三角函数的公理化定义

AN AXIOMATIC DEFINITION OF TRIGONOMETRIC FUNCTIONS

设C(x)与S(x)是R→R上的映射,若满足函数方程C(x-y)=C(x)C(y)+S(x)S(y),且ⅴx∈(0,π),S(x)>0,S(π)=0,则称C(x)与S(x)分别为余弦函数与正弦函数。这样的定义揭示了三角函数的本质属性,克服了以往定义中的不足。

Let C(x) and S(x) be two R→R mappings. If they satisfie the functional equation C(x-y)=C(x)C(y)+S(x)S(y) and (?)x∈(0, π), S(x)>0, S(π)=0, then C(x) and S(x) can be called cosine function and sine function, respectively. Using this definition the nature of the trigonometric function is exposed and the imperfect in former definitions is dismissed.