作业帮一道简单的三角比证明题
来源：学生作业帮 编辑：作业帮 分类：数学作业 时间：2024/05/13 14:28:56
一道简单的三角比证明题
求证：4sin[(b-c)/2]sin[(a-b)/2]sin[(c-a)/2]=sin(b-c)+sin(a-b)+sin(c-a)
sin2x(cos2y-1)+sin2y(cos2x-1)
=-2sin2x(siny)^2-2sin2y(sinx)^2
有一个负号啊~但是另一边化简出来没有负号，难道
cos2y=2(siny)^2+1?是cos2y=-2(siny)^2+1
求证：4sin[(b-c)/2]sin[(a-b)/2]sin[(c-a)/2]=sin(b-c)+sin(a-b)+sin(c-a)
sin2x(cos2y-1)+sin2y(cos2x-1)
=-2sin2x(siny)^2-2sin2y(sinx)^2
有一个负号啊~但是另一边化简出来没有负号，难道
cos2y=2(siny)^2+1?是cos2y=-2(siny)^2+1
令(b-c)/2=x (a-b)/2=y
则x+y=(a-c)/2 (c-a)/2=-x-y
4sin[(b-c)/2]sin[(a-b)/2]sin[(c-a)/2]=sin(b-c)+sin(a-b)+sin(c-a)
即4sinxsinysin(-x-y)=sin2x+sin2y+sin(-2x-2y)
4sinxsinysin(x+y)=sin(2x+2y)-sin2x-sin2y
sin(2x+2y)-sin2x-sin2y
=sin2xcos2y+cos2xsin2y-sin2x-sin2y
=sin2x(cos2y-1)+sin2y(cos2x-1)
=-2sin2x(siny)^2-2sin2y(sinx)^2
4sinxsinysin(x+y)
=4sinxsiny(sinxcosy+cosxsiny)
=4(sinx)^2sinycosy+4(siny)^2sinxcosx
=2(sinx)^2sin2y+2(siny)^2sin2x
sin(2x+2y)-sin2x-sin2y=-4sinxsinysin(x+y)
则x+y=(a-c)/2 (c-a)/2=-x-y
4sin[(b-c)/2]sin[(a-b)/2]sin[(c-a)/2]=sin(b-c)+sin(a-b)+sin(c-a)
即4sinxsinysin(-x-y)=sin2x+sin2y+sin(-2x-2y)
4sinxsinysin(x+y)=sin(2x+2y)-sin2x-sin2y
sin(2x+2y)-sin2x-sin2y
=sin2xcos2y+cos2xsin2y-sin2x-sin2y
=sin2x(cos2y-1)+sin2y(cos2x-1)
=-2sin2x(siny)^2-2sin2y(sinx)^2
4sinxsinysin(x+y)
=4sinxsiny(sinxcosy+cosxsiny)
=4(sinx)^2sinycosy+4(siny)^2sinxcosx
=2(sinx)^2sin2y+2(siny)^2sin2x
sin(2x+2y)-sin2x-sin2y=-4sinxsinysin(x+y)