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f(x)=cos(2x-π/3)+2sin(x-π/4)sin(x+π/4) 求最小正周期和图像的对称轴方程,

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f(x)=cos(2x-π/3)+2sin(x-π/4)sin(x+π/4) 求最小正周期和图像的对称轴方程,
f(x)=cos(2x-π/3)+2sin(x-π/4)sin(x+π/4) 求最小正周期和图像的对称轴方程,
f(x)=cos(2x-π/3)+2sin(x-π/4)sin(x+π/4)
= cos2x cosπ/3+ sin2x sinπ/3+2sin(x-π/4)cos(π/4-x)
= cos2x cosπ/3+ sin2x sinπ/3+2sin(x-π/4)cos(x-π/4)
= cos2x cosπ/3+ sin2x sinπ/3+ sin(2x-π/2)
= cos2x cosπ/3+ sin2x sinπ/3- cos2x
=1/2 cos2x+√3/2 sin2x- cos2x
=√3/2 sin2x-1/2 cos2x= sin(2x-π/6).
函数最小正周期是2π/2=π,
2x-π/6=kπ+π/2,k∈Z.
X= kπ/2+π/3,k∈Z.
对称轴方程是X= kπ/2+π/3,k∈Z.
再问: 为什么2sin(x-π/4)cos(x-π/4) 直接变成 sin(2x-π/2) ?
再答: 这是二倍角正弦公式2sinacosa=sin2a. 这里a=x-π/4, 2sin(x-π/4)cos(x-π/4) =sin[2(x-π/4)]=sin(2x-π/2)