1)求函数的最小正周期以及对称轴方程(2)求函数在区间[-pai/12,pai/2]上的值域
问题描述:
1)求函数的最小正周期以及对称轴方程(2)求函数在区间[-pai/12,pai/2]上的值域
已知函数f(x)=cos(2x-pai/3)+2sin(x-pai/4)*sin(x+pai/
最后的是sin(x+pai/4)
答
f(x)=cos(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)
=cos2xcosπ/3+sin2xsinπ/3 +2sin(x-π/4)cos(x-π/4)
=(1/2)*cos2x+(√3/2)sin2x+sin(2x-π/2)
=(1/2)*cos2x+(√3/2)sin2x-sin(π/2-2x)
=(1/2)*cos2x+(√3/2)sin2x-cos2x
=(√3/2)sin2x-(1/2)cos2x
=cos(π/6)sin2x-sin(π/6)cos2x
=sin(2x-π/6)
∴最小正周期为:2π/2=π
因为y=sinx的对称轴为:kπ+π/2,k∈Z
∴2x-π/6=kπ+π/2
x=kπ/2 +π/3即为f(x)=sin(2x-π/6)的对称轴.
2)
x∈[-π/12,π/12]
2x∈[-π/6,π/6]
2x-π/6∈[-π/3,0]
∴sin(2x-π/6)∈[-√3/2,0]
函数在区间[-π/12,π/12]上的值域为:[-√3/2,0]