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.