作业帮把函数sinx展开成（x-π/4)的幂函数
来源：学生作业帮 编辑：作业帮 分类：数学作业 时间：2024/08/28 15:16:10
把函数sinx展开成（x-π/4)的幂函数
sinx=x-x^3/3!+x^5/5!-……
cosx=1-x^2/2!+x^4/4!-……
故
sinx=sin(x-π/4+π/4)=sin(x-π/4)cosπ/4+cos(x-π/4)sinπ/4
=√2/2*[sin(x-π/4)+cos(x-π/4)]
=√2/2*[(x-π/4)-(x-π/4)^3/3!+(x-π/4)^5/5!-……+1-(x-π/4)^2/2!+(x-π/4)^4/4!-……]
=√2/2*[1+(x-π/4)-(x-π/4)^2/2!-(x-π/4)^3/3!+(x-π/4)^4/4!+(x-π/4)^5/5!-……]
cosx=1-x^2/2!+x^4/4!-……
故
sinx=sin(x-π/4+π/4)=sin(x-π/4)cosπ/4+cos(x-π/4)sinπ/4
=√2/2*[sin(x-π/4)+cos(x-π/4)]
=√2/2*[(x-π/4)-(x-π/4)^3/3!+(x-π/4)^5/5!-……+1-(x-π/4)^2/2!+(x-π/4)^4/4!-……]
=√2/2*[1+(x-π/4)-(x-π/4)^2/2!-(x-π/4)^3/3!+(x-π/4)^4/4!+(x-π/4)^5/5!-……]