∵x∈[π/4,π/2]
∴2x-π/3∈[π/6,2π/3]∴sin(2x-π/3)∈[1/2,1]∴f(x)∈[2,3]∵不等式f(x)+m>0对任意x∈[π/4,π/2]恒成立,
∴m∈[-2,+∞0