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{{Trigonometry}}
[[File:Academ Base of trigonometry.svg|thumb|upright=1.35|Basis of trigonometry: if two [[right triangle]]s have equal [[acute angle]]s, they are [[Similarity (geometry)|similar]], so their corresponding side lengths are [[Proportionality (mathematics)|proportional]].]]
In [[mathematics]], the '''trigonometric functions''' (also called '''circular functions''', '''angle functions''' or '''goniometric functions''')<ref name="Klein_1924"/><ref name="Klein_2004"/>{{r|klein}} are [[real function]]s which relate an angle of a [[right-angled triangle]] to ratios of two side lengths. They are widely used in all sciences that are related to [[geometry]], such as [[navigation]], [[solid mechanics]], [[celestial mechanics]], [[geodesy]], and many others. They are among the simplest [[periodic function]]s, and as such are also widely used for studying periodic phenomena through [[Fourier analysis]].
 
The trigonometric functions most widely used in modern mathematics are the [[sine]], the [[cosine]], and the '''tangent''' functions. Their [[multiplicative inverse|reciprocal]]s are respectively the '''cosecant''', the '''secant''', and the '''cotangent''' functions, which are less used. Each of these six trigonometric functions has a corresponding [[Inverse trigonometric functions|inverse function]], and an analog among the [[hyperbolic functions]].
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== Right-angled triangle definitions ==
[[File:TrigonometryTriangle.svg|thumb|In this right triangle, denoting the measure of angle BAC as A: {{math|1=sin ''A'' = {{sfrac|''a''|''c''}}}}; {{math|1=cos ''A'' = {{sfrac|''b''|''c''}}}}; {{math|1=tan ''A'' = {{sfrac|''a''|''b''}}}}.]]
[[File:TrigFunctionDiagram.svg|thumb|Plot of the six trigonometric functions, the unit circle, and a line for the angle {{math|1=''θ'' = 0.7 radians}}. The points labelledlabeled {{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.]]
 
If the acute angle {{mvar|θ}} is given, then any right triangles that have an angle of {{mvar|θ}} are [[similarity (geometry)|similar]] to each other. This means that the ratio of any two side lengths depends only on {{mvar|θ}}. Thus these six ratios define six functions of {{mvar|θ}}, which are the trigonometric functions. In the following definitions, the [[hypotenuse]] is the length of the side opposite the right angle, ''opposite'' represents the side opposite the given angle {{mvar|θ}}, and ''adjacent'' represents the side between the angle {{mvar|θ}} and the right angle.<ref>{{harvtxt|Protter|Morrey|1970|pp=APP-2, APP-3}}</ref><ref>{{Cite web|title=Sine, Cosine, Tangent|url=https://www.mathsisfun.com/sine-cosine-tangent.html|access-date=29 August 2020|website=www.mathsisfun.com}}</ref>
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[[mnemonics in trigonometry|Various mnemonics]] can be used to remember these definitions.
 
In a right-angled triangle, the sum of the two acute angles is a right angle, that is, {{math|90°}} or {{math|{{sfrac|π|2}} [[radian]]s}}. Therefore <math>\sin(\theta)</math> and <math>\cos(90^\circ - \theta)</math> represent the same ratio, and thus are equal. This identity and analogous relationships between the other trigonometric functions are summarized in the following table.
 
[[File:Periodic sine.svg|thumb|'''Top:''' Trigonometric function {{math|sin ''θ''}} for selected angles {{math|''θ''}}, {{math|{{pi}} − ''θ''}}, {{math|{{pi}} + ''θ''}}, and {{math|2{{pi}} − ''θ''}} in the four quadrants.<br>'''Bottom:''' Graph of sine function versus angle. Angles from the top panel are identified.]]
 
{| class="wikitable sortable"
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In geometric applications, the argument of a trigonometric function is generally the measure of an [[angle]]. For this purpose, any [[angular unit]] is convenient. One common unit is [[degree (angle)|degrees]], in which a right angle is 90° and a complete turn is 360° (particularly in [[elementary mathematics]]).
 
