I need to estimate the error in approximating of $\sin(x) \approx x-\frac{x^3}{6}$ while $|x| \le \frac{1}{2}$.
I assume that the error is somewhere between $x_0$ and $x$.
Since this is a 4th order approximation, then $(sin(x))''''$ = $sin(x)$, so $x_0 = 0$.
the error is supposed to be in the range of $[0,\frac{1}{2}]$.
lets sign the error as $c$, then $$\biggl|\frac{\sin(c)}{4!}x^4\biggr|$$ as I know, now i'm supposed to select c as the smallest value of $0$ as $c$ and $x$ as $\frac{1}{2}$ but then i get the result of $0$. Where is my mistake?