Taylor Approximation: Example

What is $\sin{x}$ near 0?

$$\begin{aligned} f(x)&=\sin{x} \\ f(x)&\approx f(0)+f'(0)(x-0)+\frac{f''(0)}{2!}(x-0)^2+\frac{f'''(0)}{3!}(x-0)^3 \\ &=\sin{0}+(\cos(0))(x-0)-\frac{\sin{0}}{2!}(x-0)^2-\frac{\cos(0)}{3!}(x-0)^3 \\ &=0+x-0-\frac{x^3}{6} \\ &=x-\frac{x^3}{6} \end{aligned}$$ Expand to higher orders: $$f(x)=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}+\dots$$