Roots of unity with trigonometric identities

Let \(\omega = \cos \frac { 1 } { 5 } \pi + \mathrm { i } \sin \frac { 1 } { 5 } \pi\). Show that \(\omega ^ { 5 } + 1 = 0\) and deduce that $$\omega ^ { 4 } - \omega ^ { 3 } + \omega ^ { 2 } - \omega = - 1$$ Show further that $$\omega - \omega ^ { 4 } = 2 \cos \frac { 1 } { 5 } \pi \quad \text { and } \quad \omega ^ { 3 } - \omega ^ { 2 } = 2 \cos \frac { 3 } { 5 } \pi$$ Hence find the values of $$\cos \frac { 1 } { 5 } \pi + \cos \frac { 3 } { 5 } \pi \quad \text { and } \quad \cos \frac { 1 } { 5 } \pi \cos \frac { 3 } { 5 } \pi$$ Find a quadratic equation having roots \(\cos \frac { 1 } { 5 } \pi\) and \(\cos \frac { 3 } { 5 } \pi\) and deduce the exact value of \(\cos \frac { 1 } { 5 } \pi\).

9 In this question you must show detailed reasoning.
You are given the complex number \(\omega = \cos \frac { 2 } { 5 } \pi + \mathrm { i } \sin \frac { 2 } { 5 } \pi\) and the equation \(z ^ { 5 } = 1\).

1. Show that \(\omega\) is a root of the equation.
2. Write down the other four roots of the equation.
3. Show that \(\omega + \omega ^ { 2 } + \omega ^ { 3 } + \omega ^ { 4 } = - 1\).
4. Hence show that \(\left( \omega + \frac { 1 } { \omega } \right) ^ { 2 } + \left( \omega + \frac { 1 } { \omega } \right) - 1 = 0\).
5. Hence determine the value of \(\cos \frac { 2 } { 5 } \pi\) in the form \(a + b \sqrt { c }\) where \(a , b\) and \(c\) are rational numbers to be found.

8

1. 1. Show that \(\omega = \mathrm { e } ^ { \frac { 2 \pi \mathrm { i } } { 7 } }\) is a root of the equation \(z ^ { 7 } = 1\).
   2. Write down the five other non-real roots in terms of \(\omega\).
2. Show that $$1 + \omega + \omega ^ { 2 } + \omega ^ { 3 } + \omega ^ { 4 } + \omega ^ { 5 } + \omega ^ { 6 } = 0$$
3. Show that:
   1. \(\quad \omega ^ { 2 } + \omega ^ { 5 } = 2 \cos \frac { 4 \pi } { 7 }\);
   2. \(\cos \frac { 2 \pi } { 7 } + \cos \frac { 4 \pi } { 7 } + \cos \frac { 6 \pi } { 7 } = - \frac { 1 } { 2 }\). There are no questions printed on this page There are no questions printed on this page \section*{There are no questions printed on this page} \end{document}

8

1. Write down the five roots of the equation \(z ^ { 5 } = 1\), giving your answers in the form \(\mathrm { e } ^ { \mathrm { i } \theta }\), where \(- \pi < \theta \leqslant \pi\).
2. Hence find the four linear factors of $$z ^ { 4 } + z ^ { 3 } + z ^ { 2 } + z + 1$$
3. Deduce that $$z ^ { 2 } + z + 1 + z ^ { - 1 } + z ^ { - 2 } = \left( z - 2 \cos \frac { 2 \pi } { 5 } + z ^ { - 1 } \right) \left( z - 2 \cos \frac { 4 \pi } { 5 } + z ^ { - 1 } \right)$$
4. Use the substitution \(z + z ^ { - 1 } = w\) to show that \(\cos \frac { 2 \pi } { 5 } = \frac { \sqrt { 5 } - 1 } { 4 }\).

6

1. Find the three roots of \(z ^ { 3 } = 1\), giving the non-real roots in the form \(\mathrm { e } ^ { \mathrm { i } \theta }\), where \(- \pi < \theta \leqslant \pi\).
2. Given that \(\omega\) is one of the non-real roots of \(z ^ { 3 } = 1\), show that $$1 + \omega + \omega ^ { 2 } = 0$$
3. By using the result in part (b), or otherwise, show that:
   1. \(\frac { \omega } { \omega + 1 } = - \frac { 1 } { \omega }\);
   2. \(\frac { \omega ^ { 2 } } { \omega ^ { 2 } + 1 } = - \omega\);
   3. \(\left( \frac { \omega } { \omega + 1 } \right) ^ { k } + \left( \frac { \omega ^ { 2 } } { \omega ^ { 2 } + 1 } \right) ^ { k } = ( - 1 ) ^ { k } 2 \cos \frac { 2 } { 3 } k \pi\), where \(k\) is an integer.

