# How do you find the Taylor polynomial of degree n=4 for x near the point a=pi for the function cosx?

May 15, 2017

$- 1 + {\left(x - \pi\right)}^{2} / 2 - {\left(x - \pi\right)}^{4} / 24$

#### Explanation:

$\cos \left(\pi\right) = - 1$
1st Derivative: $- \sin \left(x\right) \text{ then } f ' \left(\pi\right) = 0$
2nd Derivative: $- \cos \left(x\right) \text{ then } f ' ' \left(\pi\right) = - 1$
3rd Derivative: $\sin \left(x\right) \text{ then " } f ' ' ' \left(\pi\right) = 0$
4th Derivative: $\cos \left(x\right) \text{ then } f ' ' ' ' \left(\pi\right) = - 1$

Odd terms except the first is zero then use only the even terms.
Putting it all together:

-1+(x-pi)^2/(2!)-(x-pi)^4/(4!)...