## Thomas' Calculus 13th Edition

$\space Error \lt 1.67 \times 10^{-10}$
Recall that the Taylor series for $\sin x$ can be defined as: $\sin x= x-\dfrac{x^3}{3!}+\dfrac{ x^5}{5!}-....$ and $\sin x= x-\dfrac{x^3}{6}+\dfrac{ x^5}{120}-....$ Now, $|\dfrac{x^3}{6}| \lt |\dfrac{(10^{-3})^3}{6}|$ $\implies |\dfrac{x^3}{6}| \lt |\dfrac{(10^{-3})^3}{6}| = \dfrac{10^{-9}}{6}$ $\implies \space Error \lt |\dfrac{(10^{-3})^3}{3 !}| \approx 1.67 \times 10^{-10}$ So, $\space Error \lt 1.67 \times 10^{-10}$