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
- Prove that \(w ^ { n }\) is also a root of the equation \(z ^ { 7 } = 1\) for any integer \(n\).
6
- Prove that \(1 + w + w ^ { 2 } + w ^ { 3 } + w ^ { 4 } + w ^ { 5 } + w ^ { 6 } = 0\)
6 - 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 - Prove that
$$\cos \frac { 2 \pi } { 7 } + \cos \frac { 4 \pi } { 7 } + \cos \frac { 6 \pi } { 7 } = - \frac { 1 } { 2 }$$