View source for Trigonometric functions

{{short description|Functions of an angle}}
{{redir-multi|2|Logarithmic sine|Logarithmic cosine|the Clausen-related functions|log cosine function|and|log sine function}}
{{Use dmy dates|date=September 2021|cs1-dates=y}}
[[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'''){{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]].
{{Trigonometry}}

The trigonometric functions most widely used in modern mathematics are the [[sine]], the [[cosine]], and the '''tangent''' functions. {{anchor|Reciprocal_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 has an analog among the [[hyperbolic functions]].

The oldest definitions of trigonometric functions, related to right-angle triangles, define them only for [[acute angle]]s. To extend the sine and cosine functions to functions whose [[domain of a function|domain]] is the whole [[real line]], geometrical definitions using the standard [[unit circle]] (i.e., a circle with [[radius]] 1 unit) are often used; then the domain of the other functions is the real line with some isolated points removed. Modern definitions express trigonometric functions as [[Series (mathematics)|infinite series]] or as solutions of [[differential equation]]s. This allows extending the domain of sine and cosine functions to the whole [[complex plane]], and the domain of the other trigonometric functions to the complex plane with some isolated points removed.

== Notation ==
Conventionally, an abbreviation of each trigonometric function's name is used as its symbol in formulas. Today, the most common versions of these abbreviations are "{{math|sin}}" for sine, "{{math|cos}}" for cosine, "{{math|tan}}" or "{{math|tg}}" for tangent, "{{math|sec}}" for secant, "{{math|csc}}" or "{{math|cosec}}" for cosecant, and "{{math|cot}}" or "{{math|ctg}}" for cotangent. Historically, these abbreviations were first used in prose sentences to indicate particular [[line segment]]s or their lengths related to an [[circular arc|arc]] of an arbitrary circle, and later to indicate ratios of lengths, but as the [[history of the function concept|function concept developed]] in the 17th–18th century, they began to be considered as functions of real-number-valued angle measures, and written with [[functional notation]], for example {{math|sin(''x'')}}. Parentheses are still often omitted to reduce clutter, but are sometimes necessary; for example the expression <math>\sin x+y</math> would typically be interpreted to mean <math>(\sin x)+y,</math> so parentheses are required to express <math>\sin (x+y).</math>

A [[positive integer]] appearing as a superscript after the symbol of the function denotes [[exponentiation]], not [[Function composition#Functional powers|function composition]]. For example <math>\sin^2 x</math> and <math>\sin^2 (x)</math> denote <math>(\sin x)^2,</math> not <math>\sin(\sin x).</math> This differs from the (historically later) general functional notation in which <math>f^2(x) = (f \circ f)(x) = f(f(x)).</math>

In contrast, the superscript <math>-1</math> is commonly used to denote the [[inverse function]], not the [[multiplicative inverse |reciprocal]]. For example <math>\sin^{-1}x</math> and <math>\sin^{-1}(x)</math> denote the [[inverse trigonometric function]] alternatively written <math>\arcsin x\,.</math> The equation <math>\theta = \sin^{-1}x</math> implies <math>\sin \theta = x,</math> not <math>\theta \cdot \sin x = 1.</math> In this case, the superscript ''could'' be considered as denoting a composed or [[iterated function]], but negative superscripts other than <math>{-1}</math> are not in common use.

== 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 labeled {{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>
{|
| style="padding-left: 2em; padding-right: 2em; |
;sine: <math>\sin \theta = \frac \mathrm{opposite}\mathrm{hypotenuse}</math>
| style="padding-left: 2em; padding-right: 2em; |
;cosecant: <math>\csc \theta = \frac \mathrm{hypotenuse}\mathrm{opposite}</math>
|-
| style="padding-left: 2em; padding-right: 2em; |
;cosine: <math>\cos \theta = \frac \mathrm{adjacent}\mathrm{hypotenuse}</math>
| style="padding-left: 2em; padding-right: 2em; |
;secant: <math>\sec \theta = \frac \mathrm{hypotenuse}\mathrm{adjacent}</math>
|-
| style="padding-left: 2em; padding-right: 2em; |
;tangent: <math>\tan \theta = \frac \mathrm{opposite}\mathrm{adjacent}</math>
| style="padding-left: 2em; padding-right: 2em; |
;cotangent: <math>\cot \theta = \frac \mathrm{adjacent}\mathrm{opposite}</math>
|}
[[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 {{mvar|θ}}, {{math|{{pi}} − ''θ''}}, {{math|{{pi}} + ''θ''}}, and {{math|2{{pi}} − ''θ''}} in the four quadrants.<br>'''Bottom:''' Graph of sine versus angle. Angles from the top panel are identified.]]

{| class="wikitable sortable"
|+ Summary of relationships between trigonometric functions<ref>{{harvtxt|Protter|Morrey|1970|p=APP-7}}</ref>
|-
! rowspan=2 | Function
! rowspan=2 | Description
! colspan=2 | [[List of trigonometric identities|Relationship]]
|-
! using [[radian]]s
! using [[Degree (angle)|degree]]s
|-
! sine
|align=center|{{math|{{sfrac|opposite|hypotenuse}}}}
| <math>\sin \theta = \cos\left(\frac{\pi}{2} - \theta \right) = \frac{1}{\csc \theta}</math>
| <math>\sin x = \cos\left(90^\circ - x \right) = \frac{1}{\csc x}</math>
|-
! cosine
|align=center|{{math|{{sfrac|adjacent|hypotenuse}}}}
| <math>\cos \theta = \sin\left(\frac{\pi}{2} - \theta \right) = \frac{1}{\sec \theta}\,</math>
| <math>\cos x = \sin\left(90^\circ - x \right) = \frac{1}{\sec x}\,</math>
|-
! tangent
|align=center|{{math|{{sfrac|opposite|adjacent}}}}
| <math>\tan \theta = \frac{\sin \theta}{\cos \theta} = \cot\left(\frac{\pi}{2} - \theta \right) = \frac{1}{\cot \theta}</math>
| <math>\tan x = \frac{\sin x}{\cos x} = \cot\left(90^\circ - x \right) = \frac{1}{\cot x}</math>
|-
! cotangent
|align=center|{{math|{{sfrac|adjacent|opposite}}}}
| <math>\cot \theta = \frac{\cos \theta}{\sin \theta} = \tan\left(\frac{\pi}{2} - \theta \right) = \frac{1}{\tan \theta}</math>
| <math>\cot x = \frac{\cos x}{\sin x} = \tan\left(90^\circ - x \right) = \frac{1}{\tan x}</math>
|-
! secant
|align=center|{{math|{{sfrac|hypotenuse|adjacent}}}}
| <math>\sec \theta = \csc\left(\frac{\pi}{2} - \theta \right) = \frac{1}{\cos \theta}</math>
| <math>\sec x = \csc\left(90^\circ - x \right) = \frac{1}{\cos x}</math>
|-
! cosecant
|align=center|{{math|{{sfrac|hypotenuse|opposite}}}}
| <math>\csc \theta = \sec\left(\frac{\pi}{2} - \theta \right) = \frac{1}{\sin \theta}</math>
| <math>\csc x = \sec\left(90^\circ - x \right) = \frac{1}{\sin x}</math>
|}

==Radians versus degrees==
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|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|url-access=subscription}}</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 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°),<ref>{{cite oeis|A072097|Decimal expansion of 180/Pi}}</ref> and a complete [[turn (angle)|turn]] (360°) is an angle of 2{{pi}} (≈ 6.28) rad.<ref>{{cite oeis|A019692|Decimal expansion of 2*Pi}}</ref> Since radian is dimensionless, i.e. 1 rad = 1, the degree symbol can also be regarded as a [[mathematical constant]] [[Factor (arithmetic)|factor]] such that 1° = {{pi}}/180 ≈ 0.0175.{{citation needed |date=July 2025}}

==Unit-circle definitions==
[[Image:Circle-trig6.svg|right|thumb|upright=1.35|All of the trigonometric functions of the angle {{mvar|θ}} (theta) can be constructed geometrically in terms of a unit circle centered at {{mvar|O}}.]]
[[File:Periodic sine.svg|thumb|Sine function on unit circle (top) and its graph (bottom)]]
[[File:Unit Circle Definitions of Six Trigonometric Functions.svg|thumb|upright=1.2|In this illustration, the six trigonometric functions of an arbitrary angle {{mvar|θ}} are represented as [[Cartesian coordinates]] of points related to the [[unit circle]]. The {{mvar|y}}-axis ordinates of {{math|A}}, {{math|B}} and {{math|D}} are {{math|sin ''θ''}}, {{math|tan ''θ''}} and {{math|csc ''θ''}}, respectively, while the {{mvar|x}}-axis abscissas of {{math|A}}, {{math|C}} and {{math|E}} are {{math|cos ''θ''}}, {{math|cot ''θ''}} and {{math|sec ''θ''}}, respectively.]]
[[File:trigonometric function quadrant sign.svg|thumb|Signs of trigonometric functions in each quadrant. [[mnemonics in trigonometry|Mnemonics]] like "'''all''' '''s'''tudents '''t'''ake '''c'''alculus" indicates when '''s'''ine, '''c'''osine, and '''t'''angent are positive from quadrants I to IV.<ref name=steuben>{{Cite book |last1=Stueben |first1=Michael |title=Twenty years before the blackboard: the lessons and humor of a mathematics teacher |last2=Sandford |first2=Diane |date=1998 |publisher=Mathematical Association of America |isbn=978-0-88385-525-6 |series=Spectrum series |location=Washington, DC|page=119|url=https://books.google.com/books?id=qnd0P-Ja-O8C&dq=%22All+Students+Take+Calculus%22&pg=PA119}}</ref>]]

The six trigonometric functions can be defined as [[Cartesian coordinate system|coordinate values]] of points on the [[Euclidean plane]] that are related to the [[unit circle]], which is the [[circle]] of radius one centered at the origin {{math|O}} of this coordinate system. While [[#Right-angled triangle definitions|right-angled triangle definitions]] allow for the definition of the trigonometric functions for angles between {{math|0}} and <math display="inline">\frac{\pi}{2}</math> [[radian]]s {{math|(90°),}} the unit circle definitions allow the domain of trigonometric functions to be extended to all positive and negative real numbers.

