要使f(x)=sin(x+θ)+cos(x+θ)为偶函数,求θ

来源：学生作业帮助网编辑：作业帮时间：2024/11/14 22:28:18

要使f(x)=sin(x+θ)+cos(x+θ)为偶函数,求θ
要使f(x)=sin(x+θ)+cos(x+θ)为偶函数,求θ

要使f(x)=sin(x+θ)+cos(x+θ)为偶函数,求θ
f(x)
=sin(x+θ)+cos(x+θ)
=(根号2)[(根号2/2)sin(x+θ)+(根号2/2)cos(x+θ)]
=(根号2)(cos(45)sin(x+θ)+sin(45)cos(x+θ))
=(根号2)sin(x+θ+45)
=(根号2)cos(45-x-θ)
=(根号2)cos(x+θ-45)
f(x)是偶函数,所以θ-45=360k
θ=360k+45,其中k是整数

令t=x+θ
f(x)=sint+cost
=√2cos(t-π/4)
所以f(x)=√2cos(x+t-π/4)
只有t-π/4为2π的整数倍的时候f(x)=sin(x+θ)+cos(x+θ)为偶函数
所以t=2kπ+π/4 k是整数

f(x)=sin(x+θ)+cos(x+θ)
=√2[√2sin(x+θ)/2+√2cos(x+θ)/2]
=√2[sin(x+θ)cos(π/4)+cos(x+θ)sin(π/4)]
=√2sin(x+θ+π/4)
f(0)=±√2
√2sin(θ+π/4)=±√2
sin(θ+π/4)=±1
θ+π/4=2kπ±π/2
θ=2kπ+π/4或θ=2kπ-3π/4

根据偶函数性质,f(-x)=f(x)
即sin(-x+θ)+cos(-x+θ)=sin(x+θ)+cos(x+θ)
解得θ=π/4+2kπ度或π5/4+2kπ度
需要知道的是x不等于0或180度,否则题目没意义!

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