A complete MAPLE procedure is designed to implement effectively an algorithm for approximating the trigonometric functions. The algorithm gives a piecewise polynomial approximation on an arbitrary interval, presenting a special partition that we can get its parts, subintervals with ending points of finite rational numbers, together with corresponding approximate polynomials. The procedure takes a sequence of pairs of interval-polynomial as its output that we can easily explore in some useful ways. Examples on calculating approximate values of the sine function with arbitrary accuracy for both of rational and irrational arguments as well as drawing the graph of the piecewise approximate functions will be presented. Moreover, from the approximate integration of integrands of the form $x^m\sin x$ on $[a,b]$, another MAPLE procedure is proposed to find the desired polynomial estimates in norm for the best $L^2$-approximation of the sine function in the vector space $\mathcal{P}_{\ell}$ of polynomials of degree at most $\ell$, a subspace of $L^2(a,b)$.