However, in [[calculus]] and [[mathematical analysis]], the trigonometric functions are generally regarded more abstractly as functions of [[real number|real]] or [[complex number]]s, rather than angles. In fact, the functions {{math|sin}} and {{math|cos}} can be defined for all complex numbers in terms of the [[exponential function]], via power series,<ref name=":0">{{Cite book|last=Rudin, Walter, 1921–2010|url=https://www.worldcat.org/oclc/1502474|title=Principles of mathematical analysis|isbn=0-07-054235-X|edition=Third |location=New York|oclc=1502474}}</ref> or as solutions to [[differential equation]]s given particular initial values<ref>{{Cite journal|last=Diamond|first=Harvey|date=2014|title=Defining Exponential and Trigonometric Functions Using Differential Equations|url=https://www.tandfonline.com/doi/full/10.4169/math.mag.87.1.37|journal=Mathematics Magazine|language=en|volume=87|issue=1|pages=37–42|doi=10.4169/math.mag.87.1.37|s2cid=126217060|issn=0025-570X}}</ref> (''see below''), without reference to any geometric notions. The other four trigonometric functions ({{math|tan}}, {{math|cot}}, {{math|sec}}, {{math|csc}}) can be defined as quotients and reciprocals of {{math|sin}} and {{math|cos}}, except where zero occurs in the denominator. It can be proved, for real arguments, that these definitions coincide with elementary geometric definitions ''if'' ''the argument is regarded as an angle given in radians''.<ref name=":0" /> Moreover, these definitions result in simple expressions for the [[derivative]]s and [[Antiderivative|indefinite integrals]] for the trigonometric functions.<ref name=":1">{{Cite book|last=Spivak|first=Michael|title=Calculus|publisher=Addison-Wesley|year=1967|chapter=15|pages=256–257|lccn=67-20770}}</ref> Thus, in settings beyond elementary geometry, radians are regarded as the mathematically natural unit for describing angle measures.
 
When [[radian]]s (rad) are employed, the angle is given as the length of the [[arc (geometry)|arc]] of the [[unit circle]] subtended by it: the angle that subtends an arc of length 1 on the unit circle is 1 rad (≈ 57.3°), and a complete [[turn (angle)|turn]] (360°) is an angle of 2{{pi}} (≈ 6.28) rad. For real number ''x'', the notationsnotation {{math|sin ''x''}}, {{math|cos ''x''}}, etc. referrefers to the value of the trigonometric functions evaluated at an angle of ''x'' rad. If units of degrees are intended, the degree sign must be explicitly shown (e.g., {{math|sin ''x°''}}, {{math|cos ''x°''}}, etc.). Using this standard notation, the argument ''x'' for the trigonometric functions satisfies the relationship ''x'' = (180''x''/{{pi}})°, so that, for example, {{math|1=sin {{pi}} = sin 180°}} when we take ''x'' = {{pi}}. In this way, the degree symbol can be regarded as a mathematical constant such that 1° = {{pi}}/180 ≈ 0.0175.
 
==Unit-circle definitions==
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[[File:Taylorreihenentwicklung des Kosinus.svg|thumb|<math>\cos(x)</math> together with the first Taylor polynomials <math>p_n(x)=\sum_{k=0}^n (-1)^k \frac{x^{2k}}{(2k)!}</math>]]
 
[[G. H. Hardy]] noted in his 1908 work ''[[A Course of Pure Mathematics]]'' that the definition of the trigonometric functions in terms of the unit circle is not satisfactory, because it depends implicitly on a notion of angle that can be measured by a real number.<ref name="Hardy">{{citation|first=G.H.|last=Hardy|title=A 1908course pagesof 432-438pure mathematics|year=1950|edition=8th|pages=432–438}}</ref> Thus in modern analysis, trigonometric functions are usually constructed without reference to geometry.
 
