已知1/sin(120°-x)+1/sin(120°+x)=4根号3/3 求cosx的值

问题描述:

已知1/sin(120°-x)+1/sin(120°+x)=4根号3/3 求cosx的值

∵1/sin(120°-x)+1/sin(120°+x)=4√3/3,
∴3[sin(120°+x)+sin(120°-x)]=4√3sin(120°+x)sin(120°-x),
∴6sin120°cosx=2√3(cos2x-cos240°),
∴3sin(180°-60°)cosx=√3[cos2x-cos(180°+60°)],
∴3sin60°cosx=√3(cos2x+cos60°),
∴(3√3/2)cosx=√3[2(cosx)^2-1]+√3/2,
∴3cosx=4(cosx)^2-1,
∴4(cosx)^2-3cosx-1=0,
∴(4cosx+1)(cosx-1)=0,
∴cosx=-1/4,或cosx=1,