Editing Inverse trigonometric functions (section)

== Notation ==
[[File:Arcsin and arccos as actual arc lengths.svg|thumb|For a circle of radius 1, arcsin and arccos are the lengths of actual arcs determined by the quantities in question.]]
{{see also|Trigonometric functions#Notation}}
Several notations for the inverse trigonometric functions exist. The most common convention is to name inverse trigonometric functions using an arc- prefix: {{math|arcsin(''x'')}}, {{math|arccos(''x'')}}, {{math|arctan(''x'')}}, etc.<ref name=Hall_1909/> (This convention is used throughout this article.)  This notation arises from the following geometric relationships:{{citation needed|date=January 2019}}
when measuring in radians, an angle of {{mvar|θ}} radians will correspond to an [[circular arc|arc]] whose length is {{mvar|rθ}}, where {{mvar|r}} is the radius of the circle. Thus in the [[unit circle]], the cosine of x function is both the arc and the angle, because the arc of a circle of radius 1 is the same as the angle. Or, "the arc whose cosine is {{mvar|x}}" is the same as "the angle whose cosine is {{mvar|x}}", because the length of the arc of the circle in radii is the same as the measurement of the angle in radians.<ref name=Americana_1912/> In computer programming languages, the inverse trigonometric functions are often called by the abbreviated forms {{char|asin}}, {{char|acos}}, {{char|atan}}.<ref>{{cite web | author = Cook, John D. | date = 2021-02-11 | title = Trig functions across programming languages | website = johndcook.com | type = blog | url = https://www.johndcook.com/blog/2021/02/11/trig-across-languages | access-date = 2021-03-10 }}</ref>

The notations {{math|sin<sup>&minus;1</sup>(''x'')}}, {{math|cos<sup>&minus;1</sup>(''x'')}}, {{math|tan<sup>&minus;1</sup>(''x'')}}, etc., as introduced by [[John Herschel]] in 1813,<ref name=Cajori/><ref name=Herschel_1813/> are often used as well in English-language sources,<ref name=Hall_1909/> much more than the also [[Iterated function#Definition|established]] {{math|sin<sup>[&minus;1]</sup>(''x'')}}, {{math|cos<sup>[&minus;1]</sup>(''x'')}}, {{math|tan<sup>[&minus;1]</sup>(''x'')}} – conventions consistent with the notation of an [[inverse function]], that is useful (for example) to define the [[Multivalued_function|multivalued]] version of each inverse trigonometric function: <math>\tan^{-1}(x) = \{\arctan(x) + \pi k \mid k \in \mathbb Z\} ~.</math> However, this might appear to conflict logically with the common semantics for expressions such as {{math|sin<sup>2</sup>(''x'')}} (although only {{math|sin<sup>2</sup> ''x''}}, without parentheses, is the really common use), which refer to numeric power rather than function composition, and therefore may result in confusion between notation for the [[reciprocal (mathematics)|reciprocal]] ([[multiplicative inverse]]) and [[inverse function]].<ref>{{cite web |title=Inverse trigonometric functions |department=Wiki |website=Brilliant Math & Science (brilliant.org) |url=https://brilliant.org/wiki/inverse-trigonometric-functions/ |access-date=2020-08-29 |lang=en-us}}</ref>

The confusion is somewhat mitigated by the fact that each of the reciprocal trigonometric functions has its own name — for example, {{math|(cos(''x''))<sup>&minus;1</sup> {{=}} sec(''x'')}}. Nevertheless, certain authors advise against using it, since it is ambiguous.<ref name=Hall_1909/><ref name=Korn_2000/> Another precarious convention used by a small number of authors is to use an [[UPPERCASE|uppercase]] first letter, along with a “{{math|&minus;1}}” superscript: {{math|Sin<sup>&minus;1</sup>(''x'')}}, {{math|Cos<sup>&minus;1</sup>(''x'')}}, {{math|Tan<sup>&minus;1</sup>(''x'')}}, etc.<ref name=Bhatti_1999/> Although it is intended to avoid confusion with the [[reciprocal (mathematics)|reciprocal]], which should be represented by {{math|sin<sup>&minus;1</sup>(''x'')}}, {{math|cos<sup>&minus;1</sup>(''x'')}}, etc., or, better, by {{math|sin<sup>&minus;1</sup> ''x''}}, {{math|cos<sup>&minus;1</sup> ''x''}}, etc., it in turn creates yet another major source of ambiguity, especially since many popular high-level programming languages (e.g. [[Mathematica]] and [[Magma (computer algebra system)|MAGMA]]) use those very same capitalised representations for the standard trig functions, whereas others ([[Python (programming language)|Python]], [[SymPy]], [[NumPy]], [[Matlab]], [[Maple (software)|MAPLE]], etc.) use lower-case. 

Hence, since 2009, the [[ISO 80000-2#Part 2: Mathematics|ISO 80000-2]] standard has specified solely the "arc" prefix for the inverse functions.