Question

What does `arccos(cos ((-2pi)/3))` equal?

Answer 1

`(2pi)/3`

Explanation:

It would look strange how is that possible! A question usually which pops up isn't `arccos(cos(A)) = A`.

To understand this we can use `cos(-theta) = cos(theta)`
Therefore `cos(-(2pi)/3) = cos((2pi)/3)`

Following it up with `arccos(cos(-(2pi)/3)) = arccos(cos((2pi)/3))`

That leads us to our answer `(2pi)/3`.

Let us understand the same in a different manner.
The range of `arccos(x)` is `[0, pi]`.
`cos((-2pi)/3) = -1/2`

The angle `(-2pi)/3` is not in the range of the function. So we select the angle in the range `[0,pi]` which gives cos(x) = -1/2 that works out to `(2pi)/3`.

Hope this clears your doubt.