作业帮已知定义在区间[-π,2π/3]上的函数y=f（x）的图像关于直线x=-π/6对称,当下∈[-π/6,2π/3]时,函数f（x）=Asin（wx+φ）（A＞0,w＞0,-π/2＜φ＜π/2）,其图像如图所示.（1）求函数函数y=f（x）在[-π,2π/3]
来源：学生作业帮助网 编辑：作业帮 时间：2024/05/04 10:14:43
![已知定义在区间[-π,2π/3]上的函数y=f（x）的图像关于直线x=-π/6对称,当下∈[-π/6,2π/3]时,函数f（x）=Asin（wx+φ）（A＞0,w＞0,-π/2＜φ＜π/2）,其图像如图所示.（1）求函数函数y=f（x）在[-π,2π/3]](/uploads/image/z/615712-40-2.jpg?t=%E5%B7%B2%E7%9F%A5%E5%AE%9A%E4%B9%89%E5%9C%A8%E5%8C%BA%E9%97%B4%5B-%CF%80%2C2%CF%80%2F3%5D%E4%B8%8A%E7%9A%84%E5%87%BD%E6%95%B0y%3Df%EF%BC%88x%EF%BC%89%E7%9A%84%E5%9B%BE%E5%83%8F%E5%85%B3%E4%BA%8E%E7%9B%B4%E7%BA%BFx%3D-%CF%80%2F6%E5%AF%B9%E7%A7%B0%2C%E5%BD%93%E4%B8%8B%E2%88%88%5B-%CF%80%2F6%2C2%CF%80%2F3%5D%E6%97%B6%2C%E5%87%BD%E6%95%B0f%EF%BC%88x%EF%BC%89%3DAsin%EF%BC%88wx%2B%CF%86%EF%BC%89%EF%BC%88A%EF%BC%9E0%2Cw%EF%BC%9E0%2C-%CF%80%2F2%EF%BC%9C%CF%86%EF%BC%9C%CF%80%2F2%EF%BC%89%2C%E5%85%B6%E5%9B%BE%E5%83%8F%E5%A6%82%E5%9B%BE%E6%89%80%E7%A4%BA.%EF%BC%881%EF%BC%89%E6%B1%82%E5%87%BD%E6%95%B0%E5%87%BD%E6%95%B0y%3Df%EF%BC%88x%EF%BC%89%E5%9C%A8%5B-%CF%80%2C2%CF%80%2F3%5D)
已知定义在区间[-π,2π/3]上的函数y=f（x）的图像关于直线x=-π/6对称,当下∈[-π/6,2π/3]时,函数f（x）=Asin（wx+φ）（A＞0,w＞0,-π/2＜φ＜π/2）,其图像如图所示.（1）求函数函数y=f（x）在[-π,2π/3]
已知定义在区间[-π,2π/3]上的函数y=f（x）的图像关于直线x=-π/6对称,当下∈[-π/6,2π/3]时,
函数f（x）=Asin（wx+φ）（A＞0,w＞0,-π/2＜φ＜π/2）,其图像如图所示.
（1）求函数函数y=f（x）在[-π,2π/3]的表达式；
（2）求方程f（x）=（根号2）/2的解.
求详解,
已知定义在区间[-π,2π/3]上的函数y=f（x）的图像关于直线x=-π/6对称,当下∈[-π/6,2π/3]时,函数f（x）=Asin（wx+φ）（A＞0,w＞0,-π/2＜φ＜π/2）,其图像如图所示.（1）求函数函数y=f（x）在[-π,2π/3]
（1）
x∈[-π/6,2π/3]时,
f（x）=Asin（wx+φ）（A＞0,w＞0,-π/2＜φ＜π/2）
易知A=1,
由 2π/3-π/6=T/4=π/(2w) 得w=1
将x=π/2代入,π/6+φ=2kπ+π/2,
∴φ=2kπ+π/3,k∈Z
∵-π/2＜φ＜π/2
∴φ=π/3
∴f(x)=sin(x+π/3)
当x∈[-π,-π/6）时,
-π/3-x∈（-π/6,2π/3]
∴f(-π/3-x）=sin(-π/3-x+π/3)=-sinx
∵在[-π,2π/3]上,f(x)图像关于x=-π/6对称
∴x∈[-π,-π/6）时,
f(x)=f(-π/3-x）=-sinx
所以
f(x)={ sin(x+π/3),x∈[-π/6,2π/3]
{-sinx ,x∈[-π,-π/6）
（2）
x∈[-π/6,2π/3]时,f(x)=√2/2
即sin(x+π/3)=√2/2
∴x+π/3=π/4或x+π/3=3π/4
得到x=-π/12或x=5π/12
当 x∈[-π,-π/6）时,
f(x)=-sinx=√2/2,sinx=-√2/2
∴x=-π/4或x=-3π/4
∴f(x)=√2/2的解集为{-3π/4,-π/4,-π/12,5π/12}