已知函数f(x)=sin((ωx+φ),(ω>o),f(x)图象相邻最高点和最低点的横坐标相差π/2,初相为π/6..(1),求f(x)的表达式.(2)求函数f(x)在[0,π/2]上的值域

Question

已知函数f(x)=sin((ωx+φ),(ω>o),f(x)图象相邻最高点和最低点的横坐标相差π/2,初相为π/6.
.(1),求f(x)的表达式.(2)求函数f(x)在[0,π/2]上的值域

Answer 1

(1)f(x)图象相邻最高点和最低点的横坐标相差π/2 ==> f(x)最小正周期为π/2×2=π ==> ω=2π/π=2
初相为π/6 ==> φ=π/6
所以，f(x)=sin(2x+π/6)
（2）x∈[0,π/2] ==> 2x+π/6∈[π/6,7π/6]
==> f(x)∈[1/2,1]