已知sinx+siny+sinz=0,cosx+cosy+cosz=0,求证cos(x-y)=cos(y-z)=cos(

来源：学生作业帮编辑：作业帮分类：数学作业时间：2024/05/16 10:54:56

已知sinx+siny+sinz=0,cosx+cosy+cosz=0,求证cos(x-y)=cos(y-z)=cos(z-x)

由sinx+siny+sinz=0可得(sinx+siny)²=(-sinz)²,展开为sin²x+sin²y+2sinxsiny=sin²z (1)
同理cosx+cosy+cosz=0可得cos²x+cos²y+2cosxcosy=cos²z (2)
(1)(2)式相加可得2+2(cosxcosy+sinxsiny)=1
所以cos(x-y)=cosxcosy+sinxsiny=-1/2
同样的也可以得到cos(y-z)=-1/2和cos(z-x)=-1/2
所以这三个余弦是相等的.

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