There is no single-key equivalent to the calc-arcsin function.
Instead, you must first press I (calc-inverse) to set
the Inverse Flag, then press S (calc-sin).
The I key actually toggles the Inverse Flag. When this flag
is set, the word ‘Inv’ appears in the mode line.
Likewise, the H key (calc-hyperbolic) sets or clears the
Hyperbolic Flag, which transforms calc-sin into calc-sinh.
If both of these flags are set at once, the effect will be
calc-arcsinh. (The Hyperbolic flag is also used by some
non-trigonometric commands; for example H L computes a base-10,
instead of base-e, logarithm.)
Command names like calc-arcsin are provided for completeness, and
may be executed with x or M-x. Their effect is simply to
toggle the Inverse and/or Hyperbolic flags and then execute the
corresponding base command (calc-sin in this case).
The O key (calc-option) sets another flag, the
Option Flag, which also can alter the subsequent Calc command in
various ways.
The Inverse, Hyperbolic and Option flags apply only to the next Calculator command, after which they are automatically cleared. (They are also cleared if the next keystroke is not a Calc command.) Digits you type after I, H or O (or K) are treated as prefix arguments for the next command, not as numeric entries. The same is true of C-u, but not of the minus sign (K - means to subtract and keep arguments).
Another Calc prefix flag, K (keep-arguments), is discussed elsewhere. See Keep Arguments.