已知函数f(x)=asin(kx+π/3)和φ(x)=btan(kx-π/3),k>0

【解】函数f(x) 和φ(x) 的最小正周期之和是3π/2,
则2π/k+π/k=3π/2,k=2.
由f(π/2)=φ(π/2)可得,asin(π+π/3) =btan(π-π/3),
-√3a/2=-√3b,a/2= b.
由f(π/4)=-√3φ(π/4)+1可得,asin(π/2+π/3) =-√3btan(π/2-π/3)+1,
即a/2=-b+1.
由此解得：b=1/2,a=1.
∴f(x)=sin(2x+π/3),φ(x)= 1/2tan(2x-π/3).