Definition:Sine/Real Function

Definition
The real function $\sin: \R \to \R$ is defined as:


 * $\displaystyle \sin x = \sum_{n \mathop = 0}^\infty \left({-1}\right)^n \frac {x^{2n+1}}{\left({2n+1}\right)!} = x - \frac {x^3} {3!} + \frac {x^5} {5!} - \cdots$

It follows from Power Series over Factorial that these power series converge for all values of $x \in \R$.