Question

已知函数f(x)=sin(wx+φ)(w>0,0≦φ≦π)为偶函数,且其图象上相临的两对称轴之间的距离为2兀.1)求f(x)的解

已知函数f(x)=sin(wx+φ)(w>0,0≦φ≦π)为偶函数,且其图象上相临的两对称轴之间的距离为2兀（.1)求f(x)的解析式（2）若a属于(-π/3,π/2),且f(a+π/3)=1/3,求sin(2a+5π/3)

Answer 1

已知函数f(x)=sin(wx+φ)(w>0,0≦φ≦π)为偶函数,且其图象上相临的两对称轴之间的距离为2兀（.1)求f(x)的解析式（2）若a属于(-π/3,π/2),且f(a+π/3)=1/3,求sin(2a+5π/3)
(1)解析：∵函数f(x)=sin(wx+φ)(w>0,0≦φ≦π)为偶函数,且其图象上相临的两对称轴之间的距离为2兀
∴T/2=2π==>T=4π==>w=1/2
∴f(x)=sin(1/2x+φ)=sin(1/2(x+2φ))==>2φ=-π==>φ=-π/2
或f(x)=sin(1/2x+φ)=sin(1/2(x+2φ))==>2φ=π==>φ=π/2
∴f(x)=sin(1/2x+π/2)或f(x)=sin(1/2x-π/2)
(2)解析：∵a属于(-π/3,π/2),且f(a+π/3)=1/3
f(x)=sin(1/2x+π/2)==> f(a+π/3)=sin(1/2a+π/6+π/2)=1/3
设θ=1/2a+π/6+π/2==>a=2θ-4π/3
sin(2a+5π/3)= sin(2(2θ-4π/3)+5π/3)=sin(4θ-π)=-sin4θ
-π/3<2θ-4π/3<π/2==>π/2<θ<11π/12
∵sinθ=1/3==>cosθ=-2√2/3
sin4θ=2sin2θcos2θ=4sinθcosθ(1-2(sinθ)^2)=4*1/3*(-2√2/3)*(7/9)
=-8√2/9*(7/9)=-56√2/81
∴sin(2a+5π/3)= 56√2/81

Answer 2

同学。高中的？老子看了数学就烦