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 \(z ^ { n } + z ^ { - n }\) and \(z ^ { n } - z ^ { - n }\) in simplified trigonometrical form.
   2. By considering \(\left( z + z ^ { - 1 } \right) ^ { 6 }\), show that $$\cos ^ { 6 } \theta = \frac { 1 } { 32 } ( \cos 6 \theta + 6 \cos 4 \theta + 15 \cos 2 \theta + 10 )$$
   3. Obtain an expression for \(\cos ^ { 6 } \theta - \sin ^ { 6 } \theta\) in terms of \(\cos 2 \theta\) and \(\cos 6 \theta\).
2. The complex number \(w\) is \(8 \mathrm { e } ^ { \mathrm { j } \pi / 3 }\). You are given that \(z _ { 1 }\) is a square root of \(w\) and that \(z _ { 2 }\) is a cube root of \(w\). The points representing \(z _ { 1 }\) and \(z _ { 2 }\) in the Argand diagram both lie in the third quadrant.
   1. Find \(z _ { 1 }\) and \(z _ { 2 }\) in the form \(r \mathrm { e } ^ { \mathrm { j } \theta }\). Draw an Argand diagram showing \(w , z _ { 1 }\) and \(z _ { 2 }\).
   2. Find the product \(z _ { 1 } z _ { 2 }\), and determine the quadrant of the Argand diagram in which it lies.
   3. Show that the characteristic equation of the matrix $$\mathbf { M } = \left( \begin{array} { r r r } 1 & - 4 & 5 \\ 2 & 3 & - 2 \\ - 1 & 4 & 1 \end{array} \right)$$ is \(\lambda ^ { 3 } - 5 \lambda ^ { 2 } + 28 \lambda - 66 = 0\).
   4. Show that \(\lambda = 3\) is an eigenvalue of \(\mathbf { M }\), and determine whether or not \(\mathbf { M }\) has any other real eigenvalues.
   5. Find an eigenvector, \(\mathbf { v }\), of unit length corresponding to \(\lambda = 3\). State the magnitude of the vector \(\mathbf { M } ^ { n } \mathbf { v }\), where \(n\) is an integer.
   6. Using the Cayley-Hamilton theorem, obtain an equation for \(\mathbf { M } ^ { - 1 }\) in terms of \(\mathbf { M } ^ { 2 } , \mathbf { M }\) and \(\mathbf { I }\).

7

1. Solve the equation \(\cos 6 \theta = 0\), for \(0 < \theta < \pi\).
2. By using de Moivre's theorem, show that $$\cos 6 \theta \equiv \left( 2 \cos ^ { 2 } \theta - 1 \right) \left( 16 \cos ^ { 4 } \theta - 16 \cos ^ { 2 } \theta + 1 \right)$$
3. Hence find the exact value of $$\cos \left( \frac { 1 } { 12 } \pi \right) \cos \left( \frac { 5 } { 12 } \pi \right) \cos \left( \frac { 7 } { 12 } \pi \right) \cos \left( \frac { 11 } { 12 } \pi \right)$$ justifying your answer.

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 \(4 \cos ^ { 3 } 2 \theta = \cos 6 \theta + 3 \cos 2 \theta\).

5 Use de Moivre's theorem to find the constants \(A , B\) and \(C\) in the identity \(\sin ^ { 5 } \theta \equiv A \sin \theta + B \sin 3 \theta + C \sin 5 \theta\).
\(6 O\) is the origin of a coordinate system whose units are cm .
The points \(A , B , C\) and \(D\) have coordinates ( 1,0 ), ( 1,4 ), ( 6,9 ) and ( 0,9 ) respectively.
The arc \(B C\) is part of the curve with equation \(x ^ { 2 } + ( y - 10 ) ^ { 2 } = 37\).
The closed shape \(O A B C D\) is formed, in turn, from the line segments \(O A\) and \(A B\), the arc \(B C\) and the line segments \(C D\) and \(D O\) (see diagram).
A funnel can be modelled by rotating \(O A B C D\) by \(2 \pi\) radians about the \(y\)-axis.
\includegraphics[max width=\textwidth, alt={}, center]{58e9b480-f561-4a28-b911-7d9d6a80e976-3_641_1131_808_242} Find the volume of the funnel according to the model.

9. (a) (i) Expand \(\left( \cos \frac { \theta } { 3 } + i \sin \frac { \theta } { 3 } \right) ^ { 3 }\).
(ii) Hence, by using de Moivre's theorem, show that \(\cos \theta\) can be expressed as $$4 \cos ^ { 3 } \frac { \theta } { 3 } - 3 \cos \frac { \theta } { 3 }$$ (b) Hence, or otherwise, find the general solution of the equation \(\frac { \cos \theta } { \cos \frac { \theta } { 3 } } = 1\).

3. (a) Given that \(z = \cos \theta + \operatorname { isin } \theta\), use de Moivre's theorem to show that $$z ^ { n } + \frac { 1 } { z ^ { n } } = 2 \cos n \theta$$ (b) Express \(32 \cos ^ { 6 } \theta\) in the form \(a \cos 6 \theta + b \cos 4 \theta + c \cos 2 \theta + d\), where \(a , b , c , d\) are integers whose values are to be determined.

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 $$z ^ { 10 } + \frac { 1 } { z ^ { 10 } }$$