HardyVarious describesways fourexist waysin ofthe literature for defining the trigonometric functions in a manner suitable for analysis; they include:
* Using the "geometry" of the unit circle, which requires formulating the arc length of a circle (or area of a sector) analytically.<ref name="Hardy"/>
* By a power series, which is particularly well-suited to complex variables.<ref name="Hardy"/><ref name="WW">Whittaker, E. T., & Watson, G. N. (1920). A course of modern analysis: an introduction to the general theory of infinite processes and of analytic functions; with an account of the principal transcendental functions. University press.</ref>
* By using an infinite product expansion.<ref name="Hardy"/>
* By inverting the inverse trigonometric functions, which can be defined as integrals of algebraic or rational functions.<ref name="Hardy"/>
* As solutions of a differential equation.<ref name="BS">Bartle, R. G., & Sherbert, D. R. (2000). Introduction to real analysis (3rd ed). Wiley.</ref>
 
To this list of ways of defining the trigonometric functions, Bartle and Sherbert add:
* As solutions of a differential equation.
 
===Definition by differential equations===
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=== Definition via integration ===
Another way to define the trigonometric functions in analysis is using integration.<ref name="Hardy"/><ref>{{citation|last=Bartle|year=1964|title=Elements of real analysis|publisher=|pages=315–316}}</ref> For a real number <math>t</math>, put
The unit circle <math>x^2+y^2=1</math> is a rational curve under the [[tangent half-angle substitution|substitution]]<ref>Hardy (1908) ''A course in pure mathematics'', employs a similar approach but with a tangent substitution instead, which is algebraic rather than rational.</ref>
<math display="block">x\theta(t) = \frac{1-tint_0^2}{1+t^2},\quad y=\frac{2td\tau}{1+t\tau^2},=\quad -\infty <arctan t < \infty</math>
where this defines this inverse tangent function. Also, <math>\pi</math> is defined by
which covers every point once except <math>(-1,0)</math>. Then the element of arc length along the circle is
<math display="block">ds\frac12\pi = \int_0^\infty \frac{2d\,dttau}{1+t\tau^2}.</math>
a definition that goes back to [[Karl Weierstrass]].<ref>{{cite book |last=Weierstrass |first=Karl |author-link=Karl Weierstrass |chapter=Darstellung einer analytischen Function einer complexen Veränderlichen, deren absoluter Betrag zwischen zwei gegebenen Grenzen liegt |trans-chapter=Representation of an analytical function of a complex variable, whose absolute value lies between two given limits |language=de |title=Mathematische Werke |volume=1 |publication-place=Berlin |publisher=Mayer & Müller |year=1841 |publication-date=1894 |pages=51–66 |chapter-url=https://archive.org/details/mathematischewer01weieuoft/page/51/ }}</ref>
Consider the integral
<math display="block">\theta(t) = 2\arctan(t)=\int_0^t\frac{2\,d\tau}{1+\tau^2}.</math>▼
The integrand has poles at imaginary points <math>\tau=\pm i</math>, with residues <math>\pm i</math>. It follows that the integral is independent of the complex contour of integration only if taken to have values in the [[Riemann surface]] <math>\mathbb C/2\pi\mathbb Z</math>. The factor of two allows us to define <math>\pi</math> as the semiperiod:
<math display="block">\pi = \int_0^\infty\frac{2\,d\tau}{1+\tau^2}</math>
The trigonometric functions are defined by inverting <math>\theta(t)</math>. Namely, we put
<math display="block">\cos\theta = \frac{1-t^2}{1+t^2},\quad \sin\theta = \frac{2t}{1+t^2}</math>▼
for <math>(t,\theta)</math> a point of the graph of the function <math>\theta=2\arctan(t)</math>. This argument also shows that the cosine and sine, defined in this way, have semiperiod <math>\pi</math>.
 
