Question

已知tan(π/4+α)=2,求1/(2sinαcosα+cos^2α)的值

Answer 1

tan(π/4+α)=sin(π/4+α)/cos(π/4+α)

=[sin(π/4)cosα+cos(π/4)sinα]/[cos(π/4)cosα-sin(π/4)sinα]

=[cosα+sinα]/[cosα-sinα]

=2
cosα+sinα=2cosα-2sinα

cosα=3sinα

tanα=1/3

1/(2sinαcosα+cos^2α)=1/[6(sinα)^2+9(sinα)^2]

=1/[15(sinα)^2]

2(sinα)^2=1-cos(2α)
cos(2α)=[1-(tanα)^2]/[1+(tanα)^2]
=[1-1/9]/[1+1/9]

=4/5

(sinα)^2=[1-cos(2α)]/2=1/10

1/(2sinαcosα+cos^2α)=1/[15(sinα)^2]=1/[15/10]=2/3