Let <math>\mathcal L</math> be the [[Ray (geometry)|ray]] obtained by rotating by an angle {{mvar|θ}} the positive half of the {{mvar|x}}-axis ([[counterclockwise]] rotation for <math>\theta > 0,</math> and clockwise rotation for <math>\theta < 0</math>). This ray intersects the unit circle at the point <math>\mathrm{A} = (x_\mathrm{A},y_\mathrm{A}).</math> The ray <math>\mathcal L,</math> extended to a [[line (geometry)|line]] if necessary, intersects the line of equation <math>x=1</math> at point <math>\mathrm{B} = (1,y_\mathrm{B}),</math> and the line of equation <math>y=1</math> at point <math>\mathrm{C} = (x_\mathrm{C},1).</math> The [[tangent line]] to the unit circle at the point {{math|A}}, is [[perpendicular]] to <math>\mathcal L,</math> and intersects the {{mvar|y}}- and {{mvar|x}}-axes at points <math>\mathrm{D} = (0,y_\mathrm{D})</math> and <math>\mathrm{E} = (x_\mathrm{E},0).</math> The [[Cartesian coordinates|coordinates]] of these points give the values of all trigonometric functions for any arbitrary real value of {{mvar|θ}} in the following manner.

The trigonometric functions {{math|cos}} and {{math|sin}} are defined, respectively, as the {{mvar|x}}- and {{mvar|y}}-coordinate values of point {{math|A}}. That is,
<math display="block">\cos \theta = x_\mathrm{A} \quad</math> and <math>\quad \sin \theta = y_\mathrm{A}.</math><ref>{{Cite web|url=https://www.encyclopediaofmath.org/index.php/Trigonometric_functions|title=Trigonometric Functions|last=Bityutskov|first=V.I.|date=7 February 2011|website=Encyclopedia of Mathematics|language=en|archive-url=https://web.archive.org/web/20171229231821/https://www.encyclopediaofmath.org/index.php/Trigonometric_functions|archive-date=29 December 2017|url-status=live|access-date=29 December 2017}}</ref>

In the range <math>0 \le \theta \le \pi/2</math>, this definition coincides with the right-angled triangle definition, by taking the right-angled triangle to have the unit radius {{math|OA}} as [[hypotenuse]]. And since the equation <math>x^2+y^2=1</math> holds for all points <math>\mathrm{P} = (x,y)</math> on the unit circle, this definition of cosine and sine also satisfies the [[Pythagorean identity]].
<math display="block">\cos^2\theta+\sin^2\theta=1.</math>

The other trigonometric functions can be found along the unit circle as 
<math display="block">\tan \theta = y_\mathrm{B} \quad</math> and <math> \quad\cot \theta = x_\mathrm{C},</math>
<math display="block">\csc \theta\ = y_\mathrm{D} \quad</math> and <math> \quad\sec \theta = x_\mathrm{E}.</math>

By applying the Pythagorean identity and geometric proof methods, these definitions can readily be shown to coincide with the definitions of tangent, cotangent, secant and cosecant in terms of sine and cosine, that is
<math display="block">\tan \theta =\frac{\sin \theta}{\cos\theta},\quad \cot\theta=\frac{\cos\theta}{\sin\theta},\quad \sec\theta=\frac{1}{\cos\theta},\quad \csc\theta=\frac{1}{\sin\theta}.</math>

[[File:Trigonometric functions.svg|right|thumb|upright=1.35|link={{filepath:trigonometric_functions_derivation_animation.svg}}|Trigonometric functions:
{{color|#00A|Sine}},
{{color|#0A0|Cosine}},
{{color|#A00|Tangent}},
{{color|#00A|Cosecant (dotted)}},
{{color|#0A0|Secant (dotted)}},
{{color|#A00|Cotangent (dotted)}}
– [{{filepath:trigonometric_functions_derivation_animation.svg}} animation] ]]

Since a rotation of an angle of <math>\pm2\pi</math> does not change the position or size of a shape, the points {{math|A}}, {{math|B}}, {{math|C}}, {{math|D}}, and {{math|E}} are the same for two angles whose difference is an integer multiple of <math>2\pi</math>. Thus trigonometric functions are [[periodic function]]s with period <math>2\pi</math>. That is, the equalities
<math display="block">\sin\theta = \sin\left(\theta + 2 k \pi \right)\quad</math> and <math>\quad \cos\theta = \cos\left(\theta + 2 k \pi \right)</math>
hold for any angle {{mvar|θ}} and any [[integer]] {{mvar|k}}. The same is true for the four other trigonometric functions. By observing the sign and the monotonicity of the functions sine, cosine, cosecant, and secant in the four quadrants, one can show that <math>2\pi</math> is the smallest value for which they are periodic (i.e., <math>2\pi</math> is the [[periodic function|fundamental period]] of these functions). However, after a rotation by an angle <math>\pi</math>, the points {{mvar|B}} and {{mvar|C}} already return to their original position, so that the tangent function and the cotangent function have a fundamental period of <math>\pi</math>. That is, the equalities
<math display="block">\tan\theta = \tan(\theta + k\pi) \quad</math> and <math>\quad \cot\theta = \cot(\theta + k\pi)</math>
hold for any angle {{mvar|θ}} and any integer {{mvar|k}}.

==Algebraic values==
[[File:Unit circle angles color.svg|right|thumb|The [[unit circle]], with some points labeled with their cosine and sine (in this order), and the corresponding angles in radians and degrees.]]
The [[algebraic expression]]s for the most important angles are as follows:

<math display="block">\sin 0 = \sin 0^\circ \quad= \frac{\sqrt0}2 = 0</math> ([[Angle#Types of angles|zero angle]])
<math display="block">\sin \frac\pi6 = \sin 30^\circ = \frac{\sqrt1}2 = \frac{1}{2}</math>
<math display="block">\sin \frac\pi4 = \sin 45^\circ = \frac{\sqrt{2}}{2} = \frac{1}{\sqrt{2}}</math>
<math display="block">\sin \frac\pi3 = \sin 60^\circ = \frac{\sqrt{3}}{2}</math>
<math display="block">\sin \frac\pi2 = \sin 90^\circ = \frac{\sqrt4}2 = 1</math> ([[right angle]])

Writing the numerators as [[square roots]] of consecutive non-negative integers, with a denominator of 2, provides an easy way to remember the values.<ref name="Larson_2013"/>

Such simple expressions generally do not exist for other angles which are rational multiples of a right angle.
*For an angle which, measured in degrees, is a multiple of three, the [[exact trigonometric values]] of the sine and the cosine may be expressed in terms of square roots. These values of the sine and the cosine may thus be constructed by [[Compass-and-straightedge construction|ruler and compass]].
*For an angle of an integer number of degrees, the sine and the cosine may be expressed in terms of square roots and the [[cube root]] of a non-real [[complex number]]. [[Galois theory]] allows a proof that, if the angle is not a multiple of 3°, non-real cube roots are unavoidable.
*For an angle which, expressed in degrees, is a [[rational number]], the sine and the cosine are [[algebraic number]]s, which may be expressed in terms of [[nth root|{{mvar|n}}-th roots]]. This results from the fact that the [[Galois group]]s of the [[cyclotomic polynomial]]s are [[cyclic group|cyclic]].
*For an angle which, expressed in degrees, is not a rational number, then either the angle or both the sine and the cosine are [[transcendental number]]s. This is a corollary of [[Baker's theorem]], proved in 1966.
*If the sine of an angle is a rational number then the cosine is not necessarily a rational number, and vice-versa. However, if the tangent of an angle is rational then both the sine and cosine of the double angle will be rational.

===Simple algebraic values===
{{main|Exact trigonometric values#Common angles}}

The following table lists the sines, cosines, and tangents of multiples of 15 degrees from 0 to 90 degrees.
 
{| class="wikitable" style="text-align:center;"
|-
! colspan=2 | Angle, {{mvar|θ}}, in
! rowspan=2 | <math>\sin(\theta)</math>
! rowspan=2 | <math>\cos(\theta)</math>
! rowspan=2 | <math>\tan(\theta)</math>
|-
! radians
! degrees
|-
| <math>0</math>
| <math>0^\circ</math>
| <math>0</math>
| <math>1</math>
| <math>0</math>
|-
| <math>\frac{\pi}{12}</math>
| <math>15^\circ</math>
| <math>\frac{\sqrt{6}-\sqrt{2}}{4}</math>
| <math>\frac{\sqrt{6}+\sqrt{2}}{4}</math>
| <math>2-\sqrt{3}</math>
|-
| <math>\frac{\pi}{6}</math>
| <math>30^\circ</math>
| <math>\frac{1}{2}</math>
| <math>\frac{\sqrt{3}}{2}</math>
| <math>\frac{\sqrt{3}}{3}</math>
|-
| <math>\frac{\pi}{4}</math>
| <math>45^\circ</math>
| <math>\frac{\sqrt{2}}{2}</math>
| <math>\frac{\sqrt{2}}{2}</math>
| <math>1</math>
|-
| <math>\frac{\pi}{3}</math>
| <math>60^\circ</math>
| <math>\frac{\sqrt{3}}{2}</math>
| <math>\frac{1}{2}</math>
| <math>\sqrt{3}</math>
|-
| <math>\frac{5\pi}{12}</math>
| <math>75^\circ</math>
| <math>\frac{\sqrt{6}+\sqrt{2}}{4}</math>
| <math>\frac{\sqrt{6}-\sqrt{2}}{4}</math>
| <math>2 + \sqrt{3}</math>
|-
| <math>\frac{\pi}{2}</math>
| <math>90^\circ</math>
| <math>1</math>
| <math>0</math>
| {{n/a|Undefined}}
|}

==Definitions in analysis==
[[file:Trigonometrija-graf.svg|thumb|right|[[graph of a function|Graphs]] of sine, cosine and tangent]]
[[File:Taylorsine.svg|thumb|right|The sine function (blue) is closely approximated by its [[Taylor's theorem|Taylor polynomial]] of degree 7 (pink) for a full cycle centered on the origin.]]
[[File:Taylor cos.gif|thumb|Animation for the approximation of cosine via Taylor polynomials.]]
[[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.{{clarify|date=September 2025}}<ref name="Hardy">{{citation|first=G.H.|last=Hardy|title=A course of pure mathematics|year=1950|edition=8th|pages=432–438}}</ref> Thus in modern analysis, trigonometric functions are usually constructed without reference to geometry.

