知函数f(x)=2根号3sinXcosX+2cos^2X-1(X属于R)

f(x)=2根号3sinxcosx+2cos^2x-1
=√3 sin2x+cos2x
=2sin(2x+π/6)

f(A)＝2sin(2A+π/6)=2/3
sin(2A+π/6)=1/3
0<2A<π
π/6<2A+π/6<7π/6
cos(2A+π/6)=(+/-)2根号2/3
sin2A=sin(2A+π/6-π/6)
=sin(2A+π/6)cosπ/6-cos(2A+π/6)sinπ/6

=1/3*根号3/2-(+/-)2根号2/3*1/2
=(根号3（-/+)2根号2)/6

f(x)=2√3sinXcosX+2cos^2X-1=√3sin2x+cos2x=2sin（2x+π/6）
f（A）=2/3 推出√3sin2A+cos2A=2/3
cos2A=2/3-√3sin2A
两边同时平方 1-sin2Asin2A=4/9-4√3/3sin2A+3sin2Asin2A
4sin2Asin2A-4√3/3sin2A-5/9=0
解出sin2A=（√3+2√2）/6或（√3-2√2）/6
而0所以sin2A>0
所以sin2A=（√3+2√2）/6

过程见图片