作业帮2sin(x-π/4)sin(x+π/4)要怎么算?
来源：学生作业帮 编辑：拍题作业网作业帮 分类：数学作业 时间：2024/04/28 19:41:42
2sin(x-π/4)sin(x+π/4)要怎么算?
怎么得到(sinx－cosx)(sinx＋cosx)的?是把sin(x-π/4)和sin(x+π/4拆开算好再乘以2么?没有简便的方法么
怎么得到(sinx－cosx)(sinx＋cosx)的?是把sin(x-π/4)和sin(x+π/4拆开算好再乘以2么?没有简便的方法么
解
2sin(x-π/4)sin(x+π/4)
=2sin(x-π/4)sin[π/2+(x-π/4)]
=2sin(x-π/4)cos(x-π/4)
=sin[2(x-π/4)]
=sin(2x-π/2)
=sin[-(π/2-2x)]
=-sin(π/2-2x)
=-cos2x
=-(cos²x-sin²x)
=sin²x-cos²x
=(sinx-cosx)(sinx+cosx)
再问： 为什么sin[π/2+(x-π/4)= cos(x-π/4)
再答： 这是一个诱导公式 sin(π/2+a)=cosa sin(π/2-a)=cosa cos(π/2+a)=-sina cos(π/2-a)=sina 这些必须记住
2sin(x-π/4)sin(x+π/4)
=2sin(x-π/4)sin[π/2+(x-π/4)]
=2sin(x-π/4)cos(x-π/4)
=sin[2(x-π/4)]
=sin(2x-π/2)
=sin[-(π/2-2x)]
=-sin(π/2-2x)
=-cos2x
=-(cos²x-sin²x)
=sin²x-cos²x
=(sinx-cosx)(sinx+cosx)
再问： 为什么sin[π/2+(x-π/4)= cos(x-π/4)
再答： 这是一个诱导公式 sin(π/2+a)=cosa sin(π/2-a)=cosa cos(π/2+a)=-sina cos(π/2-a)=sina 这些必须记住
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