# How do you use the half angle identity to find exact value of cos(2pi/5)?

##### 1 Answer

Refer to explanation

#### Explanation:

Notice that sin(3*pi/5)) = sin(2*pi/5))

The triple angle sine formula is:$\sin \left(3 x\right) = 4 \sin \left(x\right) {\cos}^{2} \left(x\right) - \sin \left(x\right)$

The double angle sine formula is: $\sin \left(2 x\right) = 2 \sin \left(x\right) \cos \left(x\right)$

Since these are equal when $x = \frac{\pi}{5}$

$4 \sin \left(x\right) {\cos}^{2} \left(x\right) - \sin \left(x\right) = 2 \sin \left(x\right) \cos \left(x\right)$

Divide out the sin(x) term:

$4 {\cos}^{2} \left(x\right) - 1 = 2 \cos \left(x\right)$

This gives a simple quadratic in $\cos \left(x\right)$, substituting
$y = \cos \left(\frac{\pi}{5}\right)$:

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

Solving for y gives 2 solutions: $y = \frac{1 + \sqrt{5}}{4}$ and $y = \frac{1 - \sqrt{5}}{4}$. Since cos(pi/5) is positive: $\cos \left(\frac{\pi}{5}\right) = \frac{1 + \sqrt{5}}{4}$

Use the double angle formula: $\cos \left(2 x\right) = 2 {\cos}^{2} \left(x\right) - 1$

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