函数f(x)=sin(x/2+π/3)cos(x/2+π/3)的最小正周期是

Question

函数f(x)=sin(x/2+π/3)cos(x/2+π/3)的最小正周期是 请详细解答，谢谢

Answer 1

（1）f（x）=2√
3sin（x/2+π/4）cos（x/2+π/4）—sin（x+π）
=3sin(x+π/2)+sinx
=2(3/2*cosx+1/2*sinx)
=2(sinπ/3cosx+cosπ/3sinx)
=2sin(x+π/3)
所以f(x)的最小正周期为2π
（2）g(x)=2sin(x+π/3-π/6)=2sin(x+π/6)
x=π/3时，g(x)有最大值，g(x)=2sin(π/3+π/6)=2sinπ/2=2
x=π时，g(x)有最小值，g(x)=2sin(π+π/6)=-2sinπ/6=-1

Answer 2

f(x)=sin(x/2+π/3)cos(x/2+π/3)
=1/2*2sin(x/2+π/3)cos(x/2+π/3)
（正弦
倍角公式
）
=1/2*sin(x+2π/3)
所以
最小正周期
为2π/1=2π.
注：y=Asin(ωx+ψ)的最小正周期为2π/ω.