# Function y=arcsin(x)

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):

1. Domain is segment [-1,1].
2. Range is segment [-pi/2,pi/2].
3. Function is odd: arcsin(-x)=-arcsin(x).
4. 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.