Function y=sin(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=sin(x)` more than once.
However, if we consider interval `[-pi/2,pi/2]` then on this interval function passes horizontal line test. On this interval function is increasing and takes all values between -1 and 1. Therefore, for function `y=sin(x),-pi/2<=x<=pi/2` there exists inverse function. This function is denoted by `y=arcsin(x)` (sometimes also denoted by `y=asin(x)` or `y=sin^(-1)(x)`) and is read as arcsine.
We can obtain graph of the function `y=arcsin(x)` from graph of the function `y=sin(x),-pi/2<=x<=pi/2` by symmetrically transforming it about line y=x (see figure).
Properties of the function `y=arcsin(x)`:
- Domain is segment [-1,1].
- Range is segment `[-pi/2,pi/2]`.
- Function is odd: `arcsin(-x)=-arcsin(x)`.
- Function is increasing.
From above it follows that records `y=arcsin(x)` and `x=sin(y),-pi/2<=y<=pi/2` are equivalent. If we substitute expression for y from first equation into second then we will obtain that `x=sin(arcsin(x))`.
Therefore, for any x from segment [-1,1] `sin(arcsin(x))=x,-pi/2<=arcsin(x)<=pi/2`.