What is the difference between a relation and a function when finding the inverses of trig ratios?

Question

My pre calc teacher wants us to know the differance between Arcsin(1/2) and arcsin(1/2). He says the capital letter is a function and the lowercase letter is a relation.

Answer 1

The inverses of the trigonometric functions are not functions.

E.g., sin π/6 = 0.5, yet arcsin 0.5 has values of π/6 + 2kπ and 5π/6 + 2kπ.

To make the inverse a function the domain must be restricted. This is indicated by the capital A in Arcsin,