(Solved): The \( 4^{\text {th }} \) degree Taylor polynomial for \( \sin (x) \) centered at \( a=\frac{\pi}{ ...

The \( 4^{\text {th }} \) degree Taylor polynomial for \( \sin (x) \) centered at \( a=\frac{\pi}{6} \) is given by \[ \sin (x)=\frac{1}{2}+\frac{\sqrt{3}}{2}\left(x-\frac{\pi}{6}\right)-\frac{1}{4}\left(x-\frac{\pi}{6}\right)^{2}-\frac{\sqrt{3}}{12}\left(x-\frac{\pi}{6}\right)^{3}+\frac{1}{48}\left(x-\frac{\pi}{6}\right)^{4}+R_{4}(x) \] Using this, estimate \( \sin \left(39^{\circ}\right) \) correct to five decimal places.