Question

How do you calculate `arccos(cos(4))`?

Answer 1

`arccos(cos(4))` is a value, `t` between `0` and `pi` whose cosine is equal to the cosine of `4`.

Note that `4> pi`, but `4 <= (3 pi)/2 ~~ 9.42/2 = 4.71`.

Thinking of `t=4` as the radian measure of an angle,
it corresponds to an angle in quadrant 3. and the reference angle is `4 - pi`.

The second quadrant angle with the same reference angle is `pi - (4-pi)` , which is `2 pi - 4`

So
`arccos(cos(4)) = 2 pi -4`

Note
All of the above is assuming that you did not intend to ask for `arccos(cos(4^@))` , which, of course, is `4^@`.