Why arcsin is called arc, and why sin to the minus one confuses people
Sine turns an angle into a ratio; its inverse turns the ratio back into an angle. Naming that reverse step has caused two centuries of muddle, from John Herschel's 1813 superscript notation to an international standard that in 2019 finally backed a single choice: the prefix arc.
Inverse trigonometric functions undo sine, cosine, tangent and their three reciprocal cousins, recovering an angle from a ratio. Engineers, navigators and physicists rely on them constantly. The arc prefix has a tidy geometric reason. Measured in radians on a circle of radius one, an angle and the length of the arc it cuts are the same number, so the angle whose cosine is x is also the arc whose cosine is x. Programmers shorten the names further, to asin, acos and atan.
Herschel's alternative wrote the inverse as the function raised to the power minus one. The trouble is that elsewhere sine squared means the sine multiplied by itself, so sine to the minus one looks like a reciprocal, one divided by the sine, rather than an inverse. Special names for reciprocals, such as secant for one over cosine, soften the problem, but some authors still warn against the notation. Writing the minus one inside square brackets is unambiguous yet rarely seen, and capitalising the first letter only adds confusion, since software such as Mathematica already uses capitals for the ordinary functions. The ISO 80000-2 standard now specifies the arc form alone.
A deeper difficulty is that none of the six trig functions is one-to-one. Sine, for instance, hits every value over and over as the angle keeps turning, so infinitely many angles share the same sine. To define a genuine inverse, mathematicians restrict attention to a principal branch, a stretch where each ratio corresponds to exactly one angle, and call the output the principal value.
Even that choice varies. Some authors pick a different range for arcsecant and arccosecant because it keeps certain formulas free of awkward plus-or-minus signs.
Source: Inverse trigonometric functions