Various ways exist in the 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>

===Definition by differential equations===

Sine and cosine can be defined as the unique solution to the [[initial value problem]]:{{sfn|Bartle|Sherbert|1999|p=247}}
<math display="block">\frac{d}{dx}\sin x= \cos x,\ \frac{d}{dx}\cos x= -\sin x,\ \sin(0)=0,\ \cos(0)=1.</math>

Differentiating again, <math display="inline">\frac{d^2}{dx^2}\sin x = \frac{d}{dx}\cos x = -\sin x</math> and <math display="inline">\frac{d^2}{dx^2}\cos x = -\frac{d}{dx}\sin x = -\cos x</math>, so both sine and cosine are solutions of the same [[ordinary differential equation]]
<math display="block">y''+y=0\,.</math>
Sine is the unique solution with {{math|''y''(0) {{=}} 0}} and {{math|''y''′(0) {{=}} 1}}; cosine is the unique solution with {{math|''y''(0) {{=}} 1}} and {{math|''y''′(0) {{=}} 0}}.

One can then prove, as a theorem, that solutions <math>\cos,\sin</math> are periodic, having the same period. Writing this period as <math>2\pi</math> is then a definition of the real number <math>\pi</math> which is independent of geometry.

Applying the [[quotient rule]] to the tangent <math>\tan x = \sin x / \cos x</math>, 
<math display="block">\frac{d}{dx}\tan x = \frac{\cos^2 x + \sin^2 x}{\cos^2 x} = 1+\tan^2 x\,,</math>
so the tangent function satisfies the ordinary differential equation
<math display="block">y' = 1 + y^2\,.</math>
It is the unique solution with {{math|''y''(0) {{=}} 0}}.

===Power series expansion===
The basic trigonometric functions can be defined by the following [[power series]] expansions.<ref>Whitaker and Watson, p 584</ref> These series are also known as the [[Taylor series]] or [[Maclaurin series]] of these trigonometric functions:
<math display="block">
\begin{align}
\sin x &= x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots &&= \sum_{n=0}^\infty (-1)^n \frac{x^{2n+1}}{(2n+1)!} \\
\cos x & = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots &&= \sum_{n=0}^\infty (-1)^n \frac{x^{2n}}{(2n)!}
\end{align}
</math>
The [[radius of convergence]] of these series is infinite. Therefore, the sine and the cosine can be extended to [[entire function]]s (also called "sine" and "cosine"), which are (by definition) [[complex-valued function]]s that are defined and [[holomorphic]] on the whole [[complex plane]].

Term-by-term differentiation shows that the sine and cosine defined by the series obey the differential equation discussed previously, and conversely one can obtain these series from elementary recursion relations derived from the differential equation.

Being defined as fractions of entire functions, the other trigonometric functions may be extended to [[meromorphic function]]s, that is functions that are holomorphic in the whole complex plane, except some isolated points called [[zeros and poles|poles]]. Here, the poles are the numbers of the form <math display="inline">(2k+1)\frac \pi 2</math> for the tangent and the secant, or <math>k\pi</math> for the cotangent and the cosecant, where {{mvar|k}} is an arbitrary integer.

Recurrences relations may also be computed for the coefficients of the [[Taylor series]] of the other trigonometric functions. These series have a finite [[radius of convergence]]. Their coefficients have a [[combinatorics|combinatorial]] interpretation: they enumerate [[alternating permutation]]s of finite sets.<ref>Stanley, Enumerative Combinatorics, Vol I., p. 149</ref>

More precisely, defining
: {{mvar|U<sub>n</sub>}}, the {{mvar|n}}-th [[up/down number]],
: {{mvar|B<sub>n</sub>}}, the {{mvar|n}}-th [[Bernoulli number]], and
: {{mvar|E<sub>n</sub>}}, is the {{mvar|n}}-th [[Euler number]],
one has the following series expansions:<ref>Abramowitz; Weisstein.</ref> 
<math display="block">
\begin{align}
\tan x & {} = \sum_{n=0}^\infty \frac{U_{2n+1}}{(2n+1)!}x^{2n+1} \\[8mu]
& {} = \sum_{n=1}^\infty \frac{(-1)^{n-1} 2^{2n} \left(2^{2n}-1\right) B_{2n}}{(2n)!}x^{2n-1} \\[5mu]
& {} = x + \frac{1}{3}x^3 + \frac{2}{15}x^5 + \frac{17}{315}x^7 + \cdots, \qquad \text{for } |x| < \frac{\pi}{2}.
\end{align}
</math>

<math display="block">
\begin{align}
\csc x &= \sum_{n=0}^\infty \frac{(-1)^{n+1} 2 \left(2^{2n-1}-1\right) B_{2n}}{(2n)!}x^{2n-1} \\[5mu]
&= x^{-1} + \frac{1}{6}x + \frac{7}{360}x^3 + \frac{31}{15120}x^5 + \cdots, \qquad \text{for } 0 < |x| < \pi.
\end{align}
</math>

<math display="block">
\begin{align}
\sec x &= \sum_{n=0}^\infty \frac{U_{2n}}{(2n)!}x^{2n}
= \sum_{n=0}^\infty \frac{(-1)^n E_{2n}}{(2n)!}x^{2n} \\[5mu]
&= 1 + \frac{1}{2}x^2 + \frac{5}{24}x^4 + \frac{61}{720}x^6 + \cdots, \qquad \text{for } |x| < \frac{\pi}{2}.
\end{align}
</math>

<math display="block">
\begin{align}
\cot x &= \sum_{n=0}^\infty \frac{(-1)^n 2^{2n} B_{2n}}{(2n)!}x^{2n-1} \\[5mu]
&= x^{-1} - \frac{1}{3}x - \frac{1}{45}x^3 - \frac{2}{945}x^5 - \cdots, \qquad \text{for } 0 < |x| < \pi.
\end{align}
</math>

===Continued fraction expansion===

The following [[continued fraction]]s are valid in the whole complex plane:

<ref>C. D. Olds, Continued fractions, 1963, Random House, Inc., p. 138, pp 11, (without authorship)</ref>
<math display="block">\sin x =
\cfrac{x}{1 + \cfrac{x^2}{2\cdot3-x^2 +
\cfrac{2\cdot3 x^2}{4\cdot5-x^2 +
\cfrac{4\cdot5 x^2}{6\cdot7-x^2 + \ddots}}}}</math>

<math display="block">\cos x = \cfrac{1}{1 + \cfrac{x^2}{1 \cdot 2 - x^2 + \cfrac{1 \cdot 2x^2}{3 \cdot 4 - x^2 + \cfrac{3 \cdot 4x^2}{5 \cdot 6 - x^2 + \ddots}}}}</math>

{{citation needed|date=October 2025|reason=In MATHEMATICAL ANALYSIS Functions, Limits, Series,Continued Fractions. E d i t e d b y L. A. LYUSTERNIK and A. R. YANPOL’SKII
T r a n s l a t e d b y D. E. BROWN T r a n s l a t i o n e d i t e d b y E. SPENCE,1965, PERGAMON PRESS, OXFORD • LONDON • EDINBURGH • NEW YORK
p.279 "The general form of the expansion as a continued fraction of sin x
is not known."
p.280 "The general form of the expansion as a continued fraction of cos x
is unknown."

Question: Where do these continued fractions for sin(x) and cos(x) come from?}}


<math display="block">\tan x = \cfrac{x}{1 - \cfrac{x^2}{3 - \cfrac{x^2}{5 - \cfrac{x^2}{7 - \ddots}}}}=\cfrac{1}{\cfrac{1}{x} - \cfrac{1}{\cfrac{3}{x} - \cfrac{1}{\cfrac{5}{x} - \cfrac{1}{\cfrac{7}{x} - \ddots}}}}</math>

The last one was used in the historically first [[proof that π is irrational]].<ref>{{citation|editor1-last = Berggren|editor1-first = Lennart|editor2-last = Borwein|editor2-first = Jonathan M.|editor2-link = Jonathan M. Borwein| editor3-last = Borwein|editor3-first = Peter B.|editor3-link = Peter B. Borwein|last = Lambert|first = Johann Heinrich|orig-year = 1768|chapter = Mémoire sur quelques propriétés remarquables des quantités transcendantes circulaires et logarithmiques|title = Pi, a source book|place = New York|publisher = [[Springer Science+Business Media|Springer-Verlag]] |year = 2004|edition = 3rd|pages = 129&ndash;140|isbn = 0-387-20571-3}}</ref>

There is a rapidly convergent continued fraction for <math >\tan(x)</math>:
<ref>{{cite book |last1=Khovanski |first1=A. N. |title=The Applications Of Continued Fractions And Their Generalizations To Problems In Approximation Theory |publisher=Noordhoff |year=1963 |location=Groningen, The Netherlands}}</ref>

