I understand why we restrict the domain, but why do we restrict the range? Why do we necessarily care so much for the inverse trig relations to be functions? Thanks!
Basically since it is easier and simpler to work with functions than it is to work with relations. What does it even mean for a relation to be continuous for instance, let alone differentiable. It's thus convenient to consider a function rather than a relation if possible. In the case of the trigonometric functions it does very little harm to restrict them to become functions.