sin 63 degrees and sin 81 degrees 的問題

sin 18 degrees = [sqrt (5) - 1 ] / 4
sin 54 degrees = [sqrt (5) + 1 ] / 4

should be over 4 .... not over 2 ....

if sin 18 degrees is given, sin 63 degrees can be calculated by using
sin 63 = sin (45 + 18) = sin45cos18 + cos45sin18
sin 18 = [sqrt (5) - 1 ] / 4
cos 18 = sqrt[ 10 + 2sqrt(5) ] /4
sin 63 = [sqrt(2) / 8 ] {-1 + sqrt(5) + sqrt[ 10 + 2sqrt(5) ] }

sin 81 = sin (45 + 36) = sin45cos36 + cos45sin36
sin 36 = 2sin18cos18 = sqrt[ 10 - 2sqrt(5) ] /4
cos36 = 1 - 2(sin18)^2 = [sqrt (5) + 1 ] / 4
sin 81 = [sqrt(2) / 8 ] {1 + sqrt(5) + sqrt[ 10 - 2sqrt(5) ] }

sin18 can be calculated by using the equation
sin (2*18) = cos (3*18)
2sin18cos18 = 4(cos18)^3 - 3cos18
2sin18 = 4(cos18)^2 - 3
2sin18 = 4 - 4(sin18)^2 - 3
4(sin18)^2 + 2sin18 - 1 = 0
sin18 = [-2 + sqrt(20) ] / 8 = [ -1 + sqrt (5) ] / 4