已知3sinb=sin﹙2a+b﹚能否求出2tana-tan﹙a+b﹚的值?
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已知3sinb=sin﹙2a+b﹚能否求出2tana-tan﹙a+b﹚的值?
![已知3sinb=sin﹙2a+b﹚能否求出2tana-tan﹙a+b﹚的值?](/uploads/image/z/5844747-3-7.jpg?t=%E5%B7%B2%E7%9F%A53sinb%3Dsin%EF%B9%992a%2Bb%EF%B9%9A%E8%83%BD%E5%90%A6%E6%B1%82%E5%87%BA2tana-tan%EF%B9%99a%2Bb%EF%B9%9A%E7%9A%84%E5%80%BC%3F)
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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
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
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