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

2 Answers

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.

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#.