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{{Short description|Inverse functions of sin, cos, tan, etc.}}
{{use dmy dates|date=July 2022 |cs1-dates=sy }}
{{Trigonometry}}
{{redirect|Arctangent}}
In [[mathematics]], the '''inverse trigonometric functions''' (occasionally also called ''antitrigonometric'',<ref name="Hall_1909"/> ''cyclometric'',{{r|cyclometric}} or ''arcus'' functions{{r|arcus}}) are the [[inverse function]]s of the [[trigonometric functions]], under suitably restricted [[Domain of a function|domains]]. Specifically, they are the inverses of the [[sine]], [[cosine]], [[tangent (trigonometry)|tangent]], [[cotangent]], [[secant (trigonometry)|secant]], and [[cosecant]] functions,<ref>{{Cite web|last=Weisstein|first=Eric W.|title=Inverse Trigonometric Functions|url=https://mathworld.wolfram.com/InverseTrigonometricFunctions.html|access-date=2020-08-29|website=mathworld.wolfram.com|language=en}}</ref> and are used to obtain an angle from any of the angle's trigonometric ratios. Inverse trigonometric functions are widely used in [[engineering]], [[navigation]], [[physics]], and [[geometry]].
== 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>−1</sup>(''x'')}}, {{math|cos<sup>−1</sup>(''x'')}}, {{math|tan<sup>−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>[−1]</sup>(''x'')}}, {{math|cos<sup>[−1]</sup>(''x'')}}, {{math|tan<sup>[−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>−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|−1}}” superscript: {{math|Sin<sup>−1</sup>(''x'')}}, {{math|Cos<sup>−1</sup>(''x'')}}, {{math|Tan<sup>−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>−1</sup>(''x'')}}, {{math|cos<sup>−1</sup>(''x'')}}, etc., or, better, by {{math|sin<sup>−1</sup> ''x''}}, {{math|cos<sup>−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.
==Basic concepts==
[[File:TrigFunctionDiagram.svg|thumb|The points labelled {{color|#D00|1}}, {{color|#02D|Sec(''θ'')}}, {{color|#0D1|Csc(''θ'')}} represent the length of the line segment from the origin to that point. {{color|#D00|Sin(''θ'')}}, {{color|#02D|Tan(''θ'')}}, and {{color|#0D1|1}} are the heights to the line starting from the {{mvar|x}}-axis, while {{color|#D00|Cos(''θ'')}}, {{color|#02D|1}}, and {{color|#0D1|Cot(''θ'')}} are lengths along the {{mvar|x}}-axis starting from the origin.]]
===<span class="anchor" id="principal_value_anchor">Principal values</span>===
Since none of the six trigonometric functions are [[One-to-one function|one-to-one]], they must be restricted in order to have inverse functions. Therefore, the result [[Range of a function|range]]s of the inverse functions are proper (i.e. strict) [[subset]]s of the domains of the original functions.
For example, using {{em|function}} in the sense of [[multivalued function]]s, just as the [[square root]] function <math>y = \sqrt{x}</math> could be defined from <math>y^2 = x,</math> the function <math>y = \arcsin(x)</math> is defined so that <math>\sin(y) = x.</math> For a given real number <math>x,</math> with <math>-1 \leq x \leq 1,</math> there are multiple (in fact, [[countably infinite]]ly many) numbers <math>y</math> such that <math>\sin(y) = x</math>; for example, <math>\sin(0) = 0,</math> but also <math>\sin(\pi) = 0,</math> <math>\sin(2 \pi) = 0,</math> etc. When only one value is desired, the function may be restricted to its [[principal branch]]. With this restriction, for each <math>x</math> in the ___domain, the expression <math>\arcsin(x)</math> will evaluate only to a single value, called its [[principal value]]. These properties apply to all the inverse trigonometric functions.
The principal inverses are listed in the following table.
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{| class="wikitable" style="text-align:center"
|-
! scope="col" | Name
! scope="col" | Usual notation
! scope="col" | Definition
! scope="col" | Domain of {{mvar|x}} for real result
! scope="col" | Range of usual principal value <br
! scope="col" | Range of usual principal value <br
|-
! scope="row" | arcsine
| {{math|1= ''y'' = arcsin(''x'')}} || {{math|1=''x'' = [[sine|sin]](''y'')}} || {{math|−1 ≤ ''x'' ≤ 1}} || {{math|−{{sfrac|π|2}} ≤ ''y'' ≤ {{sfrac|π|2}}}} || {{math|−90° ≤ ''y'' ≤ 90°}}
|-
! scope="row" | arccosine
| {{math|1= ''y'' = arccos(''x'')}} || {{math|1=''x'' = [[cosine|cos]](''y'')}} || {{math|−1 ≤ ''x'' ≤ 1}} || {{math|0 ≤ ''y'' ≤ π}} || {{math|0° ≤ ''y'' ≤ 180°}}
|-
! scope="row" | arctangent
| {{math|1= ''y'' = arctan(''x'')}} || {{math|1=''x'' = [[Tangent (trigonometry)|tan]](''y'')}} || all real numbers || {{math|−{{sfrac|π|2}} < ''y'' < {{sfrac|π|2}}}} || {{math|−90° < ''y'' < 90°}}
|-
! scope="row" | arccotangent
| {{math|1= ''y'' = arccot(''x'')}} || {{math|1=''x'' = [[cotangent|cot]](''y'')}} || all real numbers || {{math|0 < ''y'' <
|-
! scope="row" | arcsecant
| {{math|1= ''y'' = arcsec(''x'')}} || {{math|1=''x'' = [[Secant (trigonometry)|sec]](''y'')}} || {{math|{{abs|''x''}} ≥ 1}} || {{math|0 ≤ ''y'' < {{sfrac|π|2}}}} or {{math|{{sfrac|π|2}} < ''y'' ≤ π}} || {{math|0° ≤ ''y'' < 90°}} or {{math|90° < ''y'' ≤ 180°}}
|-
! scope="row" | arccosecant
| {{math|1= ''y'' = arccsc(''x'')}} ||{{math|1=''x'' = [[cosecant|csc]](''y'')}} || {{math|{{abs|''x''}} ≥ 1}} || {{math|−{{sfrac|π|2}} ≤ ''y'' < 0}} or {{math|0 < ''y'' ≤ {{sfrac|π|2}}}} || {{math|−90° ≤ ''y'' < 0}} or {{math|0° < ''y'' ≤ 90°}}
|-
|}
====Domains====
If {{mvar|x}} is allowed to be a [[complex number]], then the range of {{mvar|y}} applies only to its real part.
