First pick a value of [tex]x[/tex] close to [tex]\dfrac{12\pi}{13}[/tex]. You should be fine with [tex]x=\pi[/tex].
The linear approximation of [tex]f(x)[/tex] at [tex]x=c[/tex] is given by
[tex]f(c)\approx f(a)+f'(a)(c-a)[/tex]
where [tex]x=a[/tex] is some fixed value close to [tex]x=c[/tex]. You have
[tex]f(x)=\cos x\implies f'(x)=-\sin x[/tex]
so
[tex]f\left(\dfrac{12\pi}{13}\right)\approx f(\pi)+f'(\pi)\left(\dfrac{12\pi}{13}-\pi\right)[/tex]
[tex]\cos\dfrac{12\pi}{13}\approx\cos\pi-\sin(\pi)\left(-\dfrac\pi{13}\right)[/tex]
[tex]\cos\dfrac{12\pi}{13}\approx-1[/tex]
The actual value is closer to -0.9709, so the approximation is decent.
