求证:函数f(x)sin(x+θ)为偶函数的充要条件是θ=kπ+π/2(k∈Z)φ=kπ+π/2(k∈Z)f(x)=sin(ωx+kπ+π/2) =coswx=cos(-wx)所以是充分条件必要条件f(x)=f(-x) sin(ωx+φ)=sin(-ωx+φ)sin(ωx+φ)+sin(-ωx+φ)=0……………………………………这步怎么来的?2sinφcoswx=0sinφ=0φ=kπ+π/2(k∈Z)
问题描述：
求证:函数f(x)sin(x+θ)为偶函数的充要条件是θ=kπ+π/2(k∈Z)
φ=kπ+π/2(k∈Z)
f(x)=sin(ωx+kπ+π/2)
=coswx=cos(-wx)所以是充分条件
必要条件f(x)=f(-x)
sin(ωx+φ)=sin(-ωx+φ)
sin(ωx+φ)+sin(-ωx+φ)=0……………………………………这步怎么来的?
2sinφcoswx=0
sinφ=0
φ=kπ+π/2(k∈Z)
答
证明函数f(x)=sin(x+θ)为偶函数,恒有f(x)=f(-x),即恒有sin(x+θ)=sin(-x+θ).恒有:sin(θ+x)-sin(θ-x)=0和差化积,可知,恒有2cosθsinx=0 恒有cosθ=0,θ=2kπ±(π/2).即θ=2kπ+(π/2)或θ=2kπ-(π/2)=2kπ-[π-...