{{DomainsImagesAndPrototypesOfTrigAndInverseTrigFunctions
|includeTableDescription=true
|includeExplanationOfNotation=true
}}
=== Solutions to elementary trigonometric equations ===
Each of the trigonometric functions is periodic in the real part of its argument, running through all its values twice in each interval of <math>2 \pi:</math>
* Sine and cosecant begin their period at <math display="inline">2 \pi k-\frac{\pi}{2}</math> (where <math>k</math> is an integer), finish it at <math display="inline">2 \pi k+\frac{\pi}{2},</math> and then reverse themselves over <math display="inline">2 \pi k+\frac{\pi}{2}</math> to <math display="inline">2 \pi k+\frac{3\pi}{2}.</math>
* Cosine and secant begin their period at <math>2 \pi k,</math> finish it at <math>2 \pi k+\pi.</math> and then reverse themselves over <math>2 \pi k+\pi</math> to <math>2 \pi k+2 \pi.</math>
* Tangent begins its period at <math display="inline">2 \pi k-\frac{\pi}{2},</math> finishes it at <math display="inline">2 \pi k+\frac{\pi}{2},</math> and then repeats it (forward) over <math display="inline">2 \pi k+\frac{\pi}{2}</math> to <math display="inline">2 \pi k+\frac{3 \pi}{2}.</math>
* Cotangent begins its period at <math>2 \pi k, </math> finishes it at <math>2 \pi k+\pi,</math> and then repeats it (forward) over <math>2 \pi k+\pi</math> to <math>2 \pi k+2 \pi.</math>
This periodicity is reflected in the general inverses, where <math>k</math> is some integer.
The following table shows how inverse trigonometric functions may be used to solve equalities involving the six standard trigonometric functions.
It is assumed that the given values <math>\theta,</math> <math>r,</math> <math>s,</math> <math>x,</math> and <math>y</math> all lie within appropriate ranges so that the relevant expressions below are [[well-defined]].
Note that "for some <math>k \in \Z</math>" is just another way of saying "for some [[integer]] <math>k.</math>"
The symbol <math>\,\iff\,</math> is [[logical equality]] and indicates that if the left hand side is true then so is the right hand side and, conversely, if the right hand side is true then so is the left hand side (see this footnote<ref group="note">The expression "LHS <math>\,\iff\,</math> RHS" indicates that {{em|either}} (a) the left hand side (i.e. LHS) and right hand side (i.e. RHS) are {{em|both}} true, or else (b) the left hand side and right hand side are {{em|both}} false; there is {{em|no}} option (c) (e.g. it is {{em|not}} possible for the LHS statement to be true and also simultaneously for the RHS statement to be false), because otherwise "LHS <math>\,\iff\,</math> RHS" would not have been written. <br/>
To clarify, suppose that it is written "LHS <math>\,\iff\,</math> RHS" where LHS (which abbreviates ''left hand side'') and RHS are both statements that can individually be either be true or false. For example, if <math>\theta</math> and <math>s</math> are some given and fixed numbers and if the following is written:
<math displaystyle="block">\tan \theta = s \,\iff\, \theta = \arctan(s)+\pi k \quad \text{ for some } k \in \Z</math>
then LHS is the statement "<math>\tan \theta = s</math>". Depending on what specific values <math>\theta</math> and <math>s</math> have, this LHS statement can either be true or false. For instance, LHS is true if <math>\theta = 0</math> and <math>s = 0</math> (because in this case <math>\tan \theta = \tan 0 = s</math>) but LHS is false if <math>\theta = 0</math> and <math>s = 2</math> (because in this case <math>\tan \theta = \tan 0 = s</math> which is not equal to <math>s = 2</math>); more generally, LHS is false if <math>\theta = 0</math> and <math>s \neq 0.</math> Similarly, RHS is the statement "<math>\theta = \arctan(s)+\pi k</math> for some <math>k \in \Z</math>". The RHS statement can also either true or false (as before, whether the RHS statement is true or false depends on what specific values <math>\theta</math> and <math>s</math> have). The logical equality symbol <math>\,\iff\,</math> means that (a) if the LHS statement is true then the RHS statement is also {{em|necessarily}} true, and moreover (b) if the LHS statement is false then the RHS statement is also {{em|necessarily}} false. Similarly, <math>\,\iff\,</math> {{em|also}} means that (c) if the RHS statement is true then the LHS statement is also {{em|necessarily}} true, and moreover (d) if the RHS statement is false then the LHS statement is also {{em|necessarily}} false.</ref> for more details and an example illustrating this concept).
{| class="wikitable" style="border: none;"
|+
|-
! Equation !! [[if and only if]] !! colspan="7" | Solution
|-
| style="text-align: center; padding: 0.5% 2em 0.5% 2em;
| style="text-align: center;" |
| style=
| style=
| style='border-style: solid none solid none; text-align: left;' |<math>\arcsin (y)</math>
| style='border-style: solid none solid none;' |<math>+</math>
| style='border-style: solid none solid none;' |
| style='border-style: solid none solid none; padding-right: 2em;' |<math>\pi k</math>
| style="text-align: center; padding-left: 1em; padding-right: 1em;" |for some <math>k \in \Z</math>
<!--------------- END: sin ''θ'' = ''x'' --------------->
|-
<!--------------- START: csc ''θ'' = ''r'' --------------->
| style="text-align: center; padding: 0.5% 2em 0.5% 2em;" |
| style="text-align: center;" |<math>\iff</math>
| style=
| style='border-style: solid none solid none; text-align: right;' |<math>(-1)^k</math>
| style='border-style: solid none solid none; text-align: left;' |<math>\arccsc (r)</math>
| style='border-style: solid none solid none;' |<math>+</math>
| style='border-style: solid none solid none;' |
| style='border-style: solid none solid none; padding-right: 2em;' |<math>\pi k</math>
| style="text-align: center; padding-left: 1em; padding-right: 1em;" |for some <math>k \in \Z</math>
<!--------------- END: csc ''θ'' = ''r'' --------------->
|-
<!--------------- START:
| style="text-align: center; padding: 0.5% 2em 0.5% 2em;
| style="text-align: center;" |
| style=
| style=
| style='border-style: solid none solid none; text-align: left;' |<math>\arccos(x)</math>
| style='border-style: solid none solid none;' |<math>+</math>
| style='border-style: solid none solid none;' |<math>2</math>
| style='border-style: solid none solid none; padding-right: 2em;' |<math>\pi k</math>
