Answer
Error $\leq 4.2 \times 10^{-6}$
Work Step by Step
The Taylor series for $\sin x $ can be defined as: $\sin x= x-\dfrac{x^3}{3!}+\dfrac{ x^5}{5!}-....$
We have $ f(x)=\sin x \\ f^{,}(x) =\cos x \\ f^{,}(x) =-\sin x\\ ......\\ f^{4}(x) =\sin x $
Need to find $|f^{4} | \leq M $.
So, $|R_n(x)| \leq M \dfrac{|x-a|^{n+1}}{(n+1)!}$
and $|R_3(0.1)| \leq (1) \times \dfrac{|0.1-0|^{4}}{4!} \approx 4.2 \times 10^{-6}$
So, Error $\leq 4.2 \times 10^{-6}$
