# How do you determine a taylor series approximation for f(x)=cos(x) where n=4 and a=3?

Feb 23, 2015

Hello,

The formula is

f(x) approx sum_{k=0}^n (f^((k))(a))/(k!) \cdot (x-a)^k

Apply these formula with $f = \cos$, $n = 4$ and $a = 3$. You get

$\cos \left(x\right) \approx \cos \left(3\right) + \cos ' \left(3\right) \left(x - 3\right) + \cos ' ' \left(3\right) {\left(x - 3\right)}^{2} / 2 + \cos ' ' ' \left(3\right) {\left(x - 3\right)}^{3} / 6 + \cos ' ' ' ' \left(3\right) {\left(x - 3\right)}^{4} / 24$.

Because you know the derivatives of $\cos$, you can write

$\cos \left(x\right) \approx \cos \left(3\right) - \sin \left(3\right) \left(x - 3\right) - \cos \left(3\right) {\left(x - 3\right)}^{2} / 2 + \sin \left(3\right) {\left(x - 3\right)}^{3} / 6 + \cos \left(3\right) {\left(x - 3\right)}^{4} / 24$.