| style="text-align: center; padding-left: 1em; padding-right: 1em;" |for some <math>k \in \Z</math>
<!--------------- END: cos ''θ'' = ''x'' --------------->
|-
<!--------------- START: sec ''θ'' = ''r'' --------------->
| style="text-align: center; padding: 0.5% 2em 0.5% 2em;" |
| style="text-align: center;" |<math>\iff</math>
| style=
| style='border-style: solid none solid none; text-align: right;' |<math>\pm\,</math>
| style='border-style: solid none solid none; text-align: left;' |<math>\arcsec (r)</math>
| style='border-style: solid none solid none;' |<math>+</math>
| style='border-style: solid none solid none;' |<math>2</math>
| style='border-style: solid none solid none; padding-right: 2em;' |<math>\pi k</math>
| style="text-align: center; padding-left: 1em; padding-right: 1em;" |for some <math>k \in \Z</math>
|-
<!---------------
|-
<!--------------- START: tan ''θ'' = ''s'' --------------->
| style="text-align: center; padding: 0.5% 2em 0.5% 2em;" |
| style="text-align: center;" |<math>\iff</math>
| style=
| style='border-style: solid none solid none; text-align: right;' |
| style='border-style: solid none solid none; text-align: left;' |<math>\arctan (s)</math>
| style='border-style: solid none solid none;' |<math>+</math>
| style='border-style: solid none solid none;' |
| style='border-style: solid none solid none; padding-right: 2em;' |<math>\pi k</math>
| style="text-align: center; padding-left: 1em; padding-right: 1em;" |for some <math>k \in \Z</math>
<!--------------- END: tan ''θ'' = ''s'' --------------->
|-
<!--------------- START:
| style="text-align: center; padding: 0.5% 2em 0.5% 2em;
| style="text-align: center;" |
| style=
| style='border-style: solid none solid none; text-align: right;' |
| style='border-style: solid none solid none; text-align: left;' |<math>\arccot (r)</math>
| style='border-style: solid none solid none;' |<math>+</math>
| style='border-style: solid none solid none;' |
| style='border-style: solid none solid none; padding-right: 2em;' |<math>\pi k</math>
| style="text-align: center; padding-left: 1em; padding-right: 1em;" |for some <math>k \in \Z</math>
<!--------------- END: cot ''θ'' = ''r'' --------------->
|}
where the first four solutions can be written in expanded form as:
{| class="wikitable" style="border: none;"
|+
|-
! Equation !! [[if and only if]] !! colspan="7" | Solution
|-
| style="text-align: center; padding: 0.5% 2em 0.5% 2em;" | <math>\sin \theta = y</math>
| style="text-align: center
| style='border-style: solid none solid none; text-align: left; padding-left: 2em;' |<math>\theta = \;\;\;\,\arcsin(y)+2 \pi k</math> <br/>{{space|10}}or <br/><math>\theta =-\arcsin(y)+2 \pi k+\pi</math>
| style="text-align:
<!--------------- END: sin ''θ'' = ''x'' --------------->
|-
<!--------------- START:
| style="text-align: center; padding: 0.5% 2em 0.5% 2em;" |
| style="text-align: center;" |
| style=
| style="text-align: center; padding-left: 1em; padding-right: 1em;" |
<!--------------- END:
|-
<!--------------- START: cos ''θ'' = ''x'' --------------->
| style="text-align: center; padding: 0.5% 2em 0.5% 2em;" | <math>\cos \theta = x</math>
| style="text-align: center;" |<math>\iff</math>
| style='border-style: solid none solid none; text-align: left; padding-left: 2em;' |<math>\theta = \;\;\;\,\arccos(x)+2 \pi k</math> <br/>{{space|9}}or <br/><math>\theta =-\arccos(x)+2 \pi k</math>
| style="text-align: center; padding-left: 1em; padding-right: 1em;" |for some <math>k \in \Z</math>
<!--------------- END: cos ''θ'' = ''x'' --------------->
|-
<!--------------- START: sec ''θ'' = ''r'' --------------->
| style="text-align: center; padding: 0.5% 2em 0.5% 2em;" | <math>\sec \theta = r</math>
| style="text-align: center;" |<math>\iff</math>
| style='border-style: solid none solid none; text-align: left; padding-left: 2em;' |<math>\theta = \;\;\;\,\arcsec(r)+2 \pi k</math> <br/>{{space|9}}or <br/><math>\theta =-\arcsec(r)+2 \pi k</math>
| style="text-align: center; padding-left: 1em; padding-right: 1em;" |for some <math>k \in \Z</math>
<!--------------- END: sec ''θ'' = ''r'' --------------->
|}
For example, if <math>\cos \theta = -1</math> then <math>\theta = \pi+2 \pi k = -\pi+2 \pi (1+k)</math> for some <math>k \in \Z.</math> While if <math>\sin \theta = \pm 1</math> then <math display="inline">\theta = \frac{\pi}{2}+\pi k =-\frac{\pi}{2}+\pi (k+1)</math> for some <math>k \in \Z,</math> where <math>k</math> will be even if <math>\sin \theta = 1</math> and it will be odd if <math>\sin \theta = -1.</math> The equations <math>\sec \theta = -1</math> and <math>\csc \theta = \pm 1</math> have the same solutions as <math>\cos \theta = -1</math> and <math>\sin \theta = \pm 1,</math> respectively. In all equations above {{em|except}} for those just solved (i.e. except for <math>\sin</math>/<math>\csc \theta = \pm 1</math> and <math>\cos</math>/<math>\sec \theta =-1</math>), the integer <math>k</math> in the solution's formula is uniquely determined by <math>\theta</math> (for fixed <math>r, s, x,</math> and <math>y</math>).
With the help of [[Parity (mathematics)|integer parity]]
<math display=block>\operatorname{Parity}(h) =
\begin{cases}
0 & \text{if } h \text{ is even } \\
1 & \text{if } h \text{ is odd } \\
\end{cases}</math>
it is possible to write a solution to <math>\cos \theta = x</math> that doesn't involve the "plus or minus" <math>\,\pm\,</math> symbol:
:<math>cos \; \theta = x \quad</math> if and only if <math>\quad \theta = (-1)^h \arccos(x) + \pi h + \pi \operatorname{Parity}(h) \quad</math> for some <math>h \in \Z.</math>
And similarly for the secant function,
:<math>sec \; \theta = r \quad</math> if and only if <math>\quad \theta = (-1)^h \arcsec(r) + \pi h + \pi \operatorname{Parity}(h) \quad</math> for some <math>h \in \Z,</math>
where <math>\pi h + \pi \operatorname{Parity}(h)</math> equals <math>\pi h</math> when the integer <math>h</math> is even, and equals <math>\pi h + \pi</math> when it's odd.
====Detailed example and explanation of the "plus or minus" symbol {{math|±}} ====
The solutions to <math>\cos \theta = x</math> and <math>\sec \theta = x</math> involve the "plus or minus" symbol <math>\,\pm,\,</math> whose meaning is now clarified. Only the solution to <math>\cos \theta = x</math> will be discussed since the discussion for <math>\sec \theta = x</math> is the same.
