问题标题:
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
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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《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|小学数学问答-字典翻译问答网》
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