# Function y=arccos(x)

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):

1. Domain is segment [-1,1].
2. Range is segment [0,pi].
3. Function is neither even, nor odd.
4. 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.