In this paper, a new class of rational quadratic/linear trigonometric Hermite functions with two shape parameters is proposed. Based on these Hermite functions, new improved first class of Side-Side (FCSS), second class of Side-Side (SCSS), first class of Side-Vertex (FCSV) and second class of Side-Vertex (SCSV) interpolation operators are proposed respectively, which can be used to construct C1 Coons surfaces over triangular domain. By altering the values of two shape parameters, the shape of the Coons surface patch can be adjusted flexibly, but without affecting the function values and partial derivatives of the boundaries. For constructing the triangular surface patches with the center of mass passing through a fixed point, we also give a center of mass function value control method, by which we can solve the corresponding shape parameter values. Moreover, we also apply these four improved interpolation operators to image interpolation. Compared with some widely used image interpolation methods, our methods achieve competitive performance.