We are given <math>x</math> between <math>-1 \leq x \leq 1</math> and we know that there is an angle <math>\theta</math> in some interval that satisfies <math>\cos \theta = x.</math> We want to find this <math>\theta.</math> The table above indicates that the solution is
<math display="block">\,\theta = \pm \arccos x+2 \pi k\, \quad \text{ for some }k \in \Z</math>
which is a shorthand way of saying that (at least) one of the following statement is true: <br />
#<math>\,\theta = \arccos x+2 \pi k\,</math> for some integer <math>k,</math> <br/>or
#<math>\,\theta =-\arccos x+2 \pi k\,</math> for some integer <math>k.</math>
As mentioned above, if <math>\,\arccos x = \pi\,</math> (which by definition only happens when <math>x = \cos \pi = -1</math>) then both statements (1) and (2) hold, although with different values for the integer <math>k</math>: if <math>K</math> is the integer from statement (1), meaning that <math>\theta = \pi+2 \pi K</math> holds, then the integer <math>k</math> for statement (2) is <math>K+1</math> (because <math>\theta = -\pi+2 \pi (1+K)</math>).
However, if <math>x \neq -1</math> then the integer <math>k</math> is unique and completely determined by <math>\theta.</math>
If <math>\,\arccos x = 0\,</math> (which by definition only happens when <math>x = \cos 0 = 1</math>) then <math>\,\pm\arccos x = 0\,</math> (because <math>\,+ \arccos x = +0 = 0\,</math> and <math>\,-\arccos x = -0 = 0\,</math> so in both cases <math>\,\pm \arccos x\,</math> is equal to <math>0</math>) and so the statements (1) and (2) happen to be identical in this particular case (and so both hold).
Having considered the cases <math>\,\arccos x = 0\,</math> and <math>\,\arccos x = \pi,\,</math> we now focus on the case where <math>\,\arccos x \neq 0\,</math> and <math>\,\arccos x \neq \pi,\,</math> So assume this from now on. The solution to <math>\cos \theta = x</math> is still
<math display="block">\,\theta = \pm \arccos x+2 \pi k\, \quad \text{ for some }k \in \Z</math>
which as before is shorthand for saying that one of statements (1) and (2) is true. However this time, because <math>\,\arccos x \neq 0\,</math> and <math>\,0 < \arccos x < \pi,\,</math> statements (1) and (2) are different and furthermore, ''exactly one'' of the two equalities holds (not both). Additional information about <math>\theta</math> is needed to determine which one holds. For example, suppose that <math>x = 0</math> and that {{em|all}} that is known about <math>\theta</math> is that <math>\,-\pi \leq \theta \leq \pi\,</math> (and nothing more is known). Then
<math display="block">\arccos x = \arccos 0 = \frac{\pi}{2}</math>
and moreover, in this particular case <math>k = 0</math> (for both the <math>\,+\,</math> case and the <math>\,-\,</math> case) and so consequently,
<math display="block">\theta ~=~ \pm \arccos x+2 \pi k ~=~ \pm \left(\frac{\pi}{2}\right)+2\pi (0) ~=~ \pm \frac{\pi}{2}.</math>
This means that <math>\theta</math> could be either <math>\,\pi/2\,</math> or <math>\,-\pi/2.</math> Without additional information it is not possible to determine which of these values <math>\theta</math> has.
An example of some additional information that could determine the value of <math>\theta</math> would be knowing that the angle is above the <math>x</math>-axis (in which case <math>\theta = \pi/2</math>) or alternatively, knowing that it is below the <math>x</math>-axis (in which case <math>\theta =-\pi/2</math>).
==== Equal identical trigonometric functions ====
{{EqualOrNegativeIdenticalTrigonometricFunctionsSolutions|includeTableDescription=true|style=}}
;Set of all solutions to elementary trigonometric equations
Thus given a single solution <math>\theta</math> to an elementary trigonometric equation (<math>\sin \theta = y</math> is such an equation, for instance, and because <math>\sin (\arcsin y) = y</math> always holds, <math>\theta := \arcsin y</math> is always a solution), the set of all solutions to it are:
{| class="wikitable" style="border: none;"
|+
|-
! If <math>\theta</math> solves !! then !! colspan="7" | Set of all solutions (in terms of <math>\theta</math>)
|-
| style="text-align: center; padding: 0.5% 2em 0.5% 2em;" | <math>\;\sin \theta = y</math>
| style="text-align: center;" |then
| style='border-style: solid none solid none; text-align:
| style='border-style: solid none solid none; text-align:
| style=
| style='border-style: solid none solid none; text-align: left; padding
| style='border-style: solid none solid none; text-align:
| style='border-style: solid none solid none; text-align: left; padding: 0;' |
| style='border-style: solid
<!--------------- END: sin ''θ'' = ''x'' --------------->
|-
<!--------------- START:
| style=
| style=
| style='border-style: solid none solid none; text-align: left; padding-
| style=
| style='border-style: solid none solid none; text-align: left; padding
| style='border-style: solid none solid none; text-align:
| style='border-style: solid none solid none; text-align: left; padding: 0;' |
| style='border-style: solid none solid none; text-align: left; padding: 0;' |
| style='border-style: solid
<!--------------- END: csc ''θ'' = ''r'' --------------->
|-
<!--------------- START:
| style=
| style=
| style='border-style: solid none solid none; text-align: left; padding-
| style=
| style='border-style: solid none solid none; text-align: left; padding
| style='border-style: solid none solid none; text-align:
| style='border-style: solid none solid none; text-align: left; padding: 0;' |
| style='border-style: solid none solid none; text-align: left; padding: 0;' |
| style='border-style: solid
<!--------------- END: cos ''θ'' = ''x'' --------------->
|-
<!--------------- START:
| style=
| style=
| style='border-style: solid none solid none; text-align: left; padding-
| style=
| style='border-style: solid none solid none; text-align: left; padding
| style='border-style: solid none solid none; text-align:
| style='border-style: solid none solid none; text-align: left; padding: 0;' |
| style='border-style: solid none solid none; text-align: left; padding: 0;' |
| style='border-style: solid
|-
<!---------------
|-
<!--------------- START:
| style=
| style=
| style='border-style: solid none solid none; text-align: left; padding-
| style=
| style='border-style: solid none solid none; text-align: left; padding
| style='border-style: solid none solid none; text-align:
| style='border-style: solid none solid none; text-align: left; padding: 0;' |
| style='border-style: solid none solid none; text-align: left; padding: 0;' |
| style='border-style: solid
<!--------------- END: tan ''θ'' = ''s'' --------------->
|-
<!--------------- START:
| style=
| style=
| style='border-style: solid none
|
|
|
|
