已知函数f(x)=2cosxsin(x+π/3)+sinxcosx-√3sin²x,x∈R
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已知函数f(x)=2cosxsin(x+π/3)+sinxcosx-√3sin²x,x∈R
(1)求函数f(x)的最小正周期
(2)若存在x∈【0,5π/12】,使不等式f(x)>m成立,求实数m的取值范围
(1)求函数f(x)的最小正周期
(2)若存在x∈【0,5π/12】,使不等式f(x)>m成立,求实数m的取值范围
f(x)=2cosxsin(x+π/3)-√3sin²x+sinxcosx
=2cosx(1/2*sinx+√3/2*cosx) -√3sin²x+sinxcosx
= sinxcosx+√3cos²x-√3sin²x+sinxcosx
=2 sinxcosx+√3(cos²x-sin²x)
=sin2x+√3cos2x
=2sin(2x+π/3)
(1)
最小正周期=2π/2=π
(2)
x∈[0,5π/12]
2x+π/3∈[π/3,7π/6]
sin(2x+π/3)∈[-1/2,1]
2sin(2x+π/3)∈[-1,2]
f(x)>m成立
m
=2cosx(1/2*sinx+√3/2*cosx) -√3sin²x+sinxcosx
= sinxcosx+√3cos²x-√3sin²x+sinxcosx
=2 sinxcosx+√3(cos²x-sin²x)
=sin2x+√3cos2x
=2sin(2x+π/3)
(1)
最小正周期=2π/2=π
(2)
x∈[0,5π/12]
2x+π/3∈[π/3,7π/6]
sin(2x+π/3)∈[-1/2,1]
2sin(2x+π/3)∈[-1,2]
f(x)>m成立
m
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