已知a属于(0,90°）,且2(sina)^2-sinacosa-3(cosa)^2=0.求sin(a+45）/(sin2a+cos2a+1)的值

题目

已知a属于(0,90°）,且2(sina)^2-sinacosa-3(cosa)^2=0.求sin(a+45）/(sin2a+cos2a+1)的值

答案

2(sina)^2-sinacosa-3(cosa)^2=0
(2sina-3cosa)(sina+cosa)=0
2sina-3cosa=0或sina+cosa=0
sina/cosa=3/2或sina/cosa=-1
tana=3/2或tana=-1(舍去)
cosa=2/√13
sin(a+45）/(sin2a+cos2a+1)
=√2/2(sina+cosa)/(sin2a+cos2a+1)
=√2/2(sina+cosa)/[(sina+cosa)^2+(cosa-sina)(sina+cosa)]
=(√2/2)/[(sina+cosa)+(cosa-sina)]
=(√2/2)/[2cosa]
=√2/(4cosa)
=√2/(4*2/√13)
=√2*√13/8
=√26/8