Question

Approximate cos (12π/ 13) by using a linear approximation with f (x) = cos x.

Answer 1

First pick a value of $x$ close to $\dfrac{12\pi}{13}$. You should be fine with $x=\pi$.

The linear approximation of $f(x)$ at $x=c$ is given by

$f(c)\approx f(a)+f'(a)(c-a)$

where $x=a$ is some fixed value close to $x=c$. You have

$f(x)=\cos x\implies f'(x)=-\sin x$

so

$f\left(\dfrac{12\pi}{13}\right)\approx f(\pi)+f'(\pi)\left(\dfrac{12\pi}{13}-\pi\right)$
$\cos\dfrac{12\pi}{13}\approx\cos\pi-\sin(\pi)\left(-\dfrac\pi{13}\right)$
$\cos\dfrac{12\pi}{13}\approx-1$

The actual value is closer to -0.9709, so the approximation is decent.