How do I evaluate cos(pi/5) without using a calculator?

2 Answers

Cos (pi /5) = cos 36° = (sqrt5 + 1)/4.

Explanation:

If theta = pi/10, then 5theta = pi/2 => cos3theta = sin2theta.[ cos (pi /2 - alpha) = sinalpha}.
=> 4 cos ^3 theta - 3costheta = 2sinthetacostheta=> 4 cos^2theta - 3 =2 sin theta.
=> 4 ( 1 - sin^2 theta) - 3 = 2 sintheta. => 4sin^2 theta+2sintheta - 1 = 0=>
sintheta =( sqrt 5 - 1 ) /4.
Now cos 2theta = cos pi/5 = 1 - 2sin^2 theta, gives the result.

Feb 13, 2016

Cos (pi/5) = (sqrt (5)+1)/4.

Explanation:

Let a = cos(pi/5), b = cos(2*pi/5). Thus cos(4*pi/5) = -a. From the double angle formulas:

b = 2a^2-1
-a = 2b^2-1

Subtracting,

a+b = 2(a^2-b^2) = 2(a+b)(a-b)

a+b is not zero, as both terms are positive, so a-b must be 1/2. Then

a-1/2 = 2a^2-1
4a^2-2a-1 = 0

and the only positive root is

a = cos (pi/5) = (sqrt(5)+1)/4.

And b = cos (2*pi/5) = a-1/2 = (sqrt(5)-1)/4.