已知向量m=(根号3,1),向量n=(cos（x/3）,-sin（x/3）),记f(x)=2*向量m*向量n*sin（x/3）

向量m.向量n=√3cos(x/3)-sin(x/3).
f(x)=2m.n*sin(x/3).
=2[√3cos(x/3)-sin(x/3)]sin(x/3).
=2[√3sin(x/3)cos(x/3-sin^2(x/3)]
=2{(√3/2)2sin(x/3)cos(x/3)-[(1-cos2x/3)/2]}.
=2[(√3/2)sin(2x/3)+(1/2)cos(2x/3)]-1.
∴f(x)=2sin(2x/3+π/6)-1.
(1)∵x∈[π/4,π], ∴2x/3+π/6∈[0,π/2], sin(2x/3+π/6)在x∈[π/4,π/2]单调递增,在x∈[π/2,π],减.
∴sin(2x/3+π/6)在x=π/2处取得最大值1,即f(x)max=2*1-1=1. sin(2x/3+π/6)在x=π处取得最小值1/2.、
f(x)min=2*1/2-1=0.
∴f(x) 在x∈[π/4,π],f(x)∈[0,1]. ---所求函数的值域.
（2）∵f(C)=2sin(2C/3+π/6)=1. ∴2C/3+π/6=π/2. 2C/3=π/2-π/6=π/3, ∴C=π/2.
∵三角形的三边a,b,c成等比数列,∴b^2=ac, .
由正弦定理,得； (sinB)^2=sinAsinC. ∵sinC=sin90°=1.∴(sinB)^2=sinA. (1).
(sinB)^2=[sin(A+C)]^2
=sinAcosC+cosAsinC.
=sinA*cos90°+cosAsin90°.
∴(sinB)^2=cosA. (2).
∵ (1)=(2), ∴sinA=cosA.
sinA/cosA=1. (cosA≠0)
tanA=1,
∴∠A=π/4.