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Roots of unity and exact trigonometric values

mathematical

This question uses the fifth roots of unity to find an exact value.

Facts you may need

Roots of $z^n = 1$ : $\text{cis}\frac{2k\pi}{n}$ — Evenly spaced on the unit circle.
$z^5 - 1 = (z - 1)(z^4 + z^3 + z^2 + z + 1)$ — Why the roots sum to zero (no $z^4$ term in $z^5 - 1$).
$\cos(-\theta) = \cos\theta$ — Pairs of roots have equal real parts.

Solve

z^5 = 1

, giving solutions in polar form.

Explain why the sum of the five roots is 0. Hence show that

\cos\frac{2\pi}{5} + \cos\frac{4\pi}{5} = -\frac12

Using

\cos\frac{4\pi}{5} = 2\cos^2\frac{2\pi}{5} - 1

, find the exact value of

\cos\frac{2\pi}{5}