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
`(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((12\pi)/(13)\right)\approx f(\pi)+f'(\pi)\left((12\pi)/(13)-\pi\right)`

`\cos(12\pi)/(13)\approx\cos\pi-\sin(\pi)\left(-\frac\pi{13}\right)`

`\cos(12\pi)/(13)\approx-1`

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