Trigonometric and inverse trigonometric functions, as the basic elementary functions, are widely used in optical measurement. In some cases, however, it is necessary to calculate their approximations in order to meet the need in hardware calculation or fast calculation. For this purpose, a method is presented in which the best approximation polynomial for a trigonometric or inverse trigonometric function, in a certain interval, is deduced based on the ∞-norm, and then the sectional approximation polynomials are obtained by use of trigonometric equations so that the values of the function in its whole domain of definition can be calculated. The coefficients and accuracies of the approximation polynomials for some typical trigonometric and inverse trigonometric functions are also provided. These results are used in optical fringe analysis, thus experimentally demonstrating the validity of the presented approach.