已知f(x)=sinwx+acoswx满足f(0)=√3且fx的图像相邻两条对称轴间的距离为π①求a与w②若f(α)=1,α∈(-π/2,π/2)求cos(α-5π/12)

Question

已知f(x)=sinwx+acoswx满足f(0)=√3且fx的图像相邻两条对称轴间的距离为π①求a与w②若f(α)=1,α∈(-π/2,π/2)求cos(α-5π/12)

Answer 1

f(x)=sinwx+acoswx
f(0)=a=√3
f(x)=sinwx+√3coswx
T=2π
w=2π/T=2π/2π=1
f(x)=sinx+√3cosx=2(1/2sinx+√3/2cosx)=2sin(x+π/3)
f(α)=2sin(α+π/3)=1
sin(α+π/3)=1/2
α∈(-π/2,π/2)
(α+π/3)∈(-π/6,5π/6)
cos(α+π/3)=√3/2 或-√3/2
cos(α-5π/12)=cos[(α+π/3)-3π/4]=cos(α+π/3)cos3π/4+sin(α+π/3)sin4π/3
=√3/2*(-√2/2)+1/2*√2/2=(√2-√6)/4或(√2+√6)/4
所以,cos(α-5π/12)=(√2+√6)/4 或 (√2-√6)/4