Note thatOn the substitutioninterval <math>t-\to 1pi/t2<\theta<\pi/2</math> transforms, the integrandtrigonometric definingfunctions <math>\theta(t)</math> toare itselfdefined whileby exchanginginverting the intervalsrelation <math>(0,1)</math>\theta and= <math>(1,\infty)arctan t</math>,. Thus andwe sodefine the quartertrigonometric periodfunctions isby
<math display="block">\frac{tan\pi}{2}theta = t,\int_0^1quad \fraccos\theta = (1+t^2)^{-1/2\,dt}{,\quad \sin\theta = t(1+t^2)^{-1/2}.</math>
where the point <math>(t,\theta)</math> is on the graph of <math>\theta=\arctan t</math> and the positive square root is taken.
Using this, one can then show that
<math display="block">\theta\left(\frac{1+t}{1-t}\right) = \theta(t) + \frac{\pi}{2}</math>▼
by making the rational substitution
<math display="block">\tau = \frac{1+\sigma}{1-\sigma}</math>▼
in the integral defining <math>\theta(t)</math>. Hence we have
<math display="block">\sin\left(\theta + \frac{\pi}{2}\right) = \frac{2\left(\frac{1+t}{1-t}\right)}{1+\left(\frac{1+t}{1-t}\right)^2} = \frac{1-t^2}{1+t^2} = \cos(\theta).</math>
Likewise,
<math display="block">\cos\left(\theta + \frac{\pi}{2}\right) = \frac{1-\left(\frac{1+t}{1-t}\right)^2}{1+\left(\frac{1+t}{1-t}\right)^2} = \frac{-2t}{1+t^2} = -\sin(\theta).</math>
 
This defines the trigonometric functions on <math>(-\pi/2,\pi/2)</math>. The definition can be extended to all real numbers by first observing that, as <math>\theta\to\pi/2</math>, <math>t\to\infty</math>, and so <math>\cos\theta = (1+t^2)^{-1/2}\to 0</math> and <math>\sin\theta = t(1+t^2)^{-1/2}\to 1</math>. Thus <math>\cos\theta</math> and <math>\sin\theta</math> are extended continuously so that <math>\cos(\pi/2)=0,\sin(\pi/2)=1</math>. Now the conditions <math>\cos(\theta+\pi)=-\cos(\theta)</math> and <math>\sin(\theta+\pi)=-\sin(\theta)</math> define the sine and cosine as periodic functions with period <math>2\pi</math>, for all real numbers.
More complicated substitutions of this kind can also be used to show the addition formulae
 
<math display="block">\cos(x + y) = \cos(x)\cos(y) - \sin(x)\sin(y),</math>
Proving the basic properties of sine and cosine, including the fact that sine and cosine are analytic, one may first establish the addition formulae. First,
<math display="block">\sin(x + y) = \cos(x)\sin(y) + \sin(x)\cos(y),</math>
<math display="block">\tan(xarctan s + y)\arctan t = \arctan(\frac{\tan x s+ \tan yt}{1-\tan x\tan yst}.)</math>
holds, provided <math>\arctan s+\arctan t\in(-\pi/2,\pi/2)</math>, since
▲<math display="block">\theta(t)arctan =s + 2\arctan( t)= \int_0int_{-s}^t\frac{d\tau}{1+\tau^2}=\,int_0^{\frac{s+t}{1-st}}\frac{d\tau}{1+\tau^2}.</math>
after the substitution <math>\tau \to \frac{s+\tau}{1-s\tau}</math>. In particular, the limiting case as <math>s\to\infty</math> gives
▲<math display="block">\tauarctan =t + \frac{1+\sigmapi}{2} = \arctan(-1/t),\quad t\in (-\sigma}infty,0).</math>
Thus we have
▲<math display="block">\thetasin\left(\theta + \frac{1+t\pi}{1-t2}\right) = \thetafrac{-1}{t\sqrt{1+(-1/t)^2}} += \frac{\pi-1}{\sqrt{1+t^2}} = -\cos(\theta)</math>
and
▲<math display="block">\cos\left(\theta + \frac{\pi}{2}\right) = \frac{1-t^2}{\sqrt{1+(-1/t)^2},\quad \sin\theta} = \frac{2tt}{\sqrt{1+t^2}} = \sin(\theta).</math>
So the sine and cosine functions are related by translation over a quarter period <math>\pi/2</math>.
 
===Definitions using functional equations===
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==Basic identities==
Many [[identity (mathematics)|identities]] interrelate the trigonometric functions. This section contains the most basic ones; for more identities, see [[List of trigonometric identities]]. These identities may be proved geometrically from the unit-circle definitions or the right-angled-triangle definitions (although, for the latter definitions, care must be taken for angles that are not in the interval {{math|[0, {{pi}}/2]}}, see [[Proofs of trigonometric identities]]). For non-geometrical proofs using only tools of [[calculus]], one may use directly the differential equations, in a way that is similar to that of the [[#Relationship to exponential function (Euler's formula) and the exponential function|above proof]] of Euler's identity. One can also use Euler's identity for expressing all trigonometric functions in terms of complex exponentials and using properties of the exponential function.
 
