△ABC中,tanA,tanB是3x2+8x-1=0的两个实数根,则4sin2C-3sinCcosC-5cos2C=_.
问题描述:
△ABC中,tanA,tanB是3x2+8x-1=0的两个实数根,则4sin2C-3sinCcosC-5cos2C=______.
答
△ABC中,tanA,tanB是3x+8x-1=0的两个实数根,
可得tanA+tanB=-
,tanAtanB=-8 3
,1 3
所以tan(A+B)=
=tanA+tanB 1-tanAtanB
=-2,即tanC=-2,-
8 3
4 3
所以4sin2C-3sinCcosC-5cos2C=9sin2C-3sinCcosC-5cos2C=
=
9sin2C-3sinCcosC-5cos2C
sin2C+cos2C
=59tan2C-3tanC-5
tan2C+1
故答案为5.