Function y=cos(x) is not invertible (not one-to-one), because it fails horizontal line test. We can draw such horizontal line that it will intersect graph of `y=cos(x)` more than once.
However, if we consider interval `[0,pi]` then on this interval function passes horizontal line test. On this interval function is decreasing and takes all values between -1 and 1. Therefore, for function `y=cos(x),0<=x<=pi` there exists inverse function. This function is denoted by `y=arccos(x)` (sometimes also denoted by `y=acos(x)` or `y=cos^(-1)(x)`) and is read as arccosine.
We can obtain graph of the function `y=arccos(x)` from graph of the function `y=cos(x),0<=x<=pi` by symmetrically transforming it about line y=x (see figure).
Properties of the function `y=arccos(x)`:
- Domain is segment [-1,1].
- Range is segment `[0,pi]`.
- Function is neither even, nor odd.
- Function is decreasing.
From above it follows that records `y=arccos(x)` and `x=cos(y),0<=y<=pi` are equivalent. If we substitute expression for y from first equation into second then we will obtain that `x=cos(arccos(x))`.
Therefore, for any x from segment [-1,1] `cos(arccos(x))=x,0<=arccos(x)<=pi`.