已知函数f（x）=sin^2ωx+√3sinωxsin(ωx+π/2)（ω＞0）的最小正周期为π（1）求f(x)(2)当x∈[-π/12,π/2]时,求函数f（x）的值域
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问题描述：
已知函数f（x）=sin^2ωx+√3sinωxsin(ωx+π/2)（ω＞0）的最小正周期为π
（1）求f(x)
(2)当x∈[-π/12,π/2]时,求函数f（x）的值域
彭远芳回答：
(1)
f(x)=sin²(ωx)+√3sin(ωx)sin(ωx+π/2)=sin²(ωx)+√3sin(ωx)cos(ωx)
=2[(1/2)sin(ωx)+(√3/2)cos(ωx)]sin(ωx)
=2[cos(π/3)sin(ωx)+sin(π/3)cos(ωx)]sin(ωx)
=2sin(ωx+π/3)sin(ωx)
=2*(1/2)[cos(ωx+π/3-ωx)-cos(ωx+π/3+ωx)]
=-cos(2ωx+π/3)+cos(π/3)
=-cos(2ωx+π/3)+1/2
=cos(2ωx+π/3-π)+1/2
=cos(2ωx-2π/3)+1/2
最小正周期为π=2π/(2ω),ω=1
f(x)=cos(2x-2π/3)+1/2
(2)
x∈[-π/12,π/2]:
f(π/3)=cos(2π/3-2π/3)+1/2=1+1/2=3/2,此为最大值
x=π/3为f(x)最大值处的对称轴
π/3-(-π/12)=5π/12
π/2-π/3=2π/12
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