The \(n^{th}\) derivative of \(sin x\) is, \[f^{(n)}(x)=sin(\frac{n \pi}{2}+x)\] which evaluates to \[n \quad at \quad x=0\] The Taylor series starts \[x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}+...;\] Therefore, the Taylor series is \[\sum_{n=0}^{\infty}(-1)^n \frac{x^{2n+1}}{(2n+1)!}, \quad x \quad \in \quad \mathbb{R}\]