设sin18=x,cos36=cos2*18=1-2(sin18)^2=1-2x^2,(cos18)^2=1-x^2
sin36=2xcos18
sin18cos36+cos18sin36=cos36
x(1-2x^2)+2x(1-x^2)=1-2x^2
4x^3-2x^-3x+1=0
4x^3-2x^-2x-x+1=02x(2x^2-x-1)-(x-1)=02x(2x+1)(x-1)-(x-1)=0
(x-1)(4x+2x-1)=0
解此方程得:sin18=x=(-1+√5)/4(x=(-1-√5)/4<0舍去)
级别:大师
3月18日23:26∵sin36°=cos54°
即sin(2×18°)=cos(3×18°)
2sin18°cos18°=4(cos18°)^3-3cos18°
∵cos18°≠0
∴2sin18°=4(cos18°)^2-3
整理得4(sin18°)^2+2sin18°-1=0
解得sin18°=(根号5-1)/4
解法2.令x=18°
∴cos3x=sin2x
∴4(cosx)^3-3cosx=2sinxcosx
∵cosx≠0
∴4(cosx)^2-3=2sinx
∴4sinx2+2sinx-1=0,
又0