91Ó°ÊÓ

$Sinx,\mathrm{Ï€}$

Since for any function$f$with a derivative of order 4 at $x=\mathrm{Ï€}$, the fourth Taylor polynomial for $x=\mathrm{Ï€}$ is given by

$P_4\left(x\right)=f\left(\mathrm{Ï€}\right)+f^{\text{'}}\left(\mathrm{Ï€}\right)(x-\mathrm{Ï€})+\frac{f^{\text{'}\text{'}}\left(\mathrm{Ï€}\right)}{2!}(x-\mathrm{Ï€}{)}^2+\frac{f^{\text{'}\text{'}}\left(\mathrm{Ï€}\right)}{3!}(x-\mathrm{Ï€}{)}^3+\frac{f^{\text{'}\text{'}\text{'}\text{'}}\left(\mathrm{Ï€}\right)}{4!}(x-\mathrm{Ï€}{)}^4$

Therefore, first find the value of the function along with $f^{\text{'}}\left(x\right),f^{\text{'}\text{'}}\left(x\right),f^{\text{'}\text{'}\text{'}}\left(x\right)$and $f^{\text{'}\text{'}\text{'}\text{'}}\left(x\right)$at $x=\mathrm{Ï€}$

$$\begin{gathered} Thus,thevalueofthethefunctionstx=\mathrm{Ï€}is \\ f\left(\mathrm{Ï€}\right)=\sin \left(\mathrm{Ï€}\right) \\ =0 \\ Thederivativesofthefunctionf\left(x\right)=\sin \left(x\right)are \\ f\text{'}\left(x\right)=\frac{d}{dx}\left[\sin x\right] \\ =\cos x \\ So,atx=\mathrm{Ï€} \\ f\text{'}\left(\mathrm{Ï€}\right)=\cos \left(\mathrm{Ï€}\right) \\ =-1 \\ Also, \\ f\text{'}\text{'}\left(x\right)=\frac{d}{dx}\left[\cos x\right] \\ =-\sin x \\ So,atx=\mathrm{Ï€} \\ f\text{'}\left(\mathrm{Ï€}\right)=-\sin \left(\mathrm{Ï€}\right) \\ =0 \\ Again, \\ f\text{'}\text{'}\text{'}\left(x\right)=-\frac{d}{dx}\left[\sin x\right] \\ =-\cos x \\ So,atx=\mathrm{Ï€} \\ \mathrm{f}\text{'}\text{'}\text{'}\left(\mathrm{Ï€}\right)=-\cos \left(\mathrm{Ï€}\right) \\ =-(-1) \\ =1 \\ f\text{'}\text{'}\text{'}\text{'}\left(x\right)=-\frac{d}{dx}\left[\cos x\right] \\ =-(-\sin x) \\ =\sin x \\ So,atx=\frac{\mathrm{Ï€}}{2} \\ f\text{'}\left(\mathrm{Ï€}\right)=\sin \left(\mathrm{Ï€}\right) \\ =0 \end{gathered}$$

Therefore, the fourth Taylor polynomial for the function $f\left(x\right)=\sin x$is

$P_4\left(x\right)=0+-1·(x-\mathrm{π})+\frac{0}{2!}(x-\mathrm{π}{)}^2+\frac{1}{3!}(x-\mathrm{π}{)}^3+\frac{0}{4!}(x-\mathrm{π}{)}^4$

$=-(x-\mathrm{Ï€})+\frac{1}{3!}(x-\mathrm{Ï€}{)}^3=-(x-\mathrm{Ï€})+\frac{1}{6}(x-\mathrm{Ï€}{)}^3$