已知函数f(x)=2sin(派x/8＋派/4)＋1

题目

已知函数f(x)=2sin(派x/8＋派/4)＋1
有三问,直接最后一问哈!令g(x)=f(x)＋f(-x),x属于R,求函数y=g(x)与x轴交点的横坐标.

答案

已知f(x)=2sin(πx/8＋π/4)＋1
g(x)=f(x)＋f(-x)=2sin(πx/8＋π/4)＋1+2sin(-πx/8＋π/4)＋1
=2[sin(πx/8＋π/4)+sin(-πx/8＋π/4)]＋2
=2*2 sin[(πx/8＋π/4-πx/8＋π/4)/2] cos[(πx/8＋π/4+πx/8-π/4)/2]＋2
=4 sin(π/4) cos(πx/8)＋2
=2*2^0.5 cos(πx/8)＋2 (2^0.5是根号2)
令g(x)=0,则2*2^0.5 cos(πx/8)＋2=0
cos(πx/8)=-2^0.5/2
πx/8=2kπ±3/4*π
x=16k±6 (k∈Z)