前面的计算太复杂了,因此设法优化一下
从前面的分析我们已知问题的关键和难点都在于证明下面三角等式:
(sin54-cos54*sin24) / (sin24*sin54) = sqrt(3)
LHS = (sin54-sin24*cos54) / (sin54*sin24)
= (sin54-sin24*cos54) / (sin54*sin24)
= (sin54 - 1/2(sin78-sin30)) / (1/2*(cos30-cos78))
= (2sin54 - sin78 + sin30) / (cos30 - cos78)
我发现可以应用如下两个等式得解:
sin54 = sin18 + 1/2 ---- (1)
sin78 = 2sin18 + sqrt(3)*cos78 ---- (2)
LHS = (2sin54 - sin78 + sin30) / (cos30 - cos78)
= (2*sin18 + 1 - 2*sin18 - sqrt(3)*cos78 + sin30) / (cos30 - cos78)
= (1 - sqrt(3)*cos78 + 1/2) / (cos30 - cos78)
= (3/2 - sqrt(3)*cos78) / (cos30 - cos78)
= sqrt(3)(sqrt(3)/2 - cos78) / (cos30 - cos78)
= sqrt(3)
当然,还需证明(1)和(2)
(2)是很容易的:
sin78 - sqrt(3)*cos78 = 2*(1/2*sin78 - sqrt(3)/2)*cos78)
= 2 * (cos60*sin78 - sin60*cos78)
= 2 * sin(78-60)
= 2 * sin18
因此 sin78 = 2sin18 + sqrt(3)*cos78
(1)需要用到我们上次得到的结果: sin18 = (sqrt(5)-1)/4
因此 sin54 - sin18 = sin(3*18) - sin18
= 2sin18 - 4(sin18)^3
= 2sin18 * (1 - 2*(sin18)^2)
= 2sin18 * (1 - 2 * (6-2sqrt(5))/16)
= 2sin18 * ((sqrt(5)+1)/4)
= 2 * ((sqrt(5)-1)/4) * ((sqrt(5)+1)/4)
= 2 * (5-1)/(4*4)
= 1/2
即 sin54 = sin18 + 1/2