已知函数f（x）= - √3sin^2x+sinxcosx 设α∈（0,π）,f（α）=0 ,求sinα的值.

题目

已知函数f（x）= - √3sin^2x+sinxcosx 设α∈（0,π）,f（α）=0 ,求sinα的值.

答案

f(x)=-√3sin²x+sinxcosx
=-√3(1-cos2x)/2+(1/2)sin2x
=(√3/2)cos2x+(1/2)sin2x-√3/2
=sin(2x+π/3)-√3/2
f(α)=sin(2α+π/3)-√3/2=0
sin(2α+π/3)=√3/2
因为α∈（0,π）,所以2α+π/3∈（π/3,2π+π/3）,
所以2α+π/3=2π/3,α=π/6
sinα=1/2