How do you find the exact values of cos 4pi/5?

Question

22949 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2016-06-09T16:55:21

$=-(sqrt5+1)/4$

We are to evalute
$cos((4pi)/5)=cos(pi-pi/5)=-cos(pi/5)$

This problem of evaluating $cos(pi/5)$ can be solved by evaluating $sin(pi/10)$ in the following way.

Let $A=pi/10$

$=>5A=pi/2$

$=>3A=pi/2-2A$

$:.cos(3A)=cos(pi/2-2A)$

$=>4cos^3A-3cosA=sin2A=2sinAcosA$

$=>4cos^2A-3=2sinA$

$=>4-4sin^2A-3=2sinA$

$=>4sin^2+2sinA-1=0$

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

$=(-2+sqrt20)/8=(sqrt5-1)/4$

$:.sin(pi/10)=(sqrt5-1)/4$ ,

Now

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

$=-(1-2sin^2(pi/10))$

$=2sin^2(pi/10)-1$

$=2*((sqrt5-1)/4)^2-1$

$=2/16(5+1-2sqrt5)-1$

$=1/8*2*(3-sqrt5)-1$

$=(3-sqrt5)/4-1$

$=(3-sqrt5-4)/4$

$=-(sqrt5+1)/4$