## Calculus: Early Transcendentals 8th Edition

$0.61566$
Given: $f(x)=\sin x; a=\dfrac{\pi}{6}$ $(38^{\circ}) \times (\dfrac{\pi}{180^{\circ}}) \approx \dfrac{19\pi}{90}$ Here, we have $|R_n(x)|\leq \dfrac{M}{(n+1)!}|x-a|^{n+1}$ $\implies |R_4(x)|\leq \dfrac{1}{5!}|\dfrac{19\pi}{90}-\dfrac{\pi}{6}|^{5}\approx 0.00000041$ and $T_4(38^{\circ})=\dfrac{1}{2}+\dfrac{\sqrt 3}{2}(x-\dfrac{\pi}{6})-\dfrac{1}{4}(x-\dfrac{\pi}{6}))^2 +\dfrac{\sqrt 3}{12}-(x-\dfrac{\pi}{2})^6+\dfrac{1}{48}(x-\dfrac{\pi}{6})^4 \approx T_4(\dfrac{19\pi}{90})$ Thus, $T_4(38^{\circ}) \approx 0.61566$