Question

a属于（π/2,π），1/sina+1/cosa=2根号2,则sin(2a+π/3)=______

Answer 1

(sina+cosa)/sinacosa=2√2
sina+cosa=2√2sinacosa
平方
sin²a+cos²a+2sinacosa=8(sinacosa)²
1+sin2a=8(1/2*sin2x)²=2sin²2a
(2sin2a+1)(sin2a-1)=0
π/2<a<π
π<2a<2π
sin2a<0
所以sin2a=-1/2
所以2a=7π/6或11π/6

1/sina+1/cosa>0
1/sina>-1/cosa
sina>0,cosa<0
所以sina<-cosa
sina+cosa<0
√2sin(a+π/4)<0
所以2a=11π/6舍去
2a=7π/6
2a+π/3=3π/2
所以sin(2a+π/3)=-1