Inverse trigonometric functions: Difference between revisions

Revision as of 22:26, 29 November 2024

Latest revision as of 16:43, 25 August 2025

(19 intermediate revisions by 15 users not shown)

Line 2:

{{use dmy dates|date=July 2022 |cs1-dates=sy }}

{{Trigonometry}}

{{redirect|Arctangent|the Earthtone9 album|arc'tan'gent|the music event|ArcTanGent Festival}}

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]].

Line 29 ⟶ 30:

{| class="wikitable" style="text-align:center"

|-

! scope="col" | Name

! scope="col" | Usual notation

! scope="col" | Definition

! scope="col" | Domain of <math>{{mvar|x</math>}} for real result

! scope="col" | Range of usual principal value <br/> ([[radian]]s)

! scope="col" | Range of usual principal value <br/> ([[Degree (angle)|degrees]])

|-

! scope="row" | arcsine

| '''arcsine''' || <math>y = \arcsin(x)</math> || {{math|1=''x'' = [[sine|sin]](''y'')}} || <math>-1 \leq x \leq 1</math> || <math>- \frac{\pi}{2} \leq y \leq \frac{\pi}{2}</math> || <math>- 90^{\circ} \leq y \leq 90^{\circ}</math>

| {{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= '''arccosine'y'' || <math>y = \arccos(''x'')</math>}} || {{math|1=''x'' = [[cosine|cos]](''y'')}} || <{{math>-1|−1 \leq≤ ''x'' \leq≤ 1</math>}} || <{{math>|0 \leq≤ ''y'' \leq≤ \pi</math>π}} || <{{math>|0^{\circ}° \leq≤ ''y'' \leq≤ 180^{\circ°}}</math>

|-

! scope="row" | arctangent

| {{math|1= ''y'arctangent''' || <math>y = \arctan(''x'')</math>}} || {{math|1=''x'' = [[Tangent (trigonometry)|tan]](''y'')}} || all real numbers || <{{math>- \frac|−{\pi}{sfrac|π|2}} < ''y'' < \frac{\pi}{sfrac|π|2}</math>}}} || <{{math>- 90^{\circ}|−90° < ''y'' < 90^{\circ°}}</math>

|-

! scope="row" | arccotangent

| {{math|1= '''arccotangent'y'' || <math>y = \arccot(''x'')</math>}} || {{math|1=''x'' = [[cotangent|cot]](''y'')}} || all real numbers || {{math|0 < ''y'' < π}} || {{math|0° < ''y'' < 180°}}

| <math>0 < y < \pi</math> || <math>0^{\circ} < y < 180^{\circ}</math>

|-

! scope="row" | arcsecant

| '''arcsecant''' || <math>y = \arcsec(x)</math> || {{math|1=''x'' = [[Secant (trigonometry)|sec]](''y'')}} || <math>{\left\vert x \right\vert} \geq 1</math>|| <math>0 \leq y < \frac{\pi}{2} \text{ or } \frac{\pi}{2} < y \leq \pi</math> || <math>0^{\circ} \leq y < 90^{\circ} \text{ or } 90^{\circ} < y \leq 180^{\circ}</math>

| {{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

| '''arccosecant''' || <math>y = \arccsc(x)</math> ||{{math|1=''x'' = [[cosecant|csc]](''y'')}} || <math>{\left\vert x \right\vert} \geq 1</math>|| <math>- \frac{\pi}{2} \leq y < 0 \text{ or } 0 < y \leq \frac{\pi}{2}</math> || <math>- 90^{\circ} \leq y < 0^{\circ} \text{ or } 0^{\circ} < y \leq 90^{\circ}</math>

| {{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°}}

|-

|}

Note: Some authors {{Citation needed|reason=Unable to find this anywhere|date=March 2021}} define the range of arcsecant to be {{nowrap|(<math display="inline">( 0 \leq y < \frac{\pi}{2} \text{</math>}} or }{{nowrap|<math display="inline">\pi \leq y < \frac{3 \pi}{2} )</math> ),}}<ref>For example: {{pb}} {{cite book |last1=Stewart |first1=James |last2=Clegg |first2=Daniel |last3=Watson |first3=Saleem |year=2021 |title=Calculus: Early Transcendentals |edition=9th |isbn=978-1-337-61392-7 |chapter=Inverse Functions and Logarithms |publisher=Cengage Learning |at=§ 1.5, {{pgs|64}} }}</ref> because the tangent function is nonnegative on this ___domain. This makes some computations more consistent. For example, using this range, <math>\tan(\arcsec(x)) = \sqrt{x^2 - 1},</math> whereas with the range {{nowrap|(<math display="inline">( 0 \leq y < \frac{\pi}{2} \text{</math>}} or }{{nowrap|<math display="inline">\frac{\pi}{2} < y \leq \pi )</math>),}} we would have to write <math>\tan(\arcsec(x)) = \pm \sqrt{x^2 - 1},</math> since tangent is nonnegative on <math display="inline">0 \leq y < \frac{\pi}{2},</math> but nonpositive on <math display="inline">\frac{\pi}{2} < y \leq \pi.</math> For a similar reason, the same authors define the range of arccosecant to be <math display="inline">( - \pi < y \leq - \frac{\pi}{2}</math> or <math display="inline">0 < y \leq \frac{\pi}{2} ) .</math>

====Domains====

If <math>{{mvar|x</math>}} is allowed to be a [[complex number]], then the range of <math>{{mvar|y</math>}} applies only to its real part.

{{DomainsImagesAndPrototypesOfTrigAndInverseTrigFunctions

Line 675 ⟶ 681:

[[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''&pi;''}} and {{math|−2''&pi;''}} respectively.]]

