△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=______.

△ABC中,tanA,tanB是3x+8x-1=0的两个实数根,
可得tanA+tanB=-

8
3
,tanAtanB=-
1
3

所以tan(A+B)=
tanA+tanB
1-tanAtanB
=
-
8
3
4
3
=-2
,即tanC=-2,
所以4sin2C-3sinCcosC-5cos2C=9sin2C-3sinCcosC-5cos2C=
9sin2C-3sinCcosC-5cos2C
sin2C+cos2C
=
9tan2C-3tanC-5
tan2C+1
=5
故答案为5.