We to find an error bound for the approximation and compare it to the actual error.
The exact value of \(\displaystyle{\sin{{\left({0.34}\right)}}}={0.33348709214081}.\)
So, the actual error is \(\displaystyle{\left({0.33348709214081}-{0.33349}\right)}={2.91}\times{10}^{{-{6}}}\)
The bound for the error on [0.30, 0.35] is given by,
\(\displaystyle{\left|{f{{\left({x}\right)}}}-{H}_{{5}}{\left({x}\right)}\right|}={\left|{\frac{{{{f}^{{6}}{\left(\xi\right)}}}}{{{6}!}}}{\left({x}={0.30}\right)}^{{2}}{\left({x}-{0.32}\right)}^{{2}}{\left({x}-{0.35}\right)}^{{2}}\right|}{N}{S}{K}={\left|{\frac{{-{\sin{{\left(\xi\right)}}}}}{{{720}}}}{\left({x}-{0.30}\right)}^{{2}}{\left({x}-{0.32}\right)}^{{2}}{\left({x}-{0.35}\right)}^{{2}}\right|}\)
When \(\displaystyle\xi\in{\left[{0.30},{0.35}\right]}.\) Evaluating this error term at x = 0.34 yields, |sin(0.34) - H_5 (0.34)| = |\frac{-sin(\xi)}{720}(0.04)^2 (0.02)^2 (-0.01)^2|

\leq |\frac{-sin(0.35)}{720}(0.04)^2 (0.02)^2 (-0.01)^2|

\leq 3.05 \times 10^{-14}ZSK This bound is not inconsistent with the actual error, because the approximation was computed using five-digit rounding arithmetic.

\leq |\frac{-sin(0.35)}{720}(0.04)^2 (0.02)^2 (-0.01)^2|

\leq 3.05 \times 10^{-14}ZSK This bound is not inconsistent with the actual error, because the approximation was computed using five-digit rounding arithmetic.