# How do you find the exact values of cos 2pi/5?

Mar 16, 2016

$\cos \left(2 \frac{\pi}{5}\right) = \frac{- 1 + \sqrt{5}}{4}$

#### Explanation:

Here the most elegant solution I found in:

http://math.stackexchange.com/questions/7695/how-to-prove-cos-frac2-pi-5-frac-1-sqrt54

$\cos \left(4 \frac{\pi}{5}\right) = \cos \left(2 \pi - 4 \frac{\pi}{5}\right) = \cos \left(6 \frac{\pi}{5}\right)$

So if $x = 2 \frac{\pi}{5}$:

$\cos \left(2 x\right) = \cos \left(3 x\right)$

Replacing the cos(2x) and cos(3x) by their general formulae:

$\textcolor{red}{\cos \left(2 x\right) = 2 {\cos}^{2} x - 1 \mathmr{and} \cos \left(3 x\right) = 4 {\cos}^{3} x - 3 \cos x}$,

we get:

$2 {\cos}^{2} x - 1 = 4 {\cos}^{3} x - 3 \cos x$

Replacing $\cos x$ by $y$:

$4 {y}^{3} - 2 {y}^{2} - 3 y - 1 = 0$

$\left(y - 1\right) \left(4 {y}^{2} + 2 y - 1\right) = 0$

We know that $y \ne 1$, so we have to solve the quadratic part:

$y = \frac{- 2 \pm \sqrt{{2}^{2} - 4 \cdot 4 \cdot \left(- 1\right)}}{2 \cdot 4}$

$y = \frac{- 2 \pm \sqrt{20}}{8}$

since $y > 0$, $y = \cos \left(2 \frac{\pi}{5}\right) = \frac{- 1 + \sqrt{5}}{4}$