Question

已知函数f(x)=cox(2x-π/3)+2sin(x-π/4)sin(x+π/4) 1.求函数f(X)的最小正周期及函数图象的对称轴方程

Answer 1

已知函数f(x)=cox(2x-π/3)+2sin(x-π/4)sin(x+π/4) 1.求函数f(X)的最小正周期及函数
图象的对称轴方程
解：f(x)=cox(2x-π/3)+2sin(x-π/4)sin(x+π/4)
=cos(2x-π/3)+2sin(x-π/4)cos[π/2-(x+π/4)]=cos(2x-π/3)+2sin(x-π/4)cos(π/4-x)
=cos(2x-π/3)+2sin(x-π/4)cos(x-π/4)=cos(2x-π/3)+sin(2x-π/2)
=cos(2x-π/3)-sin(π/2-2x)=cos(2x-π/3)-cos2x
=-2sin(2x-π/6)sin(-π/6)=sin(2x-π/6)
故Tmin=2π/2=π. 对称轴：x=π/3+kπ/2, (k∈Z).

Answer 2

f(x)=cos(2x-π/3)+2sin(x-π/4)sin(x+π/4)=0.5cos2x-0.5√3sin2x+sin²x-cos²x=cos²x-0.5√3sin2x+sin²x-cos²x-0.5=sin²x-0.5-0.5√3sin2x=-0.5cos2x-0.5√3sin2x=-0.5sin(2x+π/6)
f(X)的最小正周期=π
函数图象的对称轴方程为2x+π/6=π+kπ即x=5π/12+kπ/2