已知函数f(x)=2cosxsin(x+π/3)-根号3sin^2x+sinxcosx 当x∈[0,π/4]时,f(x)
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已知函数f(x)=2cosxsin(x+π/3)-根号3sin^2x+sinxcosx 当x∈[0,π/4]时,f(x)的值域
![已知函数f(x)=2cosxsin(x+π/3)-根号3sin^2x+sinxcosx 当x∈[0,π/4]时,f(x)](/uploads/image/z/9169403-59-3.jpg?t=%E5%B7%B2%E7%9F%A5%E5%87%BD%E6%95%B0f%28x%29%3D2cosxsin%28x%2B%CF%80%2F3%29-%E6%A0%B9%E5%8F%B73sin%5E2x%2Bsinxcosx+%E5%BD%93x%E2%88%88%5B0%2C%CF%80%2F4%5D%E6%97%B6%2Cf%28x%29)
f(x)=2cosxsin(x+π/3)-√3*(sinx)^2+sinxcosx
=2cosx[sinxcos(π/3)+cosxsin(π/3)]-√3*(sinx)^2+sinxcosx
=sinxcosx+√3*(cosx)^2-√3*(sinx)^2+sinxcosx
=2sinxcosx+√3*[(cosx)^2-(sinx)^2]
=sin2x+√3*cos2x
=2sin(2x+π/3)
x∈[0,π/4]
2x∈[0,π/2]
2x+π/3∈[π/3,5π/6]
sin(2x+π/3)∈[1/2,1]
2sin(2x+π/3)∈[1,2]
故f(x)的值域是[1,2]
如果不懂,请Hi我,祝学习愉快!
=2cosx[sinxcos(π/3)+cosxsin(π/3)]-√3*(sinx)^2+sinxcosx
=sinxcosx+√3*(cosx)^2-√3*(sinx)^2+sinxcosx
=2sinxcosx+√3*[(cosx)^2-(sinx)^2]
=sin2x+√3*cos2x
=2sin(2x+π/3)
x∈[0,π/4]
2x∈[0,π/2]
2x+π/3∈[π/3,5π/6]
sin(2x+π/3)∈[1/2,1]
2sin(2x+π/3)∈[1,2]
故f(x)的值域是[1,2]
如果不懂,请Hi我,祝学习愉快!
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