已知a属于(0,π/2),且2sina的次方-sinacosa-3cosa的次方=0,求sin(a+π/4)/sin2a+cos2a+1的值

题目

已知a属于(0,π/2),且2sina的次方-sinacosa-3cosa的次方=0,求sin(a+π/4)/sin2a+cos2a+1的值

答案

2sina的次方-sinacosa-3cosa的次方=0
=>(2sina-3cosa)(sina+cosa)=0
a属于(0,π/2) =>sina+cosa不等于0
=> 2sina-3cosa=0 =>sina=3/根号13 cosa=2/根号13
sin(a+π/4)/(sin2a+cos2a+1)
=(sina+cosa)×根号2/2/(2sinacosa+2cosacosa)
=根号2/(4cosa)
=根号26/8
如果是sin(a+π/4)/sin2a+cos2a+1=(sina+cosa)×根号2/2/(2sinacosa)+2cosacosa
=5/根号13/2/(12/13)+8/13
=5根号13/24+8/13