|
|
<!--------------- END: cot ''θ'' = ''r'' --------------->
|}
===Transforming equations===
The equations above can be transformed by using the reflection and shift identities:<ref>{{harvnb|Abramowitz|Stegun|1972|loc=p. 73, 4.3.44}}</ref>
{| class="wikitable" style=
|+ Transforming equations by shifts and reflections
|-
! scope="col" | Argument: <math>\underline{\;~~~~~~\;}=</math>
! scope="col" |<math>-\theta</math>
! scope="col" |<math>\frac{\pi}{2} \pm \theta</math>
! scope="col" |<math>\pi \pm \theta</math>
! scope="col" |<math>\frac{3\pi}{2} \pm \theta</math>
! scope="col" |<math>2 k \pi \pm \theta,</math><br/> <math>(k \in \Z)</math>
|- <!-- sin -->
! scope="row" |<math>\sin \underline{\;~~~~~~~~~~~~~~\;}=</math>
| <math>-\sin \theta</math>
| <math>\phantom{-}\cos \theta</math>
| <math>\mp\sin \theta</math>
| <math>-\cos \theta</math>
| <math>\pm\sin \theta</math>
|- <!-- csc -->
! scope="row" |<math>\csc \underline{\;~~~~~~~~~~~~~~\;}=</math>
| <math>-\csc \theta</math>
| <math>\phantom{-}\sec \theta</math>
| <math>\mp\csc \theta</math>
| <math>-\sec \theta</math>
| <math>\pm\csc \theta</math>
|- <!-- cos -->
! scope="row" |<math>\cos \underline{\;~~~~~~~~~~~~~~\;}=</math>
| <math>\phantom{-}\cos \theta</math>
| <math>\mp\sin \theta</math>
| <math>-\cos \theta</math>
| <math>\pm\sin \theta</math>
| <math>\phantom{-}\cos \theta</math>
|- <!-- sec -->
! scope="row" |<math>\sec \underline{\;~~~~~~~~~~~~~~\;}=</math>
| <math>\phantom{-}\sec \theta</math>
| <math>\mp\csc \theta</math>
| <math>-\sec \theta</math>
| <math>\pm\csc \theta</math>
| <math>\phantom{-}\sec \theta</math>
|- <!-- tan -->
! scope="row" |<math>\tan \underline{\;~~~~~~~~~~~~~~\;}=</math>
| <math>-\tan \theta</math>
| <math>\mp\cot \theta</math>
| <math>\pm\tan \theta</math>
| <math>\mp\cot \theta</math>
| <math>\pm\tan \theta</math>
|- <!-- cot -->
! scope="row" |<math>\cot \underline{\;~~~~~~~~~~~~~~\;}=</math>
| <math>-\cot \theta</math>
| <math>\mp\tan \theta</math>
| <math>\pm\cot \theta</math>
| <math>\mp\tan \theta</math>
| <math>\pm\cot \theta</math>
|}
These formulas imply, in particular, that the following hold:
<math display=block>
\begin{align}
\sin \theta
&= -\sin(-\theta)
&&= -\sin(\pi+\theta)
&&= \phantom{-}\sin(\pi-\theta) \\
&= -\cos\left(\frac{\pi}{2}+\theta\right)
&&= \phantom{-}\cos\left(\frac{\pi}{2}-\theta\right)
&&= -\cos\left(-\frac{\pi}{2}-\theta\right) \\
&= \phantom{-}\cos\left(-\frac{\pi}{2}+\theta\right)
&&= -\cos\left(\frac{3\pi}{2}-\theta\right)
&&= -\cos\left(-\frac{3\pi}{2}+\theta\right)
\\[0.3ex]
\cos \theta
&= \phantom{-}\cos(-\theta)
&&= -\cos(\pi+\theta)
&&= -\cos(\pi-\theta) \\
&= \phantom{-}\sin\left(\frac{\pi}{2}+\theta\right)
&&= \phantom{-}\sin\left(\frac{\pi}{2}-\theta\right)
&&= -\sin\left(-\frac{\pi}{2}-\theta\right) \\
&= -\sin\left(-\frac{\pi}{2}+\theta\right)
&&= -\sin\left(\frac{3\pi}{2}-\theta\right)
&&= \phantom{-}\sin\left(-\frac{3\pi}{2}+\theta\right)
\\[0.3ex]
\tan \theta
&= -\tan(-\theta)
&&= \phantom{-}\tan(\pi+\theta)
&&= -\tan(\pi-\theta) \\
&= -\cot\left(\frac{\pi}{2}+\theta\right)
&&= \phantom{-}\cot\left(\frac{\pi}{2}-\theta\right)
&&= \phantom{-}\cot\left(-\frac{\pi}{2}-\theta\right) \\
&= -\cot\left(-\frac{\pi}{2}+\theta\right)
&&= \phantom{-}\cot\left(\frac{3\pi}{2}-\theta\right)
&&= -\cot\left(-\frac{3\pi}{2}+\theta\right)
\\[0.3ex]
\end{align}
</math>
where swapping <math>\sin \leftrightarrow \csc,</math> swapping <math>\cos \leftrightarrow \sec,</math> and swapping <math>\tan \leftrightarrow \cot</math> gives the analogous equations for <math>\csc, \sec, \text{ and } \cot,</math> respectively.
So for example, by using the equality <math display="inline">\sin \left(\frac{\pi}{2}-\theta\right) = \cos \theta,</math> the equation <math>\cos \theta = x</math> can be transformed into <math display="inline">\sin \left(\frac{\pi}{2}-\theta\right) = x,</math> which allows for the solution to the equation <math>\;\sin \varphi = x\;</math> (where <math display="inline">\varphi := \frac{\pi}{2}-\theta</math>) to be used; that solution being:
<math>\varphi = (-1)^k \arcsin (x)+\pi k \; \text{ for some } k \in \Z,</math>
which becomes:
<math display="block">\frac{\pi}{2}-\theta ~=~ (-1)^k \arcsin (x)+\pi k \quad \text{ for some } k \in \Z</math>
where using the fact that <math>(-1)^{k} = (-1)^{-k}</math> and substituting <math>h :=-k</math> proves that another solution to <math>\;\cos \theta = x\;</math> is:
<math display="block">\theta ~=~ (-1)^{h+1} \arcsin (x)+\pi h+\frac{\pi}{2} \quad \text{ for some } h \in \Z.</math>
The substitution <math>\;\arcsin x = \frac{\pi}{2}-\arccos x\;</math> may be used express the right hand side of the above formula in terms of <math>\;\arccos x\;</math> instead of <math>\;\arcsin x.\;</math>
=== Relationships between trigonometric functions and inverse trigonometric functions ===
Trigonometric functions of inverse trigonometric functions are tabulated below. A quick way to derive them is by considering the geometry of a right-angled triangle, with one side of length 1 and another side of length
{|class="wikitable"
Line 345 ⟶ 493:
:<math>\begin{align}
\arcsin(-x) &= -\arcsin(x) \\
\arccsc(-x) &= -\arccsc(x) \\
\arccos(-x) &= \pi -\arccos(x) \\
\arcsec(-x) &= \pi -\arcsec(x) \\
\
\arccot(-x) &= \pi -\arccot(x)
\end{align}</math>
Reciprocal arguments:
:<math>\begin{align}
\
\
\
\
\
\
\
\
\end{align}</math>
The identities above can be used with (and derived from) the fact that <math>\sin</math> and <math>\csc</math> are [[Multiplicative inverse|reciprocals]] (i.e. <math>\csc = \tfrac1{\sin}</math>), as are <math>\cos</math> and <math>\sec,</math> and <math>\tan</math> and <math>\cot.</math>
Useful identities if one only has a fragment of a sine table:
:<math>\begin{align}
\arcsin(x) &= \frac{1}{2}\arccos\left(1-2x^2\right) \, , \text{ if } 0 \leq x \leq 1 \\
\arcsin(x) &= \arctan\left(\frac{x}{\sqrt{1 - x^2}}\right) \\
\arccos(x) &= \frac{1}{2}\arccos\left(2x^2-1\right) \, , \text{ if } 0 \leq x \leq 1 \\
\arccos(x) &= \arctan\left(\frac{\sqrt{1 - x^2}}{x}\right) \\
\arccos(x) &= \arcsin\left(\sqrt{1 - x^2}\right) \, , \text{ if } 0 \leq x \leq 1 \text{ , from which you get } \\
\arccos &\left(\frac{1-x^2}{1 + x^2}\right) = \arcsin \left (\frac{2x}{1 + x^2}\right) \, , \text{ if } 0 \leq x \leq 1 \\
\arcsin &\left(\sqrt{1 - x^2}\right) =\frac{\pi}{2}-\sgn(x)\arcsin(x) \\
\arctan(x) &= \arcsin\left(\frac{x}{\sqrt{1 + x^2}}\right) \\
\arccot(x) &= \arccos\left(\frac{x}{\sqrt{1 + x^2}}\right)
Line 380 ⟶ 530:
A useful form that follows directly from the table above is
:<math>\arctan
It is obtained by recognizing that <math>\cos\left(\arctan\left(x\right)\right) = \sqrt{\frac{1}{1+x^2}} = \cos\left(\arccos\left(\sqrt{\frac{1}{1+x^2}}\right)\right)</math>.
