Explain why sin^-1(sin(3pi/4))does not equal 3pi/4 when y=sin x and y=sin^-1 x are inverses.

Because they're not inverses, not exactly. is not invertible on its entire domain because it's not one-to-one. There are infinitely many values of such that , for example.

The standard function has a domain of and outputs values between and . This means that its inverse, , is indeed its inverse as long as .

is larger than and thus does not fall in the "invertible part" of the domain of . We have

which is a value between -1 and 1, so that

If we wanted to recover , we'd have to redefine or define a new inverse function that works on a different branch of the domain of .