<math display="block">\tan x=1+\cfrac{5x^2}{T_{0}+5x^2}, T_{k}= (4k+1)(4k+3)(4k+5)-4x^2(4k+3)+ \cfrac{x^2(4k+1)}{1+ \cfrac{x^2(4k+9)}{T_{k+1}}}</math>
Let <math>x=1</math> then the following continued fraction representation gives (asymptotically) 12.68 new correct decimal places per cycle:
<math display="block">\tan 1=1+\cfrac{5}{T_{0}+5}, T_{k}= (4k+1)(4k+3)(4k+5)-4(4k+3)+ \cfrac{4k+1}{1+ \cfrac{4k+9}{T_{k+1}}}</math>







===Partial fraction expansion===

There is a series representation as [[partial fraction expansion]] where just translated [[Multiplicative inverse|reciprocal function]]s are summed up, such that the [[Pole (complex analysis)|pole]]s of the cotangent function and the reciprocal functions match:<ref name="Aigner_2000"/>
<math display="block">
\pi \cot \pi x = \lim_{N\to\infty}\sum_{n=-N}^N \frac{1}{x+n}.
</math>
This identity can be proved with the [[Gustav Herglotz|Herglotz]] trick.<ref name="Remmert_1991"/>
Combining the {{math|(–''n'')}}-th with the {{mvar|n}}-th term lead to [[absolute convergence|absolutely convergent]] series:
<math display="block">
\pi \cot \pi x = \frac{1}{x} + 2x\sum_{n=1}^\infty \frac{1}{x^2-n^2}.
</math>
Similarly, one can find a partial fraction expansion for the secant, cosecant and tangent functions:
<math display="block">
\pi\csc\pi x = \sum_{n=-\infty}^\infty \frac{(-1)^n}{x+n}=\frac{1}{x} + 2x\sum_{n=1}^\infty \frac{(-1)^n}{x^2-n^2},
</math>
<math display="block">\pi^2\csc^2\pi x=\sum_{n=-\infty}^\infty \frac{1}{(x+n)^2},</math>
<math display="block">
\pi\sec\pi x = \sum_{n=0}^\infty (-1)^n \frac{(2n+1)}{(n+\tfrac12)^2 - x^2},
</math>
<math display="block">
\pi \tan \pi x = 2x\sum_{n=0}^\infty \frac{1}{(n+\tfrac12)^2 - x^2}.
</math>

===Infinite product expansion===
The following infinite product for the sine is due to [[Leonhard Euler]], and is of great importance in complex analysis:<ref>Whittaker and Watson, p 137</ref>
<math display="block">\sin z = z \prod_{n=1}^\infty \left(1-\frac{z^2}{n^2 \pi^2}\right), \quad z\in\mathbb C.</math>
This may be obtained from the partial fraction decomposition of <math>\cot z</math> given above, which is the logarithmic derivative of <math>\sin z</math>.<ref>Ahlfors, p 197</ref> From this, it can be deduced also that
<math display="block">\cos z = \prod_{n=1}^\infty \left(1-\frac{z^2}{(n-1/2)^2 \pi^2}\right), \quad z\in\mathbb C.</math>

=== Euler's formula and the exponential function ===
[[File:Sinus und Kosinus am Einheitskreis 3.svg|thumb|<math>\cos(\theta)</math> and <math>\sin(\theta)</math> are the real and imaginary part of <math>e^{i\theta}</math> respectively.]]

[[Euler's formula]] relates sine and cosine to the [[exponential function]]:
<math display="block">e^{ix}  = \cos x + i\sin x.</math> 
This formula is commonly considered for real values of {{mvar|x}}, but it remains true for all complex values.

''Proof'': Let <math>f_1(x)=\cos x + i\sin x,</math> and <math>f_2(x)=e^{ix}.</math> One has <math>df_j(x)/dx= if_j(x)</math> for {{math|1=''j'' = 1, 2}}. The [[quotient rule]] implies thus that <math>d/dx\, (f_1(x)/f_2(x))=0</math>. Therefore, <math>f_1(x)/f_2(x)</math> is a constant function, which equals {{val|1}}, as <math>f_1(0)=f_2(0)=1.</math> This proves the formula.

One has
<math display="block">\begin{align}
e^{ix}  &= \cos x + i\sin x\\[5pt]
e^{-ix}  &= \cos x - i\sin x.
\end{align}</math>

Solving this [[linear system]] in sine and cosine, one can express them in terms of the exponential function:
<math display="block">\begin{align}\sin x &= \frac{e^{i x} - e^{-i x}}{2i}\\[5pt]
\cos x &= \frac{e^{i x} + e^{-i x}}{2}.
\end{align}</math>

When {{mvar|x}} is real, this may be rewritten as 
<math display="block">\cos x = \operatorname{Re}\left(e^{i x}\right), \qquad \sin x = \operatorname{Im}\left(e^{i x}\right).</math>

Most [[List of trigonometric identities|trigonometric identities]] can be proved by expressing trigonometric functions in terms of the complex exponential function by using above formulas, and then using the identity <math>e^{a+b}=e^ae^b</math> for simplifying the result.

Euler's formula can also be used to define the basic trigonometric function directly, as follows, using the language of [[topological group]]s.<ref>{{cite book |last=Bourbaki |first=Nicolas |author-link=Nicolas Bourbaki |title=Topologie generale |publisher=Springer |year=1981|at=§VIII.2}}</ref> The set <math>U</math> of complex numbers of unit modulus is a compact and connected topological group, which has a neighborhood of the identity that is homeomorphic to the real line. Therefore, it is isomorphic as a topological group to the one-dimensional torus group <math>\mathbb R/\mathbb Z</math>, via an isomorphism
<math display="block">e:\mathbb R/\mathbb Z\to U.</math>
In simple terms, <math>e(t) = \exp(2\pi i t)</math>, and this isomorphism is unique up to taking complex conjugates.

For a nonzero real number <math>a</math> (the ''base''), the function <math>t\mapsto e(t/a)</math> defines an isomorphism of the group <math>\mathbb R/a\mathbb Z\to U</math>. The real and imaginary parts of <math>e(t/a)</math> are the cosine and sine, where <math>a</math> is used as the base for measuring angles. For example, when <math>a=2\pi</math>, we get the measure in radians, and the usual trigonometric functions. When <math>a=360</math>, we get the sine and cosine of angles measured in degrees.

Note that <math>a=2\pi</math> is the unique value at which the derivative 
<math display="block">\frac{d}{dt} e(t/a)</math> 
becomes a [[unit vector]] with positive imaginary part at <math>t=0</math>. This fact can, in turn, be used to define the constant <math>2\pi</math>.

=== 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
<math display="block">\theta(t) = \int_0^t \frac{d\tau}{1+\tau^2}=\arctan t</math>
where this defines this inverse tangent function. Also, <math>\pi</math> is defined by
<math display="block">\frac12\pi = \int_0^\infty \frac{d\tau}{1+\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>

On the interval <math>-\pi/2<\theta<\pi/2</math>, the trigonometric functions are defined by inverting the relation <math>\theta = \arctan t</math>. Thus we define the trigonometric functions by
<math display="block">\tan\theta = t,\quad \cos\theta = (1+t^2)^{-1/2},\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.

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.

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">\arctan s + \arctan t = \arctan \frac{s+t}{1-st}</math>
holds, provided <math>\arctan s+\arctan t\in(-\pi/2,\pi/2)</math>, since
<math display="block">\arctan s + \arctan t= \int_{-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">\arctan t + \frac{\pi}{2} = \arctan(-1/t),\quad t\in (-\infty,0).</math>
Thus we have
<math display="block">\sin\left(\theta + \frac{\pi}{2}\right) = \frac{-1}{t\sqrt{1+(-1/t)^2}} = \frac{-1}{\sqrt{1+t^2}} = -\cos(\theta)</math>
and
<math display="block">\cos\left(\theta + \frac{\pi}{2}\right) = \frac{1}{\sqrt{1+(-1/t)^2}} = \frac{t}{\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===
One can also define the trigonometric functions using various [[functional equation]]s.

For example,<ref name="Kannappan_2009"/> the sine and the cosine form the unique pair of [[continuous function]]s that satisfy the difference formula
<math display="block">\cos(x- y) = \cos x\cos y + \sin x\sin y\,</math>
and the added condition
<math display="block">0 < x\cos x < \sin x < x\quad\text{ for }\quad 0 < x < 1.</math>

===In the complex plane===
The sine and cosine of a [[complex number]] <math>z=x+iy</math> can be expressed in terms of real sines, cosines, and [[hyperbolic function]]s as follows:
<math display="block">\begin{align}\sin z &= \sin x \cosh y + i \cos x \sinh y\\[5pt]
\cos z &= \cos x \cosh y - i \sin x \sinh y\end{align}</math>

By taking advantage of [[domain coloring]], it is possible to graph the trigonometric functions as complex-valued functions. Various features unique to the complex functions can be seen from the graph; for example, the sine and cosine functions can be seen to be unbounded as the imaginary part of <math>z</math> becomes larger (since the color white represents infinity), and the fact that the functions contain simple [[Zeros and poles|zeros or poles]] is apparent from the fact that the hue cycles around each zero or pole exactly once. Comparing these graphs with those of the corresponding Hyperbolic functions highlights the relationships between the two.

{| style="text-align:center"
|+ '''Trigonometric functions in the complex plane'''
|[[File:Trig-sin.png|thumb]]
<math>
\sin z\,
</math>
[[File:Trig-cos.png|thumb]]
<math>
\cos z\,
</math>
|[[File:Trig-tan.png|thumb]]
<math>
\tan z\,
</math>
[[File:Trig-cot.png|thumb]]
<math>
\cot z\,
</math>
|[[File:Trig-sec.png|thumb]]
<math>
\sec z\,
</math>
[[File:Trig-csc.png|thumb]]
<math>
\csc z\,
</math>
|}

== Periodicity and asymptotes ==
The sine and cosine functions are [[periodic function|periodic]], with period <math>2\pi</math>, which is the smallest positive period:
<math display="block">\sin(z+2\pi) = \sin(z),\quad \cos(z+2\pi) = \cos(z).</math>
Consequently, the cosecant and secant also have <math>2\pi</math> as their period. 

