已知函数f(x)=2sin(x+α/2)cos(x+α/2)+2√3cos^2(x+α/2)-√3,α为常数

题目

已知函数f(x)=2sin(x+α/2)cos(x+α/2)+2√3cos^2(x+α/2)-√3,α为常数
求函数f(x)的最小正周期

答案

f(x)=2sin(x+α/2)cos(x+α/2)+2√3cos^2(x+α/2)-√3
f(x)=sin(2x+α)+2√3[cos(2x+α)+1]/2-√3
f(x)=sin(2x+α)+√3[cos(2x+α)+1]-√3
=sin(2x+α)+√3cos(2x+α)
=2×[1/2sin(2x+α)+√3/2cos(2x+α)]
=2[sin(2x+α)·cosπ/3+cos(2x+α)·sinπ/3]
=2sin(2x+α+π/3) ∴ ω = 2
T=2π/|ω|=π