若f(x)=asin(x+π/4)+3sin(x-π/4)是偶函数,则实数a的值为
问题描述：
若f(x)=asin(x+π/4)+3sin(x-π/4)是偶函数,则实数a的值为
答
f(x)=asin(x+π/4)+3sin(x-π/4)=a(sinxcosπ/4+cosxsinπ/4)+3(sinxcosπ/4-cosxsinπ/4)=sinx(a+3)(根号2)/2+cosx(a-3)(根号2)/2
f(-x)=sin(-x)(a+3)(根号2)/2+cos(-x)(a-3)(根号2)/2
=-sinx(a+3)(根号2)/2+cosx(a-3)(根号2)/2 f（x）偶函数
=f(x)=sinx(a+3)(根号2)/2+cosx(a-3)(根号2)/2
2sinx(a+3)(根号2)/2=0 所以a=-3
答
f(-x)=asin(-x+π/4)+3sin(-x-π/4)
=-asin(x-π/4)-3sin(x+π/4)
f(x)=asin(x+π/4)+3sin(x-π/4)是偶函数
所以f(x)=f(-x)
所以asin(x+π/4)+3sin(x-π/4)=-asin(x-π/4)-3sin(x+π/4)
对应系数相等
明显a=-3