The functions sine and cosine also have semiperiods <math>\pi</math>, and
<math display="block">\sin(z+\pi)=-\sin(z),\quad \cos(z+\pi)=-\cos(z)</math>
and consequently
<math display="block">\tan(z+\pi) = \tan(z),\quad \cot(z+\pi) = \cot(z).</math>
Also, 
<math display="block">\sin(x+\pi/2)=\cos(x),\quad \cos(x+\pi/2) = -\sin(x)</math>
(see [[Angle#Combining_angle_pairs|Complementary angles]]).

The function <math>\sin(z)</math> has a unique zero (at <math>z=0</math>) in the strip <math>-\pi < \real(z) <\pi</math>. The function <math>\cos(z)</math> has the pair of zeros <math>z=\pm\pi/2</math> in the same strip. Because of the periodicity, the zeros of sine are 
<math display="block">\pi\mathbb Z = \left\{\dots,-2\pi,-\pi,0,\pi,2\pi,\dots\right\}\subset\mathbb C.</math>
There zeros of cosine are
<math display="block">\frac{\pi}{2} + \pi\mathbb Z = \left\{\dots,-\frac{3\pi}{2},-\frac{\pi}{2},\frac{\pi}{2},\frac{3\pi}{2},\dots\right\}\subset\mathbb C.</math>
All of the zeros are simple zeros, and both functions have derivative <math>\pm 1</math> at each of the zeros.

The tangent function <math>\tan(z)=\sin(z)/\cos(z)</math> has a simple zero at <math>z=0</math> and vertical asymptotes at <math>z=\pm\pi/2</math>, where it has a simple pole of residue <math>-1</math>. Again, owing to the periodicity, the zeros are all the integer multiples of <math>\pi</math> and the poles are odd multiples of <math>\pi/2</math>, all having the same residue. The poles correspond to vertical asymptotes
<math display="block">\lim_{x\to\pi^-}\tan(x) = +\infty,\quad \lim_{x\to\pi^+}\tan(x) = -\infty.</math>

The cotangent function <math>\cot(z)=\cos(z)/\sin(z)</math> has a simple pole of residue 1 at the integer multiples of <math>\pi</math> and simple zeros at odd multiples of <math>\pi/2</math>. The poles correspond to vertical asymptotes
<math display="block">\lim_{x\to 0^-}\cot(x) = -\infty,\quad \lim_{x\to 0^+}\cot(x) = +\infty.</math>

==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 [[#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===
The cosine and the secant are [[even function]]s; the other trigonometric functions are [[odd function]]s. That is:
<math display="block">\begin{align}
\sin(-x) &=-\sin x\\
\cos(-x) &=\cos x\\
\tan(-x) &=-\tan x\\
\cot(-x) &=-\cot x\\
\csc(-x) &=-\csc x\\
\sec(-x) &=\sec x.
\end{align}</math>

===Periods===
All trigonometric functions are [[periodic function]]s of period {{math|2{{pi}}}}. This is the smallest period, except for the tangent and the cotangent, which have {{pi}} as smallest period. This means that, for every integer {{mvar|k}}, one has
<math display="block">\begin{array}{lrl}
\sin(x+&2k\pi) &=\sin x \\
\cos(x+&2k\pi) &=\cos x \\
\tan(x+&k\pi) &=\tan x \\
\cot(x+&k\pi) &=\cot x \\
\csc(x+&2k\pi) &=\csc x \\
\sec(x+&2k\pi) &=\sec x.
\end{array}</math>
See [[#Periodicity_and_asymptotes|Periodicity and asymptotes]].

===Pythagorean identity===

The Pythagorean identity, is the expression of the [[Pythagorean theorem]] in terms of trigonometric functions. It is 
<math display="block">\sin^2 x  + \cos^2 x  = 1</math>.
Dividing through by either <math>\cos^2 x</math> or <math>\sin^2 x</math> gives
<math display="block">\tan^2 x  + 1  = \sec^2 x</math>
<math display="block">1  + \cot^2 x  = \csc^2 x</math>
and
<math display="block">\sec^2 x  + \csc^2 x  = \sec^2 x \csc^2 x</math>.

===Sum and difference formulas===

The sum and difference formulas allow expanding the sine, the cosine, and the tangent of a sum or a difference of two angles in terms of sines and cosines and tangents of the angles themselves. These can be derived geometrically, using arguments that date to [[Ptolemy]] (see [[List_of_trigonometric_identities#Angle_sum_and_difference_identities|Angle sum and difference identities]]). One can also produce them algebraically using [[Euler's formula]].
; Sum
<math display="block">\begin{align}
\sin\left(x+y\right)&=\sin x \cos y + \cos x \sin y,\\[5mu]
\cos\left(x+y\right)&=\cos x \cos y - \sin x \sin y,\\[5mu]
\tan(x + y) &= \frac{\tan x + \tan y}{1 - \tan x\tan y}.
\end{align}</math>
; Difference
<math display="block">\begin{align}
\sin\left(x-y\right)&=\sin x \cos y - \cos x \sin y,\\[5mu]
\cos\left(x-y\right)&=\cos x \cos y + \sin x \sin y,\\[5mu]
\tan(x - y) &= \frac{\tan x - \tan y}{1 + \tan x\tan y}.
\end{align}</math>

When the two angles are equal, the sum formulas reduce to simpler equations known as the [[double-angle formulae]].

<math display="block">\begin{align}
\sin 2x &= 2 \sin x \cos x = \frac{2\tan x}{1+\tan^2 x}, \\[5mu]
\cos 2x &= \cos^2 x - \sin^2 x = 2 \cos^2 x - 1 = 1 - 2 \sin^2 x = \frac{1-\tan^2 x}{1+\tan^2 x},\\[5mu]
\tan 2x &= \frac{2\tan x}{1-\tan^2 x}.
\end{align}</math>

These identities can be used to derive the [[product-to-sum identities]].

By setting <math>t=\tan \tfrac12 \theta</math> (see [[List_of_trigonometric_identities#Half-angle_formulae|half-angle formulae]]), all trigonometric functions of <math>\theta</math> can be expressed as [[rational fraction]]s of <math>t</math>:
<math display="block">\begin{align}
\sin \theta &= \frac{2t}{1+t^2}, \\[5mu]
\cos \theta &= \frac{1-t^2}{1+t^2},\\[5mu]
\tan \theta &= \frac{2t}{1-t^2}.
\end{align}</math>
Together with 
<math display="block">d\theta = \frac{2}{1+t^2} \, dt,</math>
this is the [[tangent half-angle substitution]], which reduces the computation of [[integral]]s and [[antiderivative]]s of trigonometric functions to that of rational fractions.

===Derivatives and antiderivatives===
The [[derivative]]s of trigonometric functions result from those of sine and cosine by applying the [[quotient rule]]. The values given for the [[antiderivative]]s in the following table can be verified by differentiating them. The number&nbsp;{{mvar|C}} is a [[constant of integration]].

{| class="wikitable" style="text-align: center;"
!<math>f(x)</math> !! <math>f'(x)</math> !! <math display="inline">\int f(x) \, dx</math>
|-
|<math>\sin x</math>||<math>\cos x</math>||<math>-\cos x + C</math>
|-
|<math>\cos x</math>||<math>-\sin x</math>||<math>\sin x + C</math>
|-
|<math>\tan x</math>||<math>\sec^2 x</math>||<math>\ln \left| \sec x \right| + C</math>
|-
|<math>\csc x</math>||<math>-\csc x \cot x</math>||<math>\ln \left| \csc x - \cot x \right| + C</math>
|-
|<math>\sec x</math>||<math>\sec x \tan x</math>||<math>\ln \left| \sec x + \tan x \right| + C</math>
|-
|<math>\cot x</math>||<math>-\csc^2 x</math>||<math>-\ln \left| \csc x \right| + C</math>
|}
Note: For <math>0<x<\pi</math> the integral of <math>\csc x</math> can also be written as <math>-\operatorname{arsinh}(\cot x),</math> and the integral of <math>\sec x</math> for <math>-\pi/2<x<\pi/2</math> as <math>\operatorname{arsinh}(\tan x),</math> where <math>\operatorname{arsinh}</math> is the [[inverse hyperbolic sine]].

Alternatively, the derivatives of the 'co-functions' can be obtained using trigonometric identities and the chain rule:

<math display="block">
\begin{align}
\frac{d\cos x}{dx} &= \frac{d}{dx}\sin(\pi/2-x)=-\cos(\pi/2-x)=-\sin x \, , \\
\frac{d\csc x}{dx} &= \frac{d}{dx}\sec(\pi/2 - x) = -\sec(\pi/2 - x)\tan(\pi/2 - x) = -\csc x \cot x \, , \\
\frac{d\cot x}{dx} &= \frac{d}{dx}\tan(\pi/2 - x) = -\sec^2(\pi/2 - x) = -\csc^2 x \, .
\end{align}
</math>

