Often the domain of an inverse function is equal to the rang

Often the domain of an inverse function is equal to the range of its function [i.e., the domain of f^-1(x) is equal to the range of f(x)]. Each inverse trigonometric function, however, has a restricted range. Why is this necessary?

Solution

Inverse trigonometric relations are not functions ,as for any given input there exists more than one output. So, for a given number there exists more than one angle whose sine, cosine, etc., is that number.

To make it a funtion , the ranges are restricted  such that there is a one-to-one correspondence between the inputs and outputs of the inverse relations. With these restricted ranges, the inverse trigonometric relations become the inverse trigonometric functions.