θ是三角形的一个内角,且关于x的函数f(x)=sin(x+θ)+cos(x-θ)是偶函数,求θ的值

题目

θ是三角形的一个内角,且关于x的函数f(x)=sin(x+θ)+cos(x-θ)是偶函数,求θ的值

答案

∵f(x)为偶函数∴f(x)=f(-x)∴sin(-x+θ)+cos(-x-θ)=sin(x+θ)+cos(x-θ)∴-sin(x-θ)+cos(x+θ)=sin(x+θ)+cos(x-θ)∴sin(x-θ)+cos(x-θ)=cos(x+θ)-sin(x+θ)∴√2sin(x-θ+π/4)=√2sin(x+θ-π/4)∴x-θ+π/4...