函数f(x)=sin(x+θ)+根号3cos(x-θ)的图像关于y轴对称,则θ=
来源:学生作业帮助网 编辑:作业帮 时间:2024/08/17 07:51:28
![函数f(x)=sin(x+θ)+根号3cos(x-θ)的图像关于y轴对称,则θ=](/uploads/image/z/1697424-24-4.jpg?t=%E5%87%BD%E6%95%B0f%28x%29%3Dsin%28x%2B%CE%B8%29%2B%E6%A0%B9%E5%8F%B73cos%28x-%CE%B8%29%E7%9A%84%E5%9B%BE%E5%83%8F%E5%85%B3%E4%BA%8Ey%E8%BD%B4%E5%AF%B9%E7%A7%B0%2C%E5%88%99%CE%B8%3D)
函数f(x)=sin(x+θ)+根号3cos(x-θ)的图像关于y轴对称,则θ=
函数f(x)=sin(x+θ)+根号3cos(x-θ)的图像关于y轴对称,则θ=
函数f(x)=sin(x+θ)+根号3cos(x-θ)的图像关于y轴对称,则θ=
关于y轴对称就是说该函数是偶函数,就是f(x)=f(-x)即sin(x+θ)+根号3cos(x-θ)=sin(-x+θ)+根号3cos(-x-θ),sin(x+θ)+根号3cos(x-θ)=sin(-x+θ)+根号3cos(x+θ)【利用cosx= cos(-x)】两边同时除以2得(1/2)sin(x+θ)+(2分之根号3)cos(x-θ)=(1/2)sin(-x+θ)+(2分之根号3)cos(x+θ),1/2=sin30°,(2分之根号3)=cos30°,所以前式化为sin30°sin(x+θ)+cos30°cos(x-θ)=sin30°sin(-x+θ)+cos30°cos(x+θ),将sin30°sin(x+θ)移到右边sin30°sin(-x+θ)移到左边,得cos30°cos(x-θ)-sin30°sin(-x+θ)=cos30°cos(x+θ)-sin30°sin(x+θ),即cos30°cos(x-θ)+sin30°sin(x-θ)=cos30°cos(x+θ)-sin30°sin(x+θ),cos(x-θ-30°)=cos(30°+x+θ),得x-θ-30°=(30°+x+θ)+2kπ(k为整数),30°=π/6,所以θ=π/6+kπ(k为整数).
f(x)=sin(x-θ)+3cos(x-θ)的图象关于y轴对称
即偶函数,f(x)=f(-x)
sin(x-θ)+3cos(x-θ)
=sin(-x-θ)+3cos(-x-θ)
=-sin(x+θ)+3cos(x+θ)
=2sin[x+θ-(π/3)]
sinx图像对称轴π/2+kπ
y轴对称sin(x-π/2)
[x+θ-(π/3)+(π/3)-π/2]=-π/6
最后加上kπ
答案是 kπ-π/6