Solving nonlinear systems is a core issue in computer algebra. Most known algorithms are derived for polynomials, rather than more general and more complicated transcendental functions. In this paper, we investigate a class of trigonometric--polynomial systems, extending polynomial ones. To solve it, we first decompose the real solutions of the input multivariate system into those of triangular sets, in which the base univariate functions are obtained. This task is finished by Wu's method. Then, inspired by the differentiation method, we present the real root isolation algorithm for the aforementioned univariate functions. Finally we apply the complete discrimination system (for polynomials) to lift the real roots to the real solutions of the input multivariate system. Our method is entirely established on algebraic manipulation and number theory. Hence it is absolutely exact and complete.