mathbf{C} = \begin{bmatrix} a & -b \\ b & a \end{bmatrix}\), where \(a\) and \(b\) are positive real numbers, and \(\mathbf{C}^2 = \begin{bmatrix} \frac{\sqrt{3}}{2} & -\frac{1}{2} \\ \frac{1}{2} & \frac{\sqrt{3}}{2} \end{bmatrix}\) Use \(\mathbf{C}\) to show that \(\cos \frac{\pi}{12}\) can be written in the form \(\frac{\sqrt{m + n}}{2}\), where \(m\) and \(n\) are integers. [7 marks]

Use de Moivre's theorem to find the constants \(A\), \(B\) and \(C\) in the identity \(\sin^3 \theta \equiv A \sin \theta + B \sin 3\theta + C \sin 5\theta\). [4]