已知函数f(x)=sin[x+(π/3)]sin[x+(π/2)].(1)求f(x)的最小正周期.(2)若g(x)=f(x)-[(根号3)/4]..求g(x)在区间[0,π/2]上的值域.

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已知函数f(x)=sin[x+(π/3)]sin[x+(π/2)].(1)求f(x)的最小正周期.
(2)若g(x)=f(x)-[(根号3)/4]..求g(x)在区间[0,π/2]上的值域.

f(x)=sin(x+π/3)*cosx
=sinxcos(π/3)cosx+cosxsin(π/3)*cosx
=(1/4)sin2x+(√3/4)(1+cos2x)
=(1/4)sin2x+(√3/4)cos2x+√3/4
=(1/2)sin(2x+π/3)+√3/4
(1)周期是T=2π/2=π
(2) g(x)=(1/2)sin(2x+π/3)
x∈[0,π/2]
∴ 2x+π/3∈[π/3,4π/3)
∴ 2x+π/3=π/2时,y有最大值1/2
2x+π/3=4π/3时,y有最小值-√3/4
即值域是[-√3/4,1/2]