4.02r nth roots: of complex numbers

6 The complex number \(u\) is defined by $$u = \frac { 7 + \mathrm { i } } { 1 - \mathrm { i } }$$

1. Express \(u\) in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
2. Show on a sketch of an Argand diagram the points \(A , B\) and \(C\) representing \(u , 7 + \mathrm { i }\) and \(1 - \mathrm { i }\) respectively.
3. By considering the arguments of \(7 + \mathrm { i }\) and \(1 - \mathrm { i }\), show that $$\tan ^ { - 1 } \left( \frac { 4 } { 3 } \right) = \tan ^ { - 1 } \left( \frac { 1 } { 7 } \right) + \frac { 1 } { 4 } \pi$$

3

1. Find the roots of the equation \(z ^ { 3 } = - 1 - \mathrm { i }\), giving your answers in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(0 \leqslant \theta < 2 \pi\).
   Let \(\mathbf { w } = \mathbf { z } _ { 1 } ^ { 3 \mathrm { k } } + \mathbf { z } _ { 2 } ^ { 3 \mathrm { k } } + \mathbf { z } _ { 3 } ^ { 3 \mathrm { k } }\), where \(k\) is a positive integer and \(\mathrm { z } _ { 1 } , \mathrm { z } _ { 2 } , \mathrm { z } _ { 3 }\) are the roots of \(\mathrm { z } ^ { 3 } = - 1 - \mathrm { i }\).
2. Express \(w\) in the form \(R \mathrm { e } ^ { \mathrm { i } \alpha }\), where \(R > 0\), giving \(R\) and \(\alpha\) in terms of \(k\). \includegraphics[max width=\textwidth, alt={}, center]{20e14db3-0eb0-4954-91cf-027e16f8bf14-06_889_824_267_616} The diagram shows the curve with equation \(\mathrm { y } = \mathrm { x } ^ { 2 }\) for \(0 \leqslant x \leqslant 1\), together with a set of \(n\) rectangles of width \(\frac { 1 } { n }\).

3

1. Find the roots of the equation \(z ^ { 3 } = - 1 - \mathrm { i }\), giving your answers in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(0 \leqslant \theta < 2 \pi\).
   Let \(\mathbf { w } = \mathbf { z } _ { 1 } ^ { 3 \mathrm { k } } + \mathbf { z } _ { 2 } ^ { 3 \mathrm { k } } + \mathbf { z } _ { 3 } ^ { 3 \mathrm { k } }\), where \(k\) is a positive integer and \(\mathrm { z } _ { 1 } , \mathrm { z } _ { 2 } , \mathrm { z } _ { 3 }\) are the roots of \(\mathrm { z } ^ { 3 } = - 1 - \mathrm { i }\).
2. Express \(w\) in the form \(R \mathrm { e } ^ { \mathrm { i } \alpha }\), where \(R > 0\), giving \(R\) and \(\alpha\) in terms of \(k\). \includegraphics[max width=\textwidth, alt={}, center]{1de67949-6262-4ade-b986-02b6563ae404-06_889_824_267_616} The diagram shows the curve with equation \(\mathrm { y } = \mathrm { x } ^ { 2 }\) for \(0 \leqslant x \leqslant 1\), together with a set of \(n\) rectangles of width \(\frac { 1 } { n }\).

5

1. State the sum of the series \(z + z ^ { 2 } + z ^ { 3 } + \ldots + z ^ { n }\), for \(z \neq 1\).
2. Given that \(z\) is an \(n\)th root of unity and \(z \neq 1\), deduce that \(1 + z + z ^ { 2 } + \ldots + z ^ { n - 1 } = 0\).
3. Given instead that \(z = \frac { 1 } { 3 } ( \cos \theta + \mathrm { i } \sin \theta )\), use de Moivre's theorem to show that $$\sum _ { m = 1 } ^ { \infty } 3 ^ { - m } \cos m \theta = \frac { 3 \cos \theta - 1 } { 10 - 6 \cos \theta }$$

1 Find the roots of the equation \(z ^ { 3 } = 7 \sqrt { 3 } - 7 \mathrm { i }\), giving your answers in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi \leqslant \theta < \pi\).

1 Find the roots of the equation \(z ^ { 3 } = - 108 \sqrt { 3 } + 108\) i, giving your answers in the form \(r ( \cos \theta + \mathrm { i } \sin \theta )\), where \(r > 0\) and \(0 < \theta < 2 \pi\).

3 Find all the roots of the equation \(( w + 1 ) ^ { 6 } = 1\), giving your answers in the form \(\mathrm { x } + \mathrm { iy }\) where \(x\) and \(y\) are real and exact.

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.

2 Find the roots of the equation \(( z + 5 i ) ^ { 3 } = 4 + 4 \sqrt { 3 } i\), giving your answers in the form \(r \cos \theta + i ( r \sin \theta - 5 )\), where \(r > 0\) and \(0 < \theta < 2 \pi\).

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.

