已知函数f(x)=2cos(x+三分之派)【sin(x+三分之派)-根号3cos(x+三分之派)】,求函数的最小正周期和单调递减区间
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![已知函数f(x)=2cos(x+三分之派)【sin(x+三分之派)-根号3cos(x+三分之派)】,求函数的最小正周期和单调递减区间](/uploads/image/z/924780-12-0.jpg?t=%E5%B7%B2%E7%9F%A5%E5%87%BD%E6%95%B0f%28x%29%3D2cos%28x%2B%E4%B8%89%E5%88%86%E4%B9%8B%E6%B4%BE%EF%BC%89%E3%80%90sin%EF%BC%88x%2B%E4%B8%89%E5%88%86%E4%B9%8B%E6%B4%BE%EF%BC%89-%E6%A0%B9%E5%8F%B73cos%EF%BC%88x%2B%E4%B8%89%E5%88%86%E4%B9%8B%E6%B4%BE%EF%BC%89%E3%80%91%2C%E6%B1%82%E5%87%BD%E6%95%B0%E7%9A%84%E6%9C%80%E5%B0%8F%E6%AD%A3%E5%91%A8%E6%9C%9F%E5%92%8C%E5%8D%95%E8%B0%83%E9%80%92%E5%87%8F%E5%8C%BA%E9%97%B4)
已知函数f(x)=2cos(x+三分之派)【sin(x+三分之派)-根号3cos(x+三分之派)】,求函数的最小正周期和单调递减区间
已知函数f(x)=2cos(x+三分之派)【sin(x+三分之派)-根号3cos(x+三分之派)】,求函数的最小正周期和单调递减区间
已知函数f(x)=2cos(x+三分之派)【sin(x+三分之派)-根号3cos(x+三分之派)】,求函数的最小正周期和单调递减区间
f(x)=2cos(x+π/3)[sin(x+π/3)-√3cos(x+π/3)]
=2cos(x+π/3)sin(x+π/3)-2√3cos²(x+π/3)
=sin(2x+2π/3)-√3[1+cos(2x+2π/3)]
=sin(2x+2π/3)-√3cos(2x+2π/3)-√3
=2[(1/2)sin(2x+2π/3)-(√3/2)cos(2x+2π/3)]-√3
=2sin(2x+2π/3-π/3)-√3
=2sin(2x+ π/3)-√3
最小正周期Tmin=2π/2=π
当2kπ+π/2≤2x+π/3≤2kπ+3π/2 (k∈Z)时,函数单调递减,此时
kπ+π/12≤x≤kπ+7π/12 (k∈Z)
函数的单调递减区间为[kπ+π/12,kπ+7π/12] (k∈Z)
f(x)=2cos(x+π/3)[sin(x+π/3)-√3cos(x+π/3)],
=2cos(x+π/3)sin(x+π/3)-2√3cos^2(x+π/3)
=sin(2x+2π/3)-√3[2√3cos^2(x+π/3)-1]-√3
=sin(2x+2π/3)-√3cos(2x+2π/3)-√3
=2sin[(2x+2π/3)-π/3]-√3
=2s...
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f(x)=2cos(x+π/3)[sin(x+π/3)-√3cos(x+π/3)],
=2cos(x+π/3)sin(x+π/3)-2√3cos^2(x+π/3)
=sin(2x+2π/3)-√3[2√3cos^2(x+π/3)-1]-√3
=sin(2x+2π/3)-√3cos(2x+2π/3)-√3
=2sin[(2x+2π/3)-π/3]-√3
=2sin(2x+π/3)-√3
最小正周期=2π/2=π
2x+π/3∈ [2Kπ+π/2,2Kπ+3π/2]单调递减区间
x∈ [Kπ+π/12,2Kπ+7π/12]单调递减区间
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F(X)=平方根3cos ^ x +根,sinxcosx 3/2
=根号3 *(1 + cos2x)/ 2 + 3/2
所以sin2x/2-根F(发送/ 8)
=根号3 *(1 + COS派/ 4)/ 2 +的罪(派/ 4)/ 2 - 3/2的平方根
=平方根3 *(1/2 +的平方根的2/4)+ 2/4-平方根的2分之3的根
=(平方根6 + 2的平方根)/ 4