Line 401 ⟶ 551:
==={{anchor|Derivatives}}Derivatives of inverse trigonometric functions===
The [[derivative]]s for complex values of ''z'' are as follows:
Line 418 ⟶ 568:
\end{align}</math>
These formulas can be derived in terms of the derivatives of trigonometric functions. For example, if <math>x = \sin \theta</math>, then <math display=inline>dx/d\theta = \cos \theta = \sqrt{1-x^2},</math> so
:<math>\frac{d
===Expression as definite integrals===
Line 428 ⟶ 578:
\arctan(x) &{}= \int_0^x \frac{1}{z^2 + 1} \, dz \; ,\\
\arccot(x) &{}= \int_x^\infty \frac{1}{z^2 + 1} \, dz \; ,\\
\arcsec(x) &{}= \int_1^x \frac{1}{z \sqrt{z^2 - 1}} \, dz = \pi + \
\arccsc(x) &{}= \int_x^\infty \frac{1}{z \sqrt{z^2 - 1}} \, dz = \int_{-\infty}^{-x} \frac{1}{z \sqrt{z^2 - 1}} \, dz \; , & x &{} \geq 1\\
\end{align}</math>
When ''x'' equals 1, the integrals with limited domains are [[improper integral]]s, but still well-defined.
Line 454 ⟶ 604:
[[Leonhard Euler]] found a series for the arctangent that converges more quickly than its [[Taylor series]]:
: <math>\arctan(z) = \frac z {1 + z^2} \sum_{n=0}^\infty \prod_{k=1}^n \frac{2k z^2}{(2k + 1)(1 + z^2)}.</math><ref>{{citation|title= An elementary derivation of Euler's series for the arctangent function|journal = The Mathematical Gazette | author = Hwang Chien-Lih | doi = 10.1017/S0025557200178404 | year = 2005 | volume = 89 | issue = 516|pages = 469–470 |s2cid = 123395287 }}</ref>
(The term in the sum for ''n'' = 0 is the [[empty product]], so is 1.)
Line 465 ⟶ 615:
:<math>\arctan(z) = i\sum_{n=1}^\infty\frac{1}{2n - 1}\left(\frac{1}{(1 + 2i/z)^{2n-1}} - \frac{1}{(1 - 2i/z)^{2n - 1}}\right),</math>
where <math>i=\sqrt{-1}</math> is the [[imaginary unit]].<ref>{{citation
====Continued fractions for arctangent====
Line 497 ⟶ 647:
For all real ''x'' not between -1 and 1:
:<math>\begin{align}
\int \arcsec(x) \, dx &{}= x \, \arcsec(x) - \sgn(x) \ln
\int \arccsc(x) \, dx &{}= x \, \arccsc(x) + \sgn(x) \ln
\end{align}</math>
Line 527 ⟶ 677:
:<math>\int \arcsin(x) \, dx = x \arcsin(x) + \sqrt{1-x^2} + C </math>
== Extension to the complex plane ==
[[File:Riemann surface for Arg of ArcTan of x.svg|thumb|A [[Riemann surface]] for the argument of the relation {{math|1=tan ''z'' = ''x''}}. The orange sheet in the middle is the principal sheet representing {{math|arctan ''x''}}. The blue sheet above and green sheet below are displaced by {{math|2''π''}} and {{math|
Since the inverse trigonometric functions are [[analytic function]]s, they can be extended from the [[Number line|real line]] to the complex plane. This results in functions with multiple sheets and [[branch point]]s. One possible way of defining the extension is:
:<math>\arctan(z) = \int_0^z \frac{dx}{1 + x^2} \quad z \neq -i, +i </math>
where the part of the imaginary axis which does not lie strictly between the branch points (
The arcsine function may then be defined as:
Line 546 ⟶ 696:
where the part of the real axis between −1 and +1 inclusive is the cut between the principal sheet of arcsec and other sheets;
:<math>\arccsc(z) = \arcsin\left(\frac{1}{z}\right) \quad z \neq -1, 0, +1 </math>
which has the same cut as
===Logarithmic forms===
Line 553 ⟶ 703:
:<math>\begin{align}
\arcsin(z) &{}= -i \ln \left( \sqrt{1-z^2} + iz \right) = i \ln \left( \sqrt{1-z^2} - iz \right) &{}= \arccsc\left(\frac{1}{z}\right) \\[10pt]
\arccos(z) &{}= -i \ln \left( i \sqrt {1-z^2} + z \right) = \frac{\pi}{2} - \arcsin(z) &{}= \arcsec\left(\frac{1}{z}\right) \\[10pt]
\arctan(z) &{}= -\frac{i}{2}\ln \left(\frac{i - z}{i + z}\right) = -\frac{i}{2}\ln \left(\frac{1 + iz}{1 - iz}\right) &{}= \arccot\left(\frac{1}{z}\right) \\[10pt]
\arccot(z) &{}= -\frac{i}{2}\ln\left( \frac{z + i}{z - i} \right) = -\frac{i}{2}\ln\left( \frac{iz - 1}{iz + 1} \right) &{}= \arctan\left(\frac{1}{z}\right) \\[10pt]
Line 564 ⟶ 714:
Because all of the inverse trigonometric functions output an angle of a right triangle, they can be generalized by using [[Euler's formula]] to form a right triangle in the complex plane. Algebraically, this gives us:
:<math>ce^{i\theta} = c\cos(\theta) + ic\sin(\theta)</math>
or
:<math>ce^{i\theta} = a + ib</math>
where <math>a</math> is the adjacent side, <math>b</math> is the opposite side, and <math>c</math> is the [[hypotenuse]]. From here, we can solve for <math>\theta</math>.
