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

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Answer 1 · 2016-03-16T22:03:54

$cos(2pi/5)=(-1+sqrt(5))/4$

Here the most elegant solution I found in:

$cos(4pi/5)=cos(2pi-4pi/5)=cos(6pi/5)$

So if $x=2pi/5$ :

$cos(2x)=cos(3x)$

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

$color(red)(cos(2x)=2cos^2x-1 and cos(3x)=4cos^3x-3cosx)$ ,

we get:

$2cos^2x-1=4cos^3x-3cosx$

Replacing $cosx$ by $y$ :

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

$(y-1)(4y^2+2y-1)=0$

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

$y=(-2+-sqrt(2^2-4*4*(-1)))/(2*4)$

$y=(-2+-sqrt(20))/8$

since $y>0$ , $y=cos(2pi/5)=(-1+sqrt(5))/4$