解sin36

Since sin36 = cos54

2(sin18)(cos18) = 4(cos18)^3 - 3cos18

2(sin18) = 4(cos18)^2 - 3

2(sin18) = 4[1-(sin18)^2] - 3

4(sin18)^2 + 2(sin18) - 1 = 0

sin18 = (√5 - 1) / 4

(cos18)^2 = 1 - (sin18)^2 = 1 - [(√5 - 1) / 4]^2

(cos18)^2 = (5 + √5) / 8
cos18 = (1/2)√ [(5 + √5) / 2 ]


So sin36 = sin (2*18)

= 2sin18cos18

= (√5 - 1) / 4 * √[(5+√5)/2]

化簡後可得 :

sin 36 = [ √(10 - 2√5)]/4

let x=sin36,
(4/9)^2+(5/9)^2=x^2
16/81+25/81=x^2
41/81=x^2
41^(1/2)/9=x

0.587785252