:<math>\begin{align}
e^{\ln(c) + i\theta
\ln c + i\theta
\theta & = \operatorname{Im}\left( \ln(a +
\end{align}</math>
or
:<math>\theta = -i\ln\left(\frac{a + ib}{c}\right)</math>
Simply taking the imaginary part works for any real-valued <math>a</math> and <math>b</math>, but if <math>a</math> or <math>b</math> is complex-valued, we have to use the final equation so that the real part of the result isn't excluded. Since the length of the hypotenuse doesn't change the angle, ignoring the real part of <math>\ln(a+bi)</math> also removes <math>c</math> from the equation. In the final equation, we see that the angle of the triangle in the complex plane can be found by inputting the lengths of each side. By setting one of the three sides equal to 1 and one of the remaining sides equal to our input <math>z</math>, we obtain a formula for one of the inverse trig functions, for a total of six equations. Because the inverse trig functions require only one input, we must put the final side of the triangle in terms of the other two using the [[Pythagorean Theorem]] relation
:<math>a^2 + b^2 = c^2</math>
The table below shows the values of a, b, and c for each of the inverse trig functions and the equivalent expressions for <math>\theta</math> that result from plugging the values into the equations <math>\theta = -i\ln\left(\tfrac{a + ib}{c}\right)</math> above and simplifying.
:<math>\begin{align}
& a & & b & & c && -i\
\arcsin(z)\ \ & \sqrt{1 - z^2} & & z & & 1 & & -i\ln\left( \frac{\sqrt{1 - z^2} +
\arccos(z)\ \ & z & & \sqrt{1 - z^2} & & 1 & & -i\ln\left( \frac{z + i\sqrt{1 - z^2}}{1} \right) && = -i\ln\left( z + \sqrt{z^2 - 1} \right) && \operatorname{Im}\left(\ln\left( z + \sqrt{z^2 - 1} \right)\right) \\
\arctan(z)\ \ & 1 & & z & & \sqrt{1 + z^2} & & -i\ln\left( \frac{1 + iz}{\sqrt{1 + z^2}
\arccot(z)\ \ & z & & 1 & & \sqrt{z^2 + 1} & & -i\ln\left( \frac{z + i}{\sqrt{z^2 + 1}
\arcsec(z)\ \ & 1 & & \sqrt{z^2 - 1} & & z & & -i\ln\left( \frac{1 + i\sqrt{z^2 - 1}}{z} \right) && = -i\ln\left( \frac{1}{z} + \sqrt{\frac{1}{z^2}-1} \right) && \operatorname{Im}\left(\ln\left( \frac{1}{z} + \sqrt{\frac{1
\arccsc(z)\ \ & \sqrt{z^2 - 1} & & 1 & & z & & -i\ln\left( \frac{\sqrt{z^2 - 1} + i}{z} \right) && = -i\ln\left( \sqrt{1-\frac{1}{z^2}} + \frac{i}{z} \right) && \operatorname{Im}\left(\ln\left( \sqrt{
\end{align}</math>
The particular form of the simplified expression can cause the output to differ from the [[#Principal_values|usual principal branch]] of each of the inverse trig functions. The formulations given will output the usual principal branch when using the <math> \operatorname{Im}\left( \ln z \right) \in (-\pi,\pi] </math> and <math> \operatorname{Re}\left(\sqrt{z}\right) \ge 0 </math> principal branch for every function except arccotangent in the <math>\theta</math> column. Arccotangent in the <math>\theta</math> column will output on its usual principal branch by using the <math> \operatorname{Im}\left( \ln z \right) \in [0,2\pi) </math> and <math> \operatorname{Im}\left(\sqrt{z}\right) \ge 0 </math> convention.
In this sense, all of the inverse trig functions can be thought of as specific cases of the complex-valued log function. Since these definition work for any complex-valued <math>z</math>, the definitions allow for [[hyperbolic angle]]s as outputs and can be used to further define the [[inverse hyperbolic functions]]. It's possible to algebraically prove these relations by starting with the exponential forms of the trigonometric functions and solving for the inverse function.
====Example proof====
:<math>\
\sin(\phi) &= z \\
\phi &= \arcsin(z)
\end{align}</math>
Using the [[Trigonometric functions#Relationship to exponential function (Euler's formula)|exponential definition of sine]], and letting <math>\xi = e^{i \phi}, </math>
:<math>\begin{align}
z &= \frac{e^{i \phi} - e^{-i \phi}}{2i} \\[10mu]
0 &= \xi^2 - 2i z \xi - 1 \\[5mu]
\xi &= iz \pm \sqrt{1 - z^2} \\[5mu]
\phi &= -i \ln \left(iz \pm \sqrt{1 - z^2}\right)
(the positive branch is chosen)
Line 636 ⟶ 767:
:<math>\phi= \arcsin(z) = -i \ln \left(iz + \sqrt{1-z^2} \right)</math>
{| style="text-align:center;"
|+ [[Domain coloring|Color wheel graphs]] of '''inverse trigonometric functions in the [[complex plane]]'''
| [[Image:Complex
| [[Image:Complex
| [[Image:Complex
|-
| <math>\arcsin(z)</math>
| <math>\arccos(z)</math>
| <math>\arctan(z)</math>
|}
{| style="text-align:center;"
|+
| [[Image:Complex Arccosecant.svg|275x275px|Arccosecant of z in the complex plane.]]
| [[Image:Complex Arcsecant.svg|275x275px|Arcsecant of z in the complex plane.]]
| [[Image:Complex Arccotangent.svg|275x275px|Arccotangent of z in the complex plane.]]
|-
| <math>\arccsc(z)</math>
| <math>\arcsec(z)</math>
| <math>\arccot(z)</math>
|}
== Applications ==
===
[[Image:Trigonometry triangle.svg|right|thumb|A [[right triangle]] with sides relative to an angle at the <math>A</math> point.]]
