Question: Why is sin^(-1) the reciprocal of sin but sin^(-1)(x) is arcsin(x) and other related questions.

In the Programming Guide, Ch. 3 "Maple Expressions", subsection 3.13 "Other Expressions" there is a section called "Composition".

There is the following snippet

In particular, although the inverses of the circular functions are commonly denoted by a power-like notation in written mathematics, in Maple, for example, sin^(-1) denotes the reciprocal of the sin function, while sin@@(-1) denotes the arcsine (arcsin).

I opened a new worksheet to check this. I found the results confusing.

Consider (1) and (2). 

(1) is in accordance with the quoted snippet: sin^(-1) is the reciprocal of the sine function. But when we use this same expression as a function call, the function that is called is arcsine. Does this make sense to be this way?

Next, consider (3), (4), and (5), which I expected to be the same expressions as (1), (2) and (2), respectively. The only difference is that (1) and (2) use 2D math and (3), (4), and (5) use Maple input.

Both results using Maple input give as output the reciprocal of sine.

Why is there this difference between the 2D version and the Maple input version?

Consider (4) and (5). Why is it that we need to add parentheses for the argument x to be applied to the function?

Finally, what is the reasoning behind the syntax sin@@(-1) denoting arcsine?

@@ represents repeated composition. sin@@3 represents the function sin(sin(sin))). 

Is sin@@(-1) equal to arcsine simply by convention or is there some logical reason?

Download Repeated_Composition.mw