Q14 (c) (ii)

Criteria	Marks
Provides correct solution	3
Finds the solutions to the degree 5 polynomial, or equivalent merit	2
Uses $\theta=\frac\pi{10}$ in the equation from part (i), or equivalent merit	1

Sample answer:

$\operatorname{Re}\left(e^{i\pi/10}\right)=\cos\frac\pi{10}$

When $\theta=\frac\pi{10}$, $\cos5\theta=\cos\frac\pi2=0$.

$16\cos^5\theta-20\cos^3\theta+5\cos\theta=0$
$\cos\theta(16\cos^4\theta-20\cos^2\theta+5)=0$

$\cos^2\theta=\frac{20\pm\sqrt{400-4(16)(5)}}{32}=\frac{5\pm\sqrt5}{8}$
$\therefore\cos\frac\pi{10}=\pm\sqrt{\frac{5\pm\sqrt5}{8}}$

The roots of $\cos5\theta=0$ are
$5\theta=\frac\pi2,\frac{3\pi}2,\frac{5\pi}2,\frac{7\pi}2,\frac{9\pi}2$
$\theta=\frac\pi{10},\frac{3\pi}{10},\frac\pi2,\frac{7\pi}{10},\frac{9\pi}{10}$

Thus the largest is
$\therefore\operatorname{Re}\left(e^{i\pi/10}\right)=\cos\frac\pi{10}=\sqrt{\frac{5+\sqrt5}{8}}.$