3. Solve the equation $$z ^ { 5 } = 16 - 16 \mathrm { i } \sqrt { 3 }$$ giving your answers in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\) where \(\theta\) is in terms of \(\pi\) and \(0 \leqslant \theta < 2 \pi\).

1. Solve the equation

$$z ^ { 5 } = 32$$ Give your answers in the form \(r ( \cos \theta + \mathrm { i } \sin \theta )\), where \(r > 0\) and \(0 \leqslant \theta < 2 \pi\)

4. (a) Express the complex number \(18 \sqrt { 3 } - 18 \mathrm { i }\) in the form $$r ( \cos \theta + \mathrm { i } \sin \theta ) \quad - \pi < \theta \leqslant \pi$$ (b) Solve the equation $$z ^ { 4 } = 18 \sqrt { 3 } - 18 i$$ giving your answers in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\) where \(- \pi < \theta \leqslant \pi\)

5. Solve the equation \(z ^ { 5 } = \mathrm { i }\) giving your answers in the form \(\cos \theta + \mathrm { i } \sin \theta\).
(Total 5 marks)

4. $$z = - 8 + ( 8 \sqrt { } 3 ) i$$

1. Find the modulus of \(z\) and the argument of \(z\). Using de Moivre's theorem,
2. find \(z ^ { 3 }\),
3. find the values of \(w\) such that \(w ^ { 4 } = z\), giving your answers in the form \(a + \mathrm { i } b\), where \(a , b \in \mathbb { R }\).

3. (a) Express the complex number \(- 2 + ( 2 \sqrt { 3 } ) \mathrm { i }\) in the form \(r ( \cos \theta + \mathrm { i } \sin \theta ) , - \pi < \theta \leqslant \pi\).
(b) Solve the equation $$z ^ { 4 } = - 2 + ( 2 \sqrt { } 3 ) i$$ giving the roots in the form \(r ( \cos \theta + \mathrm { i } \sin \theta ) , - \pi < \theta \leqslant \pi\).

6. Solve the equation $$z ^ { 5 } = - 16 \sqrt { } 3 + 16 i$$ giving your answers in the form \(r ( \cos \theta + \mathrm { i } \sin \theta )\), where \(r > 0\) and \(- \pi < \theta < \pi\).

4. (a) Use de Moivre's theorem to show that $$\cos 6 \theta = 32 \cos ^ { 6 } \theta - 48 \cos ^ { 4 } \theta + 18 \cos ^ { 2 } \theta - 1$$ (b) Hence solve for \(0 \leqslant \theta \leqslant \frac { \pi } { 2 }\) $$64 \cos ^ { 6 } \theta - 96 \cos ^ { 4 } \theta + 36 \cos ^ { 2 } \theta - 3 = 0$$ giving your answers as exact multiples of \(\pi\).

2. $$z = - 2 + ( 2 \sqrt { 3 } ) \mathrm { i }$$

1. Find the modulus and the argument of \(z\). Using de Moivre's theorem,
2. find \(z ^ { 6 }\), simplifying your answer,
3. find the values of \(w\) such that \(w ^ { 4 } = z ^ { 3 }\), giving your answers in the form \(a + \mathrm { i } b\) where \(a , b \in \mathbb { R }\).

3. (a) Find the four roots of the equation \(z ^ { 4 } = 8 ( \sqrt { 3 } + \mathrm { i } )\) in the form \(z = r \mathrm { e } ^ { \mathrm { i } \theta }\) (b) Show these roots on an Argand diagram.

3. Solve the equation $$z ^ { 3 } + 32 + 32 i \sqrt { 3 } = 0$$ giving your answers in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\) where \(r > 0\) and \(- \pi < \theta \leqslant \pi\)

3. (a) By writing \(\frac { \pi } { 12 } = \frac { \pi } { 3 } - \frac { \pi } { 4 }\), show that

1. \(\sin \left( \frac { \pi } { 12 } \right) = \frac { 1 } { 4 } ( \sqrt { 6 } - \sqrt { 2 } )\)
2. \(\cos \left( \frac { \pi } { 12 } \right) = \frac { 1 } { 4 } ( \sqrt { 6 } + \sqrt { 2 } )\) (b) Hence find the exact values of \(z\) for which $$z ^ { 4 } = 4 \left( \cos \frac { \pi } { 3 } + i \sin \frac { \pi } { 3 } \right)$$ Give your answers in the form \(z = a + i b\) where \(a , b \in \mathbb { R }\)

1. Solve the equation

$$z ^ { 5 } - 32 i = 0$$ giving each answer in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\) where \(0 < \theta < 2 \pi\)

4. $$z = - 8 + ( 8 \sqrt { } 3 ) \mathrm { i }$$

1. Find the modulus of \(z\) and the argument of \(z\). Using de Moivre's theorem,
2. find \(z ^ { 3 }\),
3. find the values of \(w\) such that \(w ^ { 4 } = z\), giving your answers in the form \(a + \mathrm { i } b\), where \(a , b \in \mathbb { R }\).