In 1958, Vietoris proved that σn(x) is positive for all n ≥ 1 and x ∈ (0, π). We establish the following refinement. The inequalitieshold for all natural numbers n and real numbers x ∈ (0, π) if and only if a0 = 0, a1 = −π 3 a4, a2 = 3π 2 a4, a3 = −3πa4, −1/π 3 ≤ a4 < 0.