Inverse trigonometric functions are useful when trying to determine the remaining two angles of a [[right triangle]] when the lengths of the sides of the triangle are known. Recalling the right-triangle definitions of sine and cosine, it follows that
Line 677 ⟶ 812:
{{anchor|Two-argument variant of arctangent}}
{{main|atan2}}
The two-argument [[atan2|{{math|atan2}}]] function computes the arctangent of {{math|''y''
In terms of the standard '''arctan''' function, that is with range of {{open-open|−π/2, π/2}}, it can be expressed as follows:
<math display="block">\operatorname{atan2}(y, x) = \begin{cases}
\arctan\left(\frac y x\right)
\arctan\left(\frac
\
-\frac{\pi}{2} & \quad y < 0,\; x = 0 \\
\text{undefined} & \quad y = 0,\; x = 0
\end{cases}</math>
It also equals the [[principal value]] of the [[arg (mathematics)|arg]]ument of the [[complex number]] {{math|''x''
This limited version of the function above may also be defined using the [[tangent half-angle formula]]e as follows:
provided that either {{math|''x''
The above argument order (
====Arctangent function with ___location parameter====
In many applications<ref>when a time varying angle crossing <math>\pm\pi/2</math> should be mapped by a smooth line instead of a saw toothed one (robotics,
:<math>
y = \arctan_\eta(x) := \arctan(x) + \pi \
</math>
The function <math>\operatorname{rni}</math> rounds to the nearest integer.
====Numerical accuracy====
For angles near 0 and {{pi}}, arccosine is [[ill-conditioned]], and
==See also==
{{cols}}
*[[Arcsine distribution]]
*[[Inverse exsecant]]
*[[Inverse versine]]
Line 717 ⟶ 855:
*[[Trigonometric function]]
*[[Trigonometric functions of matrices]]
{{colend}}
== Notes ==
Line 722 ⟶ 861:
== References==
* {{Cite book|editor1-last=Abramowitz|editor1-first=Milton|editor1-link=Milton Abramowitz|editor2-last=Stegun|editor2-first=Irene A.|editor2-link=Irene Stegun|title=Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables|publisher=[[Dover Publications]]|___location=New York|isbn=978-0-486-61272-0|year=1972|url=https://archive.org/details/handbookofmathe000abra}}
{{reflist
|refs =
<ref name="Hall_1909">{{cite book
<ref name=
<ref name="arcus">{{cite book|title=Encyclopaedia of Mathematics|title-link=Encyclopedia of Mathematics|author-first=Michiel|author-last=Hazewinkel|author-link=Michiel Hazewinkel|publisher=[[Kluwer Academic Publishers]] / [[Springer Science & Business Media]]|orig-year=1987|date=1994|edition=unabridged reprint|isbn=978-155608010-4}} {{pb}} {{cite book |last1=Bronshtein |first1=I. N. |last2=Semendyayev |first2=K. A. |last3=Musiol |first3=Gerhard |last4=Mühlig |first4=Heiner |title=Handbook of Mathematics |edition=6th |publisher=Springer |___location=Berlin |doi=10.1007/978-3-663-46221-8 |chapter=Cyclometric or Inverse Trigonometric Functions |doi-broken-date=1 July 2025 |at={{nobr|§ 2.8}}, {{pgs|85–89}} }} {{pb}} However, the term "arcus function" can also refer to the function giving the [[Argument (complex analysis)|argument]] of a complex number, sometimes called the ''arcus''.</ref>
<ref name="Americana_1912">{{cite book|chapter=Inverse trigonometric functions|title=The Americana: a universal reference library|volume=21|editor-first1=Frederick Converse|editor-last1=Beach|editor-first2=George Edwin|editor-last2=Rines|date=1912|title-link=The Americana}}</ref>
<ref name="Cajori">{{cite book|url = https://archive.org/details/ahistorymathema02cajogoog|author-last=Cajori|author-first=Florian|author-link=Florian Cajori|title=A History of Mathematics|page=[https://archive.org/details/ahistorymathema02cajogoog/page/n284 272]|edition=2|year=1919|publisher=[[The Macmillan Company]]|___location=New York, NY}}</ref>
<ref name="Herschel_1813">{{cite journal|url=https://books.google.com/books?id=qpRJAAAAYAAJ&pg=PA8|author-last=Herschel|author-first=John Frederick William|author-link=John Frederick William Herschel|title=On a remarkable Application of Cotes's Theorem|journal=Philosophical Transactions|page=8|volume=103|number=1|date=1813|publisher=Royal Society, London|doi=10.1098/rstl.1813.0005|doi-access=free}}</ref>
<ref name="Korn_2000">{{cite book|title=Mathematical handbook for scientists and engineers: Definitions, theorems, and formulars for reference and review|url=https://archive.org/details/mathematicalhand00korn_849|url-access=limited|first1=Grandino Arthur|last1=Korn|first2=Theresa M.|last2=Korn|author2-link= Theresa M. Korn|edition=3<!-- (based on 1968 edition by McGrawHill, Inc.) -->|year=2000|orig-year=1961|publisher=[[Dover Publications, Inc.]]|___location=Mineola, New York, USA|chapter=21.2.-4. Inverse Trigonometric Functions|page=[https://archive.org/details/mathematicalhand00korn_849/page/n828 811]|isbn=978-0-486-41147-7}}</ref>
<ref name="Bhatti_1999">{{cite book|title=Calculus and Analytic Geometry|date=1999|publisher=Punjab Textbook Board|___location=[[Lahore]]|edition=1|page=140|author-first1=Sanaullah|author-last1=Bhatti|author-last2=Nawab-ud-Din|author-first3=Bashir|author-last3=Ahmed|author-first4=S. M.|author-last4=Yousuf|author-first5=Allah Bukhsh|author-last5=Taheem|editor-first1=Mohammad Maqbool|editor-last1=Ellahi|editor-first2=Karamat Hussain|editor-last2=Dar|editor-first3=Faheem|editor-last3=Hussain|language=en-PK|chapter=Differentiation of Trigonometric, Logarithmic and Exponential Functions}}</ref>
<ref name="Borwein_2004">{{cite book|title=Experimentation in Mathematics: Computational Paths to Discovery|url=https://archive.org/details/experimentationm00borw_656|url-access=limited|author-first1=Jonathan|author-last1=Borwein|author-first2=David|author-last2=Bailey|author-first3=Roland|author-last3=Gingersohn|edition=1|date=2004|publisher=[[A. K. Peters]]|page=[https://archive.org/details/experimentationm00borw_656/page/n60 51]|___location=Wellesley, MA, USA|isbn=978-1-56881-136-9}}</ref>
<ref name="Gade_2010">{{cite journal|author-last=Gade|author-first=Kenneth|date=2010|title=A non-singular horizontal position representation|journal=The Journal of Navigation|publisher=[[Cambridge University Press]]|volume=63|issue=3|pages=395–417|url=http://www.navlab.net/Publications/A_Nonsingular_Horizontal_Position_Representation.pdf|doi=10.1017/S0373463309990415|bibcode=2010JNav...63..395G}}</ref>
}}
==External links==
*{{MathWorld|urlname=InverseTangent|title=Inverse Tangent}}
{{Trigonometric and hyperbolic functions}}
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