高考问答snh48严佼君生写黑历史:一道三角函数题
来源：百度文库 编辑：高考问答 时间：2024/05/07 12:15:13
求sin（A+B）的值
∵sin(B+3π/4)= sin(π-π/4+B)= sin(B-π/4)=5/13
∴cos(B-π/4)=12/13
又∵cos(π/4-A)=3/5
π/2<A<3/4π/4
∴sin(π/4-A)=-4/5
sin(B-A)=sin[(B-π/4)+( π/4-A)]=sin(B-π/4)cos(π/4-A)+cos(B-π/4)sin(π/4-A)
=5/13*3/5+12/13*(-4/5)
=3/13-48/65=-33/65
Sin(3π/4+B)cos(π/4-A)=1/2[sin(3π/4+B+π/4-A)+sin(3π/4+b-π/4+A)]
=1/2[-sin(B-A)+cos(B+A)]=3/5*5/13=3/13
∴-sin(B-A)+cos(B+A)=6/13
Cos(B+A)=6/13-33/65=-3/65
∴sin(B+A)=√1-(3/65)^2或-√1-(3/65)^2
先求sin(A+π/4)
因为A+π/4 = π/2-(π/4-A)
所以sin(A+π/4)=cos(π/4-A)=3/5
因为 π/2<A<3π/4 所以 3π/4<A<π 所以 cos(A+π/4)=-4/5
因为0<B<π/2 所以，3π/4<3π/4+B<5π/4 所以cos(3π/4+B)<0 再根据 sin(3π/4+B)=5/13 得,cos(3π/4+B)=-12/13
sin(A+B+π)=sin(A+π/4 + 3π/4+B)=sin(A+π/4)cos(3π/4+B)+cos(A+π/4)sin(3π+B)=3/5*(-12/13)+(-4/5)*(5/13) =-56/65
所以sin(A+B)=56/65