8 The complex number \(\omega\) is given by \(\omega = \cos \frac { 2 \pi } { 5 } + \mathrm { i } \sin \frac { 2 \pi } { 5 }\).

1. 1. Verify that \(\omega\) is a root of the equation \(z ^ { 5 } = 1\).
   2. Write down the three other non-real roots of \(z ^ { 5 } = 1\), in terms of \(\omega\).
2. 1. Show that \(1 + \omega + \omega ^ { 2 } + \omega ^ { 3 } + \omega ^ { 4 } = 0\).
   2. Hence show that \(\left( \omega + \frac { 1 } { \omega } \right) ^ { 2 } + \left( \omega + \frac { 1 } { \omega } \right) - 1 = 0\).
3. Hence show that \(\cos \frac { 2 \pi } { 5 } = \frac { \sqrt { 5 } - 1 } { 4 }\).

6 Let \(w\) be the root of the equation \(z ^ { 7 } = 1\) that has the smallest argument \(\alpha\) in the interval \(0 < \alpha < \pi\) 6

1. Prove that \(w ^ { n }\) is also a root of the equation \(z ^ { 7 } = 1\) for any integer \(n\). 6
2. Prove that \(1 + w + w ^ { 2 } + w ^ { 3 } + w ^ { 4 } + w ^ { 5 } + w ^ { 6 } = 0\) 6
3. Show the positions of \(w , w ^ { 2 } , w ^ { 3 } , w ^ { 4 } , w ^ { 5 }\), and \(w ^ { 6 }\) on the Argand diagram below.
   [0pt] [2 marks] \includegraphics[max width=\textwidth, alt={}, center]{44e22a98-6424-4fb1-8a37-c965773cb7b6-08_835_898_1802_571} 6
4. Prove that $$\cos \frac { 2 \pi } { 7 } + \cos \frac { 4 \pi } { 7 } + \cos \frac { 6 \pi } { 7 } = - \frac { 1 } { 2 }$$

6 In this question you must show detailed reasoning.
You are given the complex number \(\omega = \cos \frac { 2 } { 5 } \pi + i \sin \frac { 2 } { 5 } \pi\) and the equation \(z ^ { 5 } = 1\).

1. Show that \(\omega\) is a root of the equation.
2. Write down the other four roots of the equation.
3. Show that \(\omega + \omega ^ { 2 } + \omega ^ { 3 } + \omega ^ { 4 } = - 1\).
4. Hence show that \(\left( \omega + \frac { 1 } { \omega } \right) ^ { 2 } + \left( \omega + \frac { 1 } { \omega } \right) - 1 = 0\).
5. Hence determine the value of \(\cos \frac { 2 } { 5 } \pi\) in the form \(a + b \sqrt { c }\) where \(a , b\) and \(c\) are rational numbers to be found. Total Marks for Question Set 4: 38

10 One root of the equation \(z ^ { 5 } - 1 = 0\) is the complex number \(\omega = \mathrm { e } ^ { \frac { 2 } { 5 } \pi \mathrm { i } }\).

1. Show that
   1. \(\quad \omega ^ { 5 } = 1\),
   2. \(\quad \omega + \omega ^ { 2 } + \omega ^ { 3 } + \omega ^ { 4 } = - 1\),
   3. \(\quad \omega + \omega ^ { 4 } = 2 \cos \frac { 2 } { 5 } \pi\), and write down a similar expression for \(\omega ^ { 2 } + \omega ^ { 3 }\).
   4. Using these results, find the values of \(\cos \frac { 2 } { 5 } \pi + \cos \frac { 4 } { 5 } \pi\) and \(\cos \frac { 2 } { 5 } \pi \times \cos \frac { 4 } { 5 } \pi\), and deduce a quadratic equation, with integer coefficients, which has roots $$\cos \frac { 2 } { 5 } \pi \quad \text { and } \quad \cos \frac { 4 } { 5 } \pi$$