麻煩高手解一下這題數學,謝謝!!?

∠B = ∠C = ( 180° - 108° ) / 2 = 36°
由正弦定理得 m / n = sin 108° / sin 36°
令 θ = 36°
m / n = sin 3θ / sin θ = ( 3*sin θ - 4*sin^3 θ ) / sin θ = 3 - 4*sin^2 θ
其次, 想辦法算出 m / n 之比值 :

5θ = 180°
cos 3θ = cos ( 180° - 2θ ) = - cos 2θ
4*cos^3 θ - 3*cos θ = 1 - 2*cos^2 θ

令 x = cos θ = cos 36°
4x^3 - 3x = 1 - 2x^2
4x^3 + 2x^2 - 3x - 1 = 0
(x+1)( 4x^2 - 2x - 1 ) = 0
x = - 1 (不合)
x = [ 2 ± √(4+16) ] / 8 = ( 1 ± √5 ) / 4 (負不合)

又 4x^2 - 2x - 1 = 0
所以 4x^2 = 2x + 1

m / n
= 3 - 4*sin^2 θ
= 3 - 4( 1 - cos^2 θ )
= 3 - 4( 1 - x^2 )
= - 1 + 4x^2
= - 1 + 2x + 1
= 2x
= 2 * ( 1 + √5 ) / 4
= ( 1 + √5 ) / 2

即 m : n = ( 1 + √5 ) : 2
故可令 m = ( 1 + √5 )k , n = 2k , 其中 k ≠ 0

x
= (m+n) / (m-n)
= (3+√5)k / [ (√5 - 1)k ]
= (3+√5)(√5 + 1) / 4
= ( 3√5 + 3 + 5 + √5 ) / 4
= 2 + √5

y
= (m+n)^2 / ( m n )
= [ (3+√5)k ]^2 / [ 2( 1 + √5 )k^2 ]
= ( 14 + 6√5 ) / ( 2 + 2√5 )
= ( 7 + 3√5 ) / ( 1 + √5 )
= ( 7 + 3√5 )( √5 - 1 ) / 4
= ( 7√5 - 7 + 15 - 3√5 ) / 4
= 2 + √5

z
= ( m / n )^3
= [ ( 1 + √5 ) / 2 ]^3
= ( 1 + 3*1*√5 + 3*1*5 + 5√5 ) / 8
= 2 + √5

所以 :
x = y = z ..... Ans