(2013•梅州二模)已知函数f(x)=2cos2x+23sinxcosx.
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(2013•梅州二模)已知函数f(x)=2cos
![(2013•梅州二模)已知函数f(x)=2cos2x+23sinxcosx.](/uploads/image/z/16482657-57-7.jpg?t=%EF%BC%882013%E2%80%A2%E6%A2%85%E5%B7%9E%E4%BA%8C%E6%A8%A1%EF%BC%89%E5%B7%B2%E7%9F%A5%E5%87%BD%E6%95%B0f%28x%29%EF%BC%9D2cos2x%2B23sinxcosx%EF%BC%8E)
(1)函数f(x)=2cos2x+2
3sinxcosx=1+cos2x+
3sin2x=2sin(2x+
π
6)+1,
∴函数的最小正周期为:
2π
2=π.
(2)∵f(C)=2,∴2sin(2C+
π
6)+1=2,
∴sin(2C+
π
6)=
1
2
∵0<C<π,
∴
π
6<2C+
π
6<2π+
π
6,
2C+
π
6=
5π
6,C=
π
3;
∵2sinB=cos(A-C)-cos(A+C)=2sinAsinC,
∴sin(A+C)=sinAsinC,
即:sinAcosC+cosAsinC=sinAsinC,
即:tanA=
sinC
sinC−cosC
=
sin
π
3
sin
π
3−cos
π
3
=
3
2
3sinxcosx=1+cos2x+
3sin2x=2sin(2x+
π
6)+1,
∴函数的最小正周期为:
2π
2=π.
(2)∵f(C)=2,∴2sin(2C+
π
6)+1=2,
∴sin(2C+
π
6)=
1
2
∵0<C<π,
∴
π
6<2C+
π
6<2π+
π
6,
2C+
π
6=
5π
6,C=
π
3;
∵2sinB=cos(A-C)-cos(A+C)=2sinAsinC,
∴sin(A+C)=sinAsinC,
即:sinAcosC+cosAsinC=sinAsinC,
即:tanA=
sinC
sinC−cosC
=
sin
π
3
sin
π
3−cos
π
3
=
3
2
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