已知函数fx=sin(2x-π/6)+2cosx^2-1,在三角形abc中,内角ABC的对边分别

f(x)=sin(2x-π/6)+2(cosx)^2-1
=(√3/2)sin2x-(1/2)cos2x+cos2x
=(√3/2)sin2x+(1/2)cos2x
=sin(2x+π/6),
∴f(A)=sin(2A+π/6)=1/2,
∴2A+π/6=5π/6,
∴A=π/3.
b,a,c成等差数列,
∴a=(b+c)/2,
向量AB×向量AC=cbcosA=bc/2=9,
∴bc=18,
由余弦定理,a^2=b^2+c^2-bc=(b+c)^2-3bc=4a^2-54,
∴54=3a^2,a^2=18,a=3√2.