已知函数f(x)=2cos^2+2根号3sinxcosx(1)求函数f(x)在【-π/6,π/3】上的值域(2)在三角形ABC中,若f(C)=2,2sinB=cos(A-C)-cos(A+C),求tanA的值
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![已知函数f(x)=2cos^2+2根号3sinxcosx(1)求函数f(x)在【-π/6,π/3】上的值域(2)在三角形ABC中,若f(C)=2,2sinB=cos(A-C)-cos(A+C),求tanA的值](/uploads/image/z/1628076-12-6.jpg?t=%E5%B7%B2%E7%9F%A5%E5%87%BD%E6%95%B0f%28x%29%3D2cos%5E2%2B2%E6%A0%B9%E5%8F%B73sinxcosx%281%29%E6%B1%82%E5%87%BD%E6%95%B0f%28x%29%E5%9C%A8%E3%80%90-%CF%80%2F6%2C%CF%80%2F3%E3%80%91%E4%B8%8A%E7%9A%84%E5%80%BC%E5%9F%9F%EF%BC%882%EF%BC%89%E5%9C%A8%E4%B8%89%E8%A7%92%E5%BD%A2ABC%E4%B8%AD%2C%E8%8B%A5f%28C%29%3D2%2C2sinB%3Dcos%28A-C%29-cos%28A%2BC%29%2C%E6%B1%82tanA%E7%9A%84%E5%80%BC)
已知函数f(x)=2cos^2+2根号3sinxcosx(1)求函数f(x)在【-π/6,π/3】上的值域(2)在三角形ABC中,若f(C)=2,2sinB=cos(A-C)-cos(A+C),求tanA的值
已知函数f(x)=2cos^2+2根号3sinxcosx
(1)求函数f(x)在【-π/6,π/3】上的值域
(2)在三角形ABC中,若f(C)=2,2sinB=cos(A-C)-cos(A+C),求tanA的值
已知函数f(x)=2cos^2+2根号3sinxcosx(1)求函数f(x)在【-π/6,π/3】上的值域(2)在三角形ABC中,若f(C)=2,2sinB=cos(A-C)-cos(A+C),求tanA的值
(1)f(x)=2cos²x+2√3sinxcosx
=2cos²x+√3sin2x
=cos2x+1+√3sin2x
=2sin(2x+π/6)+1
∵x∈[-π/6,π/3].∴2x+π/6∈[-π/6,π]
∴f(x)∈[0,3]
(2)∵cos(A-C)-cos(A+C),
=cosAcosC+sinAsinC-(cosAcosC-sinAsinC)
=2sinAsinC
=2sinB
∴sinAsinC=sinB
f(C)=2sin(2C+π/6)+1=2
所以sin(2C+π/6)=1/2
即2C+π/6=π/6或5π/6
当2C+π/6=π/6,C=0
不符舍去
所以2C+π/6=5π/6,C=π/3
sinAsinC=sinB=sin(A+C)
即(√3sinA)/2=sin(A+π/3)=(sinA)/2+(√3cosA)/2
(√3-1)sinA=√3cosA
tanA=√3/(√3-1)= 3-√3
f(x)=2根号3sinxcosx+2cos^2x-1=根号3*sin2x+cos2x=2(cos派/6*sin2x+sin派/6*cos2x)=2sin(2x+派/6) g(x)=|f(x)|=|2sin(2x+派/6)|