已知函数f(x)=2sin²(π/4+X)+√3cos2x-1,x∈R (1)求函数f(x)的最小正周期和单调增区间(2)在三角形ABC中,若f(C)=√3,2sinB=cos(A-C)-cos(A+C),求tanA的值
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![已知函数f(x)=2sin²(π/4+X)+√3cos2x-1,x∈R (1)求函数f(x)的最小正周期和单调增区间(2)在三角形ABC中,若f(C)=√3,2sinB=cos(A-C)-cos(A+C),求tanA的值](/uploads/image/z/1857567-39-7.jpg?t=%E5%B7%B2%E7%9F%A5%E5%87%BD%E6%95%B0f%28x%29%3D2sin%26sup2%3B%28%CF%80%2F4%2BX%29%2B%E2%88%9A3cos2x-1%2Cx%E2%88%88R+%EF%BC%881%EF%BC%89%E6%B1%82%E5%87%BD%E6%95%B0f%EF%BC%88x%EF%BC%89%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%E5%A2%9E%E5%8C%BA%E9%97%B4%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%3D%E2%88%9A3%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)=2sin²(π/4+X)+√3cos2x-1,x∈R (1)求函数f(x)的最小正周期和单调增区间(2)在三角形ABC中,若f(C)=√3,2sinB=cos(A-C)-cos(A+C),求tanA的值
已知函数f(x)=2sin²(π/4+X)+√3cos2x-1,x∈R (1)求函数f(x)的最小正周期和单调增区间
(2)在三角形ABC中,若f(C)=√3,2sinB=cos(A-C)-cos(A+C),求tanA的值
已知函数f(x)=2sin²(π/4+X)+√3cos2x-1,x∈R (1)求函数f(x)的最小正周期和单调增区间(2)在三角形ABC中,若f(C)=√3,2sinB=cos(A-C)-cos(A+C),求tanA的值
f(x) = 2sin²(π/4+x)+√3cos2x-1
= {1-cos[2(π/4+x)] + √3cos2x - 1
= -cos(π/2+2x) + √3cos2x
= -sin2x + √3cos2x
= -2(sin2xcosπ/3 - cos2xsinπ/3)
= -2sin(2x-π/3)
最小正周期 = 2π/2 = π
2x-π/3∈(2kπ+π/2,2kπ+3π/2),其中k∈Z时单调增
单调增区间:(kπ+5π/12,kπ+11π/12),其中k∈Z-2sin(2x-π/3)
C∈(0,π)
2C-π/3∈(-π/3,5π/3)
f(C)=√3
-2sin(2x-π/3)=√3
sin(2x-π/3)=-√3/2
2C-π/3=4π/3
C=5π/6
B=π-(A+C)
sinB=sin(A+C)
2sinB=cos(A-C)-cos(A+C)
2sin(A+C)=cos(A-C)-cos(A+C)
2sinAcosC+2cosAsinC = cosAcosC+sinAsinC-(cosAcosC-sinAsinC)
2sinAcosC+2cosAsinC = 2sinAsinC
-√3sinA+cosA= sinA
(√3+1)sinA=cosA
tanA = 1/(√3+1) = (√3-1)/2