【50分】函数f(x)=sin^2(x/2)+√3sin(x/2)cos(x/2)-1/2,求单调区间!

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【50分】函数f(x)=sin^2(x/2)+√3sin(x/2)cos(x/2)-1/2,求单调区间!
【50分】函数f(x)=sin^2(x/2)+√3sin(x/2)cos(x/2)-1/2,求单调区间!

【50分】函数f(x)=sin^2(x/2)+√3sin(x/2)cos(x/2)-1/2,求单调区间!
f(x)=sin^2(x/2)+√3sin(x/2)cos(x/2)-1/2
=(1/2)(1-cosx)+(√3/2)sinx-1/2
=(√3/2)sinx-(1/2)cosx
=sinxcos30°-cosxsin30°
=sin(x-30°)=sin(x-π/6）
由2kπ-π/2≤x-π/6≤2kπ+π/2得2kπ-π/3≤x≤2kπ+2π/3
[2kπ-π/3,2kπ+2π/3]是增区间
由2kπ+π/2≤x-π/6≤2kπ+3π/2得2kπ+2π/3≤x≤2kπ+5π/3
[2kπ+2π/3,2kπ+5π/3 ]是减区间

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