We explicitly determine the best uniform polynomial approximation p∗n−1 to a class of rational functions of the form 1/(x − c)2 + K(a,b, c,n)/(x − c) on [a, b] represented by their Chebyshev expansion, where a, b, and c are real numbers, n − 1 denotes the degree of the best approximating polynomial, and K is a constant determined by a, b, c, and n. Our result is based on the explicit determination of a phase angle η in the representation of the approximation error by a trigonometric function. Moreover, we formulate an ansatz which offers a heuristic strategies to determine the best approximating polynomial to a function represented by its Chebyshev expansion. Combined with the phase angle method, this ansatz can be used to find the best uniform approximation to some more functions.