==Inverse functions==
{{Main|Inverse trigonometric functions}}
The trigonometric functions are periodic, and hence not [[injective function|injective]], so strictly speaking, they do not have an [[inverse function]]. However, on each interval on which a trigonometric function is [[monotonic]], one can define an inverse function, and this defines inverse trigonometric functions as [[multivalued function]]s. To define a true inverse function, one must restrict the domain to an interval where the function is monotonic, and is thus [[bijection|bijective]] from this interval to its image by the function. The common choice for this interval, called the set of [[principal value]]s, is given in the following table. As usual, the inverse trigonometric functions are denoted with the prefix "arc" before the name or its abbreviation of the function.
{| class="wikitable" style="text-align: center;"
! Function !! Definition !! Domain !! Set of principal values
|-
| <math>y = \arcsin x</math> || <math>\sin y = x</math> || <math>-1 \le x \le 1</math> || <math display="inline">-\frac{\pi}{2} \le y \le \frac{\pi}{2}</math>
|-
| <math>y = \arccos x</math> || <math>\cos y = x</math> || <math>-1 \le x \le 1</math> || <math display="inline">0 \le y \le \pi</math>
|-
| <math>y = \arctan x</math> || <math>\tan y = x</math> || <math>-\infty < x < \infty</math> || <math display="inline">-\frac{\pi}{2} < y < \frac{\pi}{2}</math>
|-
| <math>y = \arccot x</math> || <math>\cot y = x</math> || <math>-\infty < x < \infty</math> || <math display="inline">0 < y < \pi</math>
|- 
| <math>y = \arcsec x</math> || <math>\sec y = x</math> || <math>x<-1 \text{ or } x>1</math> || <math display="inline">0 \le y \le \pi,\; y \ne \frac{\pi}{2}</math>
|-
| <math>y = \arccsc x</math> || <math>\csc y = x</math> || <math>x<-1 \text{ or } x>1</math> || <math display="inline">-\frac{\pi}{2} \le y \le \frac{\pi}{2},\; y \ne 0</math>
|}

The notations {{math|sin<sup>−1</sup>}}, {{math|cos<sup>−1</sup>}}, etc. are often used for {{math|arcsin}} and {{math|arccos}}, etc. When this notation is used, inverse functions could be confused with multiplicative inverses. The notation with the "arc" prefix avoids such a confusion, though "arcsec" for arcsecant can be confused with "[[arcsecond]]".

Just like the sine and cosine, the inverse trigonometric functions can also be expressed in terms of infinite series. They can also be expressed in terms of [[complex logarithm]]s.

==Applications==
{{Main|Uses of trigonometry}}

===Angles and sides of a triangle===
In this section {{mvar|A}}, {{mvar|B}}, {{mvar|C}} denote the three (interior) angles of a triangle, and {{mvar|a}}, {{mvar|b}}, {{mvar|c}} denote the lengths of the respective opposite edges. They are related by various formulas, which are named by the trigonometric functions they involve.

====Law of sines====
{{Main|Law of sines}}
The law of sines states that for an arbitrary triangle with sides {{mvar|a}}, {{mvar|b}}, and {{mvar|c}} and angles opposite those sides {{mvar|A}}, {{mvar|B}} and {{mvar|C}}:
<math display="block">\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c} = \frac{2\Delta}{abc},</math>
where {{math|Δ}} is the area of the triangle,
or, equivalently,
<math display="block">\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R,</math>
where {{mvar|R}} is the triangle's [[circumscribed circle|circumradius]].

It can be proved by dividing the triangle into two right ones and using the above definition of sine. The law of sines is useful for computing the lengths of the unknown sides in a triangle if two angles and one side are known. This is a common situation occurring in ''[[triangulation]]'', a technique to determine unknown distances by measuring two angles and an accessible enclosed distance.

====Law of cosines====
{{Main|Law of cosines}}
The law of cosines (also known as the cosine formula or cosine rule) is an extension of the [[Pythagorean theorem]]:
<math display="block">c^2=a^2+b^2-2ab\cos C,</math>
or equivalently,
<math display="block">\cos C=\frac{a^2+b^2-c^2}{2ab}.</math>

In this formula the angle at {{mvar|C}} is opposite to the side&nbsp;{{mvar|c}}. This theorem can be proved by dividing the triangle into two right ones and using the [[Pythagorean theorem]].

The law of cosines can be used to determine a side of a triangle if two sides and the angle between them are known. It can also be used to find the cosines of an angle (and consequently the angles themselves) if the lengths of all the sides are known.

====Law of tangents====
{{main|Law of tangents}}
The law of tangents says that:
<math display="block">\frac{\tan \frac{A-B}{2 }}{\tan \frac{A+B}{2 } } = \frac{a-b}{a+b}</math>.

====Law of cotangents====
{{main|Law of cotangents}}
If {{mvar|s}} is the triangle's semiperimeter, {{math|(''a'' + ''b'' + ''c'')/2}}, and {{mvar|r}} is the radius of the triangle's [[incircle]], then {{math|''rs''}} is the triangle's area. Therefore [[Heron's formula]] implies that:

<math display="block">r = \sqrt{\frac{1}{s} (s-a)(s-b)(s-c)}</math>.

The law of cotangents says that:<ref name="Allen_1976"/>
<math display="block">\cot{ \frac{A}{2}} = \frac{s-a}{r}</math>
It follows that
<math display="block">\frac{\cot \dfrac{A}{2}}{s-a}=\frac{\cot \dfrac{B}{2}}{s-b}=\frac{\cot \dfrac{C}{2}}{s-c}=\frac{1}{r}.</math>

===Periodic functions===
[[File:Lissajous curve 5by4.svg|thumb|right|A [[Lissajous curve]], a figure formed with a trigonometry-based function.]]
[[File:Synthesis square.gif|thumb|upright=1.5|right|An animation of the [[additive synthesis]] of a [[Square wave (waveform)|square wave]] with an increasing number of harmonics]]
[[File:Sawtooth Fourier Animation.gif|thumb|upright=1.3|Sinusoidal basis functions (bottom) can form a sawtooth wave (top) when added. All the basis functions have nodes at the nodes of the sawtooth, and all but the fundamental ({{math|1=''k'' = 1}}) have additional nodes. The oscillation seen about the sawtooth when {{mvar|k}} is large is called the [[Gibbs phenomenon]].]]

The trigonometric functions are also important in physics. The sine and the cosine functions, for example, are used to describe [[simple harmonic motion]], which models many natural phenomena, such as the movement of a mass attached to a spring and, for small angles, the pendular motion of a mass hanging by a string. The sine and cosine functions are one-dimensional projections of [[uniform circular motion]].

Trigonometric functions also prove to be useful in the study of general [[periodic function]]s. The characteristic wave patterns of periodic functions are useful for modeling recurring phenomena such as sound or light [[wave]]s.<ref name="Farlow_1993"/>

Under rather general conditions, a periodic function {{math|1=''f''&hairsp;(''x'')}} can be expressed as a sum of sine waves or cosine waves in a [[Fourier series]].<ref name="Folland_1992"/> Denoting the sine or cosine [[basis functions]] by {{mvar|φ<sub>k</sub>}}, the expansion of the periodic function {{math|1=''f''&hairsp;(''t'')}} takes the form:
<math display="block">f(t) = \sum _{k=1}^\infty c_k \varphi_k(t).</math>

For example, the [[Square wave (waveform)|square wave]] can be written as the [[Fourier series]]
<math display="block">f_\text{square}(t) = \frac{4}{\pi} \sum_{k=1}^\infty {\sin \big( (2k-1)t \big) \over 2k-1}.</math>

In the animation of a square wave at top right it can be seen that just a few terms already produce a fairly good approximation. The superposition of several terms in the expansion of a [[sawtooth wave]] are shown underneath.

==History==
{{Main|History of trigonometry}}
While the early study of trigonometry can be traced to antiquity, the trigonometric functions as they are in use today were developed in the medieval period. The [[Chord (geometry)|chord]] function was defined by [[Hipparchus]] of [[İznik|Nicaea]] (180–125&nbsp;BCE) and [[Ptolemy]] of [[Egypt (Roman province)|Roman Egypt]] (90–165&nbsp;CE). The functions of sine and [[versine]] (1 − cosine) are closely related to the [[Jyā, koti-jyā and utkrama-jyā|''jyā'' and ''koti-jyā'']] functions used in [[Gupta period]] [[Indian astronomy]] (''[[Aryabhatiya]]'', ''[[Surya Siddhanta]]''), via translation from Sanskrit to Arabic and then from Arabic to Latin.<ref name="Boyer_1991"/> (See [[Aryabhata's sine table]].)

All six trigonometric functions in current use were known in [[Islamic mathematics]] by the 9th century, as was the [[law of sines]], used in [[solving triangles]].<ref name="Gingerich_1986"/> [[Al-Khwārizmī]] (c. 780–850) produced tables of sines and cosines. Circa 860, [[Habash al-Hasib al-Marwazi]] defined the tangent and the cotangent, and produced their tables.<ref name="Sesiano">Jacques Sesiano, "Islamic mathematics", p. 157, in {{Cite book |title=Mathematics Across Cultures: The History of Non-western Mathematics |editor1-first=Helaine |editor1-last=Selin |editor1-link=Helaine Selin |editor2-first=Ubiratan |editor2-last=D'Ambrosio |editor2-link=Ubiratan D'Ambrosio |year=2000 |publisher=[[Springer Science+Business Media]] |isbn=978-1-4020-0260-1}}</ref><ref name="Britannica">{{cite web |title=trigonometry |date=17 November 2023 |url=http://www.britannica.com/EBchecked/topic/605281/trigonometry |publisher=Encyclopedia Britannica}}</ref> [[Muhammad ibn Jābir al-Harrānī al-Battānī]] (853–929) defined the reciprocal functions of secant and cosecant, and produced the first table of cosecants for each degree from 1° to 90°.<ref name="Britannica"/> The trigonometric functions were later studied by mathematicians including [[Omar Khayyám]], [[Bhāskara II]], [[Nasir al-Din al-Tusi]], [[Jamshīd al-Kāshī]] (14th century), [[Ulugh Beg]] (14th century), [[Regiomontanus]] (1464), [[Georg Joachim Rheticus|Rheticus]], and Rheticus' student [[Valentinus Otho]].

[[Madhava of Sangamagrama]] (c. 1400) made early strides in the [[mathematical analysis|analysis]] of trigonometric functions in terms of [[series (mathematics)|infinite series]].<ref name="mact-biog"/> (See [[Madhava series]] and [[Madhava's sine table]].)

