已知函数f(x)=cos(2x-3.14/3)+2sin(x-3.14/4)sin(x+3.14/4)

题目

已知函数f(x)=cos(2x-3.14/3)+2sin(x-3.14/4)sin(x+3.14/4)
(1)求函数f(x)的最小正周期和图像的对称轴方程
（2）求函数f(x)在区间[-3.14/12,3.14/2]上的值域

答案

已知函数f(x)=cos(2x-π/3)+2sin(x-π/4)sin(x+π/4)
(1)求函数f(x)的最小正周期和图像的对称轴方程
（2）求函数f(x)在区间[-π/12,π/2]上的值域
⑴f(x)=cos(2x-π/3)+2sin(x-π/4)sin(x+π/4)
=cos(2x-π/3)+√2sinx(x-π/4)√2sin(x+π/4)
=cos(2x-π/3)+(sinx-cosx)(sinx+cosx)
=cos(2x-π/3)+(sinx^2-cosx^2)
=cos(2x-π/3)+cos2x
=1/2cos2x-√3/2sin2x+cos2x
=3/2cos2x-√3/2sin2x
=√3(√3/2cos2x-1/2sin2x)
=√3sin(2x-π/3)
∴函数f(x)的最小正周期为π,图像的对称轴方程 x=kπ/2+π/6
（2）函数f(x)在区间[-π/12,π/2]上的值域 [-√3,√3]