## Thomas' Calculus 13th Edition

$\space Error \leq 4.2 \times 10^{-6}$
Recall the Taylor series for $\sin x= x-\dfrac{x^3}{3!}+\dfrac{ x^5}{5!}-....$ $f(x)=\sin x \\ f^{,}(x) =\cos x \\ f^{,}(x) =-\sin x\\ ......\\ f^{4}(x) =\sin x$ We will compute $|f^{4} | \leq M$. $|R_n(x)| \leq M \dfrac{|x-a|^{n+1}}{(n+1)!} \implies |R_3(0.1)| \leq (1) \times \dfrac{|0.1-0|^{4}}{4!} \approx 4.2 \times 10^{-6}$ Now, $\space Error \leq 4.2 \times 10^{-6}$