Formal definition of the function $Sin$ Which is the formal definition of the function $Sin$, starting from axioms of real numbers ?

I never found it in any book.

Not its Taylor serie, that is based upon the intuitive definition of $Sin$; I want the formal definition of $Sin$ without confusing drawings of circles, angles and straight lines.
One formal definition is via complex analysis, as used, e.g., by metamath (df-sin):
$$
\sin z=\frac{e^{iz}-e^{-iz}}{2i}.
$$

This, in turn, rests on the definition of $\exp$ (df-ef):
$$
\exp z=\sum_{k=0}^\infty\frac{z^k}{k!}.
$$

For more details see the definition of sum (df-sum) which, in the case of infinite sums, is defined in terms of limits (df-clim).
Maybe I found an answer in this web page:

http://math2.org/math/algebra/functions/trig/sin-cos-definition.htm