已知函数f(x)=1-cos^2+2根号3sinxcosx-1/2cos2x,x属于R

Question

已知函数f(x)=1-cos^2+2根号3sinxcosx-1/2cos2x,x属于R

Answer 1

原式=1-(1+cos2x)/2+√3sin2x-1/2cos2x. x∈R.
=√3sin2x-cos2x+1/2.
=2[(√3/2sin2x-1/2cos2x+1/2.
∴f(x)=2sin(2x-π/6)+1/2.
f(x)的最小证周期T=2π/2=π;

当sin(2x-π/6)=1时,f(x)max=2*1+1/2=5/2；
当sin(2x-π/6)=-1时,f(x)min=2*(-1)+1/2=-3/2.
∴f(x)的值域：f(x)∈[-3/2,5/2}.