The tangent function was brought to Europe by [[Giovanni Bianchini]] in 1467 in trigonometry tables he created to support the calculation of stellar coordinates.<ref>{{cite journal | url=https://www.jstor.org/stable/45211959 | jstor=45211959 | title=The end of an error: Bianchini, Regiomontanus, and the tabulation of stellar coordinates | last1=Van Brummelen | first1=Glen | journal=Archive for History of Exact Sciences | year=2018 | volume=72 | issue=5 | pages=547–563 | doi=10.1007/s00407-018-0214-2 | s2cid=240294796 }}</ref> 

The terms ''tangent'' and ''secant'' were first introduced by the Danish mathematician [[Thomas Fincke]] in his book ''Geometria rotundi'' (1583).<ref name="Fincke"/>

The 17th century French mathematician [[Albert Girard]] made the first published use of the abbreviations {{math|sin}}, {{math|cos}}, and {{math|tan}} in his book ''Trigonométrie''.<ref name=MacTutor>{{MacTutor|id=Girard_Albert}}</ref>

In a paper published in 1682, [[Gottfried Leibniz]] proved that {{math|sin ''x''}} is not an [[algebraic function]] of {{mvar|x}}.<ref name="Bourbaki_1994"/> Though defined as ratios of sides of a [[right triangle]], and thus appearing to be [[rational function]]s, Leibnitz result established that they are actually [[transcendental function]]s of their argument. The task of assimilating circular functions into algebraic expressions was accomplished by Euler in his ''[[Introduction to the Analysis of the Infinite]]'' (1748). His method was to show that the sine and cosine functions are [[alternating series]] formed from the even and odd terms respectively of the [[exponential function|exponential series]]. He presented "[[Euler's formula]]", as well as near-modern abbreviations ({{math|sin.}}, {{math|cos.}}, {{math|tang.}}, {{math|cot.}}, {{math|sec.}}, and {{math|cosec.}}).<ref name="Boyer_1991"/>

A few functions were common historically, but are now seldom used, such as the [[chord (geometry)|chord]], [[versine]] (which appeared in the earliest tables<ref name="Boyer_1991"/>), [[haversine]], [[coversine]],<ref>{{harvtxt|Nielsen|1966|pp=xxiii–xxiv}}</ref> half-tangent (tangent of half an angle), and [[exsecant]]. [[List of trigonometric identities]] shows more relations between these functions.

<math display="block">\begin{align}
\operatorname{crd}\theta &= 2 \sin\tfrac12\theta, \\[5mu]
\operatorname{vers}\theta&=1-\cos \theta = 2\sin^2\tfrac12\theta, \\[5mu]
\operatorname{hav}\theta&=\tfrac{1}{2}\operatorname{vers}\theta = \sin^2\tfrac12\theta, \\[5mu]
\operatorname{covers}\theta&=1-\sin\theta = \operatorname{vers}\bigl(\tfrac12\pi - \theta\bigr), \\[5mu]
\operatorname{exsec}\theta &= \sec\theta - 1.
\end{align}</math>

{{anchor|Logarithmic sine|Logarithmic cosine|Logarithmic secant|Logarithmic cosecant|Logarithmic tangent|Logarithmic cotangent}}Historically, trigonometric functions were often combined with [[logarithm]]s in compound functions like the logarithmic sine, logarithmic cosine, logarithmic secant, logarithmic cosecant, logarithmic tangent and logarithmic cotangent.<ref name="Hammer_1897">{{cite book |title=Lehrbuch der ebenen und sphärischen Trigonometrie. Zum Gebrauch bei Selbstunterricht und in Schulen, besonders als Vorbereitung auf Geodäsie und sphärische Astronomie |language=de |trans-title= |editor-first=Ernst Hermann Heinrich |editor-last=von Hammer |editor-link=:de:Ernst von Hammer |location=Stuttgart, Germany |publisher=[[J. B. Metzlerscher Verlag]] |date=1897 |edition=2 |url=https://quod.lib.umich.edu/u/umhistmath/ABN6964.0001.001/?view=toc |access-date=2024-02-06}}</ref><ref name="Heß_1916">{{cite book |title=Trigonometrie für Maschinenbauer und Elektrotechniker - Ein Lehr- und Aufgabenbuch für den Unterricht und zum Selbststudium |language=de |trans-title= |author-first=Adolf |author-last=Heß |location=Winterthur, Switzerland |publisher=Springer |edition=6 |date=1926 |orig-date=1916 |doi=10.1007/978-3-662-36585-4 |isbn=978-3-662-35755-2}}</ref><ref name="Lötzbeyer_1950">{{cite book |title=Erläuterungen und Beispiele für den Gebrauch der vierstelligen Tafeln zum praktischen Rechnen |language=de |trans-title= |chapter=§ 14. Erläuterungen u. Beispiele zu T. 13: lg sin X; lg cos X und T. 14: lg tg x; lg ctg X |trans-chapter= |author-first=Philipp |author-last=Lötzbeyer |date=1950 |edition=1 |isbn=978-3-11114038-4 |id=Archive ID 541650 |publication-place=Berlin, Germany |publisher=[[Walter de Gruyter & Co.]] |doi=10.1515/9783111507545 |url=https://www.degruyter.com/document/doi/10.1515/9783111507545/html |chapter-url=https://www.degruyter.com/document/doi/10.1515/9783111507545-015/html |access-date=2024-02-06}}</ref><ref name="Roegel_2016">{{cite book |title=A reconstruction of Peters's table of 7-place logarithms (volume 2, 1940) |date=2016-08-30 |editor-first=Denis |editor-last=Roegel |id=hal-01357842 |url=https://inria.hal.science/hal-01357842/document |location=Vandoeuvre-lès-Nancy, France |publisher=[[Université de Lorraine]] |access-date=2024-02-06 |url-status=live |archive-url=https://web.archive.org/web/20240206211422/https://inria.hal.science/hal-01357842/document |archive-date=2024-02-06}}</ref>

==Etymology==
{{main|History of trigonometry#Etymology}}
The word {{Lang|la-x-medieval|sine}} derives<ref>The anglicized form is first recorded in 1593 in [[Thomas Fale]]'s ''Horologiographia, the Art of Dialling''.</ref> from [[Latin]] ''[[wikt:sinus|sinus]]'', meaning "bend; bay", and more specifically "the hanging fold of the upper part of a [[toga]]", "the bosom of a garment", which was chosen as the translation of what was interpreted as the Arabic word ''jaib'', meaning "pocket" or "fold" in the twelfth-century translations of works by [[Al-Battani]] and [[Muḥammad ibn Mūsā al-Khwārizmī|al-Khwārizmī]] into [[Medieval Latin]].<ref>Various sources credit the first use of {{Lang|la-x-medieval|sinus}} to either 
* [[Plato Tiburtinus]]'s 1116 translation of the ''Astronomy'' of [[Al-Battani]]
* [[Gerard of Cremona]]'s translation of the ''Algebra'' of [[Muḥammad ibn Mūsā al-Khwārizmī|al-Khwārizmī]]
* [[Robert of Chester]]'s 1145 translation of the tables of al-Khwārizmī
See Merlet, [https://link.springer.com/chapter/10.1007/1-4020-2204-2_16#page-1 ''A Note on the History of the Trigonometric Functions''] in Ceccarelli (ed.), ''International Symposium on History of Machines and Mechanisms'', Springer, 2004<br>See Maor (1998), chapter 3, for an earlier etymology crediting Gerard.<br>See {{cite book |last=Katx |first=Victor |date=July 2008 |title=A history of mathematics |edition=3rd |location=Boston |publisher=[[Pearson (publisher)|Pearson]] |page=210 (sidebar) |isbn= 978-0321387004 |language=en }}</ref>
The choice was based on a misreading of the Arabic written form ''j-y-b'' ({{lang|ar|[[:wikt:جيب|جيب]]}}), which itself originated as a [[transliteration]] from Sanskrit ''{{IAST|jīvā}}'', which along with its synonym ''{{IAST|jyā}}'' (the standard Sanskrit term for the sine) translates to "bowstring", being in turn adopted from [[Ancient Greek language|Ancient Greek]] {{lang|grc|[[Chord (geometry)|χορδή]]}} "string".<ref name="Plofker_2009"/>

The word ''tangent'' comes from Latin ''tangens'' meaning "touching", since the line ''touches'' the circle of unit radius, whereas ''secant'' stems from Latin ''secans''—"cutting"—since the line ''cuts'' the circle.<ref>Oxford English Dictionary</ref>

The prefix "[[co (function prefix)|co-]]" (in "cosine", "cotangent", "cosecant") is found in [[Edmund Gunter]]'s ''Canon triangulorum'' (1620), which defines the ''cosinus'' as an abbreviation of the ''sinus complementi'' (sine of the [[complementary angle]]) and proceeds to define the ''cotangens'' similarly.<ref name="Gunter_1620"/><ref name="Roegel_2010"/>

==See also==
{{colbegin|colwidth=25em}}
* [[Bhāskara I's sine approximation formula]]
* [[Small-angle approximation]]
* [[Differentiation of trigonometric functions]]
* [[Generalized trigonometry]]
* [[Generating trigonometric tables]]
* [[List of integrals of trigonometric functions]]
* [[List of periodic functions]]
* [[Polar sine]] – a generalization to vertex angles
* [[Sinc function]]
{{colend}}

