Question

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

Answer 1

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

Explanation:

If `theta` = `pi`/10, then 5`theta` = `pi`/2 `=>` cos3`theta` = sin2`theta`.[ cos (`pi` /2 - `alpha`) = sin`alpha`}.
`=>` 4`cos ^3` `theta` - 3cos`theta` = 2sin`theta`cos`theta` `=>` 4 `cos^2` `theta` - 3 =2 sin `theta`.
`=>` 4 ( 1 - `sin^2` `theta`) - 3 = 2 sin`theta`. `=>` 4`sin^2` `theta`+2sin`theta` - 1 = 0`=>`
sin`theta` =( `sqrt` 5 - 1 ) /4.
Now cos 2`theta` = cos `pi`/5 = 1 - 2`sin^2` `theta`, gives the result.

Answer 2

#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#.