3sinb=sin(2a+b)可得sin(2a+b)-sinb=2sinb
由两角正弦差的公式:
sin(2a+b)-sinb=2cos[(2a+b+b)/2]sin[(2a+b-b)/2]=2cos(a+b)sina
因此:cos(a+b)sina=sinb,即:cos(a+b)=sinb/sina
则:sin(2a+b)=sin[a+(a+b)]=sinacos(a+b)+cosasin(a+b)……(*)
将sin(2a+b)=3sinb,cos(a+b)sina=sinb代入等式(*):
3sinb=sinb+cosasin(a+b),因此sin(a+b)=2sinb/cosa
则:tan(a+b)=sin(a+b)/cos(a+b)=(2sinb/cosa)/(sinb/sina)
=2(sinb/cosa)*(sina/sinb)=2sina/cosa=2tana