Sharpness and generalization of Jordan’s inequality and its application

Let θ≥2θ≥2 be a given real number, and aa, b∈Rb∈R be two parameters, and let Q(x;a,b,θ)=2π+a(πθ−(2x)θ)+b(πθ−(2x)θ)2. We determine the values a=2π−θ−1θ,b=(−π2+4+4θ)π−2θ−14θ2, which provide the best approximation: sinxx≈Q(x;2π−θ−1θ,(−π2+4+4θ)π−2θ−14θ2,θ),0