The standard basic real sine function is the D4364: Real function \begin{equation} \mathbb{R} \to [-1, 1], \quad x \mapsto \lim_{N \to \infty} \sum_{n = 0}^N (-1)^n \frac{x^{2n + 1}}{(2n + 1)!} \end{equation}

The standard basic real sine function is the D4364: Real function \begin{equation} \mathbb{R} \to [-1, 1], \quad x \mapsto \sum_{n = 0}^{\infty} (-1)^n \frac{x^{2n + 1}}{(2n + 1)!} \end{equation}

The standard basic real sine function is the D4364: Real function \begin{equation} \mathbb{R} \to [-1, 1], \quad x \mapsto x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \frac{x^9}{9!} - \dots \end{equation}