confusing. I hated how "inverse sin" aka "(sin^-1)" and 1/sin were completely different things.
That's a problem with how you were taught algebra, not trig. A function takes one or more values and produces an answer. Some functions only take a single value, like sine:
y = sin(
x). Another is the function for (scalar) multiplication:
y =
s(
x) =
s*
x.
An inverse for a function is one such that if
y =
f(
x) then
x =
f-1(
y). Clearly your experience with the sine function fit that mold. But if algebra never taught that multiplication is a function, then you wouldn't think about the fact that
x =
s-1(
y) =
y/
s. Then 1/sin is just the case where
s is sin(
z) but used in multiplication like a variable.