Sine, cosine and tangent each take an angle as input and produce a single number as output. Their inverses — arcsine, arccosine and arctangent — are supposed to run that process backward, taking a number and returning the angle that produced it. There's an immediate problem: sine, cosine and tangent are all periodic, meaning they repeat the same output for infinitely many different angles, so a given output number doesn't correspond to just one angle, it corresponds to infinitely many. An inverse function, by definition, has to return exactly one output for each input, which means arcsine can't actually undo sine in full — it has to make an arbitrary choice about which one of the infinitely many correct angles to return, and then stick to it consistently.
A restricted domain is what makes an inverse possible at all
The standard fix is to first restrict sine itself to a narrower domain — conventionally, angles from negative ninety degrees to positive ninety degrees — over which sine happens to be one-to-one, meaning each angle in that narrower range does produce a genuinely distinct output, with no two different angles in the restricted range sharing the same result. Only once sine has been artificially restricted this way does it actually have a proper, well-defined inverse: arcsine is then defined as the inverse of this restricted version of sine, not of the full, unrestricted, endlessly repeating original function. Cosine and tangent go through the same kind of restriction, each to its own conventionally chosen narrower range, before their own inverses, arccosine and arctangent, can be properly defined in the same way.
What arcsine gives back is a convention, not the only correct answer
This means arcsine's output should be understood as "the one specific angle, from within this particular chosen range, whose sine happens to equal this number" rather than "the actual angle that originally produced this number" in some more general, unrestricted sense — if the original angle that produced a given sine value actually fell outside arcsine's restricted range, arcsine will still return some other angle within its own range that happens to share the same sine value, not the original one. This matters in practice whenever a real problem's underlying angle could plausibly fall outside the conventional restricted range — a calculator's arcsine button always returns an angle within that fixed range, and a person interpreting the result has to think separately about whether the range's arbitrary convention actually matches the real angle their particular problem is asking about.
What we're still unsure about
The mathematical necessity of restricting sine's domain before it has a proper inverse, and the specific conventional range chosen for arcsine, arccosine and arctangent, are settled, precisely defined mathematics with no genuine ambiguity in the formal definitions themselves. Where real, if minor, inconsistency does show up across different fields and even different calculators is in exactly how some closely related conventions are handled — some disciplines and tools differ slightly in how they define related quantities like the range for arctangent of two separate arguments, used to correctly determine an angle's quadrant in applications like navigation or robotics — a genuinely practical source of occasional confusion, though it involves related conventions rather than any actual disagreement about the core inverse trigonometric functions themselves.
This sits inside Inverse Trigonometric Functions, one of seven topics in Trigonometry, one of seven domains in Mathematics, one of seventeen subjects the app can quiz you on.