===Parity===
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==See also==
{{colbegin|colwidth=25em}}
* [[Mnemonics in trigonometry]]
* [[Bhaskara I's sine approximation formula]]
* [[Small-angle approximation]]
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* [[Generalized trigonometry]]
* [[Generating trigonometric tables]]
* [[Hyperbolic functions]]
* [[List of integrals of trigonometric functions]]
* [[List of periodic functions]]
* [[List of trigonometric identities]]
* [[Polar sine]] – a generalization to vertex angles
* [[Proofs of trigonometric identities]]
* [[Versine]] – for several less used trigonometric functions and unit circle diagrams of all functions
* [[Chord (geometry)#In trigonometry]]
{{colend}}
 
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{{notelist}}
{{reflist|refs=
<ref name="Klein_1924"klein>{{cite book |chapter=Die goniometrischen Funktionen |at={{nobr|Ch. 3.2}}, {{pgs|175 ff.}} |title=Elementarmathematik vom höheren Standpunkt aus: Arithmetik, Algebra, Analysis |volume=1 |author-first=Christian Felix |author-last=Klein |author-link=Christian Felix Klein |date=1924<!-- 1927 --> |orig-year=1902<!-- 1908 --> |edition=3rd |publisher=[[J. Springer]] |location=Berlin |language=de |chapter-url=https://books.google.com/books?id=5t8fAAAAIAAJ&pg=PA175 }} 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 |translator-first1=E. R. |translator-last1=Hedrick |translator-first2=C. A. |translator-last2=Noble |chapter-url=https://archive.org/details/geometryelementa0000feli/page/162/?q=%22ii.+the+goniometric+functions%22 |chapter=The Goniometric Functions |at=Ch. 3.2, {{pgs|162 ff.}} }}</ref>
<ref name="Klein_2004">{{cite book |title=Elementary Mathematics from an Advanced Standpoint: Arithmetic, Algebra, Analysis |author-first=Christian Felix |author-last=Klein |author-link=Christian Felix Klein |date=2004 |orig-year=1932 |edition=Translation of 3rd German |publisher=[[Dover Publications, Inc.]] / [[The Macmillan Company]] |translator-first1=E. R. |translator-last1=Hedrick |translator-first2=C. A. |translator-last2=Noble |isbn=978-0-48643480-3 |url=https://books.google.com/books?id=8KuoxgykfbkC |access-date=13 August 2017 |url-status=live |archive-url=https://web.archive.org/web/20180215144848/https://books.google.com/books?id=8KuoxgykfbkC |archive-date=15 February 2018 }}</ref>
<ref name="Larson_2013">{{cite book |title=Trigonometry |edition=9th |first1=Ron |last1=Larson |publisher=Cengage Learning |date=2013 |isbn=978-1-285-60718-4 |page=153 |url=https://books.google.com/books?id=zbgWAAAAQBAJ |url-status=live |archive-url=https://web.archive.org/web/20180215144848/https://books.google.com/books?id=zbgWAAAAQBAJ |archive-date=15 February 2018 }} [https://books.google.com/books?id=zbgWAAAAQBAJ&pg=PA153 Extract of page 153] {{webarchive|url=https://web.archive.org/web/20180215144848/https://books.google.com/books?id=zbgWAAAAQBAJ&pg=PA153 |date=15 February 2018 }}</ref>
<ref name="Aigner_2000">{{cite book |author-last1=Aigner |author-first1=Martin |author1-link=Martin Aigner |author-last2=Ziegler |author-first2=Günter M. |author-link2=Günter Ziegler |title=Proofs from THE BOOK |publisher=[[Springer-Verlag]] |edition=Second |date=2000 |isbn=978-3-642-00855-9 |page=149 |url=https://www.springer.com/mathematics/book/978-3-642-00855-9 |url-status=live |archive-url=https://web.archive.org/web/20140308034453/http://www.springer.com/mathematics/book/978-3-642-00855-9 |archive-date=8 March 2014 }}</ref>
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