A complete MAPLE procedure is designed to effectively implement an algorithm for approximating 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 exploit in some useful ways. Examples on calculating approximate values of the sine function with arbitrary accuracy for both rational and irrational arguments as well as drawing the graph of the piecewise approximate functions are presented. Moreover, from the approximate integration on [ a , b ] with integrands of the form x m sin x , 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 P ℓ of polynomials of degree at most ℓ, a subspace of L 2 ( a , b ) .
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机译：完整的枫木程序旨在有效地实现一种近似三角函数的算法。该算法给出了任意间隔的分段多项式近似，呈现了我们可以获得其部件的特殊分区，其中具有有限的有限数字的结束点的子内部，以及相应的近似多项式。该过程采用一系列的间隔多项式作为其输出，我们可以以某种有用的方式轻松利用。呈现了在rational和非理性参数的任意精度计算正弦函数的近似值的示例以及绘制分段近似函数的图形。此外，从XM SiN X的整体的近似积分，提出了另一个枫木过程，以找到最佳L 2的最佳L 2中所需的多项式估计值 - 在矢量空间P中的正弦函数最多的多项式的多项式，L 2的子空间（A，B）。
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