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 (−i and +i) is the [[branch cut]] between the principal sheet and other sheets. The path of the integral must not cross a branch cut. For ''z'' not on a branch cut, a straight line path from 0 to ''z'' is such a path. For ''z'' on a branch cut, the path must approach from {{nowrap|Re[x] > 0}} for the upper branch cut and from {{nowrap|Re[x] < 0}} for the lower branch cut.

Line 712 ⟶ 718:

:<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}

Line 806 ⟶ 812:

{{anchor|Two-argument variant of arctangent}}

{{main|atan2}}

The two-argument [[atan2|{{math|atan2}}]] function computes the arctangent of {{math|''y'' / ''x''}} given ''{{mvar|y''}} and ''{{mvar|x''}}, but with a range of (−{{pi}}open-closed|−π,&nbsp;{{pi π}}]. In other words, {{math|atan2(''y'',&nbsp; ''x'')}} is the angle between the positive ''{{mvar|x''}}-axis of a plane and the point {{math|(''x'',&nbsp; ''y'')}} on it, with positive sign for counter-clockwise angles ([[upper half-plane]], {{math|''y''&nbsp; >&nbsp; 0}}), and negative sign for clockwise angles (lower half-plane, {{math|''y''&nbsp; <&nbsp; 0}}). It was first introduced in many computer programming languages, but it is now also common in other fields of science and engineering.

In terms of the standard '''arctan''' function, that is with range of (−{{sfrac|{{pi}}open-open|−π/2}}, {{sfrac|{{pi}}|π/2}}), it can be expressed as follows:

:<math display="block">\operatorname{atan2}(y, x) = \begin{cases}

\arctan\left(\frac y x\right) & \quad x > 0 \\

\arctan\left(\frac y x\right) + \pi & \quad y \ge 0,\; x < 0 \\

Line 819 ⟶ 825:

\end{cases}</math>

It also equals the [[principal value]] of the [[arg (mathematics)|arg]]ument of the [[complex number]] {{math|''x''&nbsp; +&nbsp; '''i'''''yiy''}}.

This limited version of the function above may also be defined using the [[tangent half-angle formula]]e as follows:

:<math display="block">\operatorname{atan2}(y, x) = 2\arctan\left(\frac{y}{\sqrt{x^2 + y^2} + x}\right)</math>

provided that either {{math|''x''&nbsp; >&nbsp; 0}} or {{math|''y''&nbsp; ≠&nbsp; 0}}. However this fails if given {{math|''x&nbsp;'' ≤&nbsp; 0}} and {{math|1= ''y&nbsp;'' =&nbsp; 0}} so the expression is unsuitable for computational use.

The above argument order (''{{mvar|y''}}, ''{{mvar|x''}}) seems to be the most common, and in particular is used in [[ISO standard]]s such as the [[C (programming language)|C programming language]], but a few authors may use the opposite convention (''{{mvar|x''}}, ''{{mvar|y''}}) so some caution is warranted. These{{crossref|(See variations are detailed at [[{{slink|atan2#|Realizations of the function in common computer languages|atan2]]}}.)}}

====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, astromomyastronomy, angular movement in general){{citation needed|date=March 2020}}</ref> the solution <math>y</math> of the equation <math>x=\tan(y)</math> is to come as close as possible to a given value <math>-\infty < \eta < \infty</math>. The adequate solution is produced by the parameter modified arctangent function

:<math>

y = \arctan_\eta(x) := \arctan(x) + \pi \, \operatorname{rni}\left(\frac{\eta - \arctan(x)}{\pi} \right)\, .

Line 861 ⟶ 867:

<ref name="Hall_1909">{{cite book|title=Trigonometry|volume=Part I: Plane Trigonometry|author1-first=Arthur Graham|author1-last=Hall|author2-first=Fred Goodrich|author2-last=Frink|date=January 1909|___location=Ann Arbor, Michigan, USA|chapter=Chapter II. The Acute Angle [14] Inverse trigonometric functions|publisher=[[Henry Holt and Company]] / Norwood Press / J. S. Cushing Co. - Berwick & Smith Co., Norwood, Massachusetts, USA|publication-place=New York, USA|page=15|chapter-url = https://archive.org/stream/planetrigonometr00hallrich#page/n30/mode/1up|access-date=2017-08-12|quote=[…] {{mono|1=α&nbsp;= arcsin&nbsp;''m''}}: It is frequently read "[[arc-sine]] ''m''" or "[[anti-sine]] ''m''," since two mutually inverse functions are said each to be the [[anti-function]] of the other. […] A similar symbolic relation holds for the other [[trigonometric function]]s. […] This notation is universally used in Europe and is fast gaining ground in this country. A less desirable symbol, {{mono|1=α&nbsp;= sin{{sup|-1}}''m''}}, is still found in English and American texts. The notation {{mono|1=α&nbsp;= inv sin ''m''}} is perhaps better still on account of its general applicability. […]}}</ref>

<ref name=cyclometric>{{cite book|title=Elementarmathematik vom höheren Standpunkt aus: Arithmetik, Algebra, Analysis|volume=1|author-first=Felix|author-last=Klein|author-link=Felix Klein|date=1924<!-- 1927 -->|orig-year=1902<!-- 1908 -->|edition=3rd|publisher=J. Springer|___location=Berlin|language=de}} Translated as {{cite book|title=Elementary Mathematics from an Advanced Standpoint: Arithmetic, Algebra, Analysis |author-first=Felix |author-last=Klein |author-link=Felix Klein |display-authors=0 |year=1932 |publisher=Macmillan |isbn=978-0-486-43480-3 |translator-first1=E. R. |translator-last1=Hedrick |translator-first2=C. A. |translator-last2=Noble |url=https://books.google.com/books?id=8KuoxgykfbkC }}</ref>

<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 NovemberJuly 20242025 |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>