已知cosα是方程3x2-x-2=0的根，且α是第三象限角，则sin(-α+3π2)cos(3π2+α)tan2(π-α)cos(π2+α)sin(π2-α)=（）A.916B.-916C.-54D.54

Question

已知cosα是方程3x²-x-2=0的根，且α是第三象限角，则

(sin(-α+(3π)/(2))cos((3π)/(2)+α)tan2(π-α))/(cos((π)/(2)+α)sin((π)/(2)-α))=（　　）

A.

(9)/(16)

B. -

(9)/(16)

C. -

(5)/(4)

D.

(5)/(4)

Answer 1

由3x²-x-2=0，得x=1或x=-(2)/(3)，
∵cosα是方程3x²-x-2=0的根，且α是第三象限角，
∴cosα=-(2)/(3)，sinα=-√(1-cos2α)=-√(1-(-(2)/(3))2)=-(√(5))/(3)，tanα=(√(5))/(2)．
则(sin(-α+(3π)/(2))cos((3π)/(2)+α)tan2(π-α))/(cos((π)/(2)+α)sin((π)/(2)-α))=(cosα•sinα•tan2α)/(-sinα•cosα)=-tan²α=-(5)/(4)．
故选：C．