已知tan(a 派/4)=1/2.且-派/2

题目

已知tan(a 派/4)=1/2.且-派/2

答案

tan(α+π/4)=(tanα+tan(π/4))/(1-tanα*tan(π/4))=(tanα+1)/(1-tanα)=1/2
=> tanα=-1/3
-π/2<α<0 => sinα=-1/√10
∴ (2sin²α+sin2α)/cos(α-π/4)
=(2sin²α+2sinαcosα)/(cosαcos(π/4)+sinαsin(π/4))
=2sinα(sinα+cosα)/[sin(π/4)*(cosα+sinα)]
=2sinα/sin(π/4)
=-2/√10*2/√2
=-2√5/5
选A