==Notes==
{{notelist}}
<references>
<ref name=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=Felix |author-last=Klein |author-link=Felix Klein |date=1924 |orig-year=1902 |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="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>
<ref name="Remmert_1991">{{cite book |title=Theory of complex functions |author-first1=Reinhold |author-last1=Remmert |publisher=Springer |date=1991 |isbn=978-0-387-97195-7 |page=327 |url=https://books.google.com/books?id=CC0dQxtYb6kC |url-status=live |archive-url=https://web.archive.org/web/20150320010718/http://books.google.com/books?id=CC0dQxtYb6kC |archive-date=20 March 2015 }} [https://books.google.com/books?id=CC0dQxtYb6kC&pg=PA327 Extract of page 327] {{webarchive|url=https://web.archive.org/web/20150320010448/http://books.google.com/books?id=CC0dQxtYb6kC&pg=PA327 |date=20 March 2015 }}</ref>
<ref name="Kannappan_2009">{{cite book |author-last=Kannappan |author-first=Palaniappan |title=Functional Equations and Inequalities with Applications |date=2009 |publisher=Springer |isbn=978-0387894911}}</ref>
<ref name="Allen_1976">The Universal Encyclopaedia of Mathematics, Pan Reference Books, 1976, pp. 529–530. English version George Allen and Unwin, 1964. Translated from the German version Meyers Rechenduden, 1960.</ref>
<ref name="Farlow_1993">{{cite book |title=Partial differential equations for scientists and engineers |url=https://books.google.com/books?id=DLUYeSb49eAC&pg=PA82 |author-first=Stanley J. |author-last=Farlow|author-link= Stanley Farlow |page=82 |isbn=978-0-486-67620-3 |publisher=Courier Dover Publications |edition=Reprint of Wiley 1982 |date=1993 |url-status=live |archive-url=https://web.archive.org/web/20150320011420/http://books.google.com/books?id=DLUYeSb49eAC&pg=PA82 |archive-date=20 March 2015 }}</ref>
<ref name="Folland_1992">See for example, {{cite book |author-first=Gerald B. |author-last=Folland |title=Fourier Analysis and its Applications |publisher=American Mathematical Society |edition=Reprint of Wadsworth & Brooks/Cole 1992 |chapter-url=https://books.google.com/books?id=idAomhpwI8MC&pg=PA77 |pages=77ff |chapter=Convergence and completeness |date=2009 |isbn=978-0-8218-4790-9 |url-status=live |archive-url=https://web.archive.org/web/20150319230954/http://books.google.com/books?id=idAomhpwI8MC&pg=PA77 |archive-date=19 March 2015 }}</ref>
<ref name="Boyer_1991">Boyer, Carl B. (1991). A History of Mathematics (Second ed.). John Wiley & Sons, Inc. {{isbn|0-471-54397-7}}, p. 210.</ref>
<ref name="Gingerich_1986">{{cite magazine |title=Islamic Astronomy |author-first=Owen |author-last=Gingerich |magazine=[[Scientific American]] |date=1986 |volume=254 |page=74 |url=http://faculty.kfupm.edu.sa/PHYS/alshukri/PHYS215/Islamic_astronomy.htm |access-date=13 July 2010 |archive-url=https://web.archive.org/web/20131019140821/http://faculty.kfupm.edu.sa/PHYS/alshukri/PHYS215/Islamic_astronomy.htm |archive-date=19 October 2013}}</ref>
<ref name="mact-biog">{{cite web |publisher=School of Mathematics and Statistics University of St Andrews, Scotland |title=Madhava of Sangamagrama |author-first1=J. J. |author-last1=O'Connor |author-first2=E. F. |author-last2=Robertson |url=http://www-gap.dcs.st-and.ac.uk/~history/Biographies/Madhava.html |archive-url=https://web.archive.org/web/20060514012903/http://www-gap.dcs.st-and.ac.uk/~history/Biographies/Madhava.html |url-status=dead |archive-date=14 May 2006 |access-date=8 September 2007 }}</ref>
<ref name="Fincke">{{cite web |url=http://www-history.mcs.st-andrews.ac.uk/Biographies/Fincke.html |title=Fincke biography |access-date=15 March 2017 |url-status=live |archive-url=https://web.archive.org/web/20170107035144/http://www-history.mcs.st-andrews.ac.uk/Biographies/Fincke.html |archive-date=7 January 2017 }}</ref>
<ref name="Bourbaki_1994">{{cite book |title=Elements of the History of Mathematics |url=https://archive.org/details/elementsofhistor0000bour |url-access=registration |author-first=Nicolás |author-last=Bourbaki |publisher=Springer |date=1994|isbn=9783540647676 }}</ref>
<ref name="Gunter_1620">{{cite book |author-first=Edmund |author-last=Gunter |author-link=Edmund Gunter |title=Canon triangulorum |date=1620}}</ref>
<ref name="Roegel_2010">{{cite web |title=A reconstruction of Gunter's Canon triangulorum (1620) |editor-first=Denis |editor-last=Roegel |type=Research report |publisher=HAL |date=6 December 2010 |id=inria-00543938 |url=https://hal.inria.fr/inria-00543938/document |access-date=28 July 2017 |url-status=live |archive-url=https://web.archive.org/web/20170728192238/https://hal.inria.fr/inria-00543938/document |archive-date=28 July 2017}}</ref>
<ref name="Plofker_2009">See Plofker, ''[[Mathematics in India (book)|Mathematics in India]]'', Princeton University Press, 2009, p. 257<br>See {{cite web |url=http://www.clarku.edu/~djoyce/trig/ |title=Clark University |url-status=live |archive-url=https://web.archive.org/web/20080615133310/http://www.clarku.edu/~djoyce/trig/ |archive-date=15 June 2008 }}<br>See Maor (1998), chapter 3, regarding the etymology.</ref>
</references>

==References==
{{refbegin}}
* {{AS ref}}
* [[Lars Ahlfors]], ''Complex Analysis: an introduction to the theory of analytic functions of one complex variable'', second edition, [[McGraw-Hill Book Company]], New York, 1966.
* {{cite book |last1=Bartle |first1=Robert G. |last2=Sherbert |first2=Donald R. |author-link1=Robert G. Bartle |title=Introduction to Real Analysis |year=1999 |edition=3rd |publisher=Wiley |isbn=9780471321484}}
* [[Carl Benjamin Boyer|Boyer, Carl B.]], ''A History of Mathematics'', John Wiley & Sons, Inc., 2nd edition. (1991). {{isbn|0-471-54397-7}}.
* {{cite book |last=Cajori |first=Florian |author-link=Florian Cajori |title=A History of Mathematical Notations |volume=2 |publisher=Open Court |year=1929 |chapter=§2.2.1. Trigonometric Notations |pages=142–179 (¶511–537) |chapter-url=https://archive.org/details/b29980343_0002/page/142/}}
* Gal, Shmuel and Bachelis, Boris. An accurate elementary mathematical library for the IEEE floating point standard, ACM Transactions on Mathematical Software (1991).
* Joseph, George G., ''The Crest of the Peacock: Non-European Roots of Mathematics'', 2nd ed. [[Penguin Books]], London. (2000). {{isbn|0-691-00659-8}}.
* Kantabutra, Vitit, "On hardware for computing exponential and trigonometric functions," ''IEEE Trans. Computers'' '''45''' (3), 328–339 (1996).
* Maor, Eli, ''[https://web.archive.org/web/20040404234808/http://www.pupress.princeton.edu/books/maor/ Trigonometric Delights]'', Princeton Univ. Press. (1998). Reprint edition (2002): {{isbn|0-691-09541-8}}.
* Needham, Tristan, [https://web.archive.org/web/20040602145226/http://www.usfca.edu/vca/PDF/vca-preface.pdf "Preface"]" to ''[http://www.usfca.edu/vca/ Visual Complex Analysis]''. Oxford University Press, (1999). {{isbn|0-19-853446-9}}.
* {{citation |last1=Nielsen |first1=Kaj L. |title=Logarithmic and Trigonometric Tables to Five Places |edition=2nd |location=New York|publisher=[[Barnes & Noble]] |date=1966 |lccn=61-9103}}
* O'Connor, J. J., and E. F. Robertson, [https://web.archive.org/web/20130120084848/http://www-gap.dcs.st-and.ac.uk/~history/HistTopics/Trigonometric_functions.html "Trigonometric functions"], ''[[MacTutor History of Mathematics archive]]''. (1996).
* O'Connor, J. J., and E. F. Robertson, [http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Madhava.html "Madhava of Sangamagramma"], ''[[MacTutor History of Mathematics archive]]''. (2000).
* Pearce, Ian G., [http://www-history.mcs.st-andrews.ac.uk/history/Projects/Pearce/Chapters/Ch9_3.html "Madhava of Sangamagramma"] {{Webarchive|url=https://web.archive.org/web/20060505201342/http://www-history.mcs.st-andrews.ac.uk/history/Projects/Pearce/Chapters/Ch9_3.html |date=2006-05-05 }}, ''[[MacTutor History of Mathematics archive]]''. (2002).
* {{ citation | last1 = Protter | first1 = Murray H. | last2 = Morrey | first2 = Charles B. Jr. | year = 1970 | lccn = 76087042 | title = College Calculus with Analytic Geometry | edition = 2nd | publisher = [[Addison-Wesley]] | location = Reading }}
* Weisstein, Eric W., [http://mathworld.wolfram.com/Tangent.html "Tangent"] from ''[[MathWorld]]'', accessed 21 January 2006.
{{refend}}

==External links==
{{Wikibooks|Trigonometry}}
* {{springer |title=Trigonometric functions|id=p/t094210}}
* [http://www.visionlearning.com/library/module_viewer.php?mid=131&l=&c3= Visionlearning Module on Wave Mathematics]
* [https://web.archive.org/web/20071006172054/http://glab.trixon.se/ GonioLab] Visualization of the unit circle, trigonometric and hyperbolic functions
* [http://mathworld.wolfram.com/q-Sine.html q-Sine] Article about the [[q-analog]] of sin at [[MathWorld]]
* [http://mathworld.wolfram.com/q-Cosine.html q-Cosine] Article about the [[q-analog]] of cos at [[MathWorld]]

{{Trigonometric and hyperbolic functions}}

{{Authority control}}

{{DEFAULTSORT:Trigonometric Functions}}
[[Category:Analytic functions]]
[[Category:Angle]]
[[Category:Ratios]]
[[Category:Trigonometric functions| ]]