Q13 (c) (iii)

Criteria	Marks
Provides correct solution	3
Uses the connection between one of the solutions in part (i) and $\cos\frac{3\pi}{5}$	2
Finds the values of $u$, or equivalent merit	1

Sample answer:
By part (i), $z=e^{3i\pi/5}$ is a solution of $z^5+1=0$.

Since $e^{3i\pi/5}\ne-1$, by part (ii),
$u=e^{3i\pi/5}+\frac1{e^{3i\pi/5}}=2\cos\frac{3\pi}{5}$
is a solution of $u^2-u-1=0$.

This equation has two solutions, $\frac{1+\sqrt5}{2}$ and $\frac{1-\sqrt5}{2}$.

$\cos\frac{3\pi}{5}<0\quad\text{whereas}\quad\frac{1+\sqrt5}{2}>0$
$2\cos\frac{3\pi}{5}=\frac{1-\sqrt5}{2}$
$\therefore\cos\frac{3\pi}{5}=\frac{1-\sqrt5}{4}$