Trigonometric functions#Infinite product expansion

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 segments or their lengths related to an arc of an arbitrary circle, and later to indicate ratios of lengths, but as the 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 $\backslash sin\; x+y$ would typically be interpreted to mean $(\backslash sin\; x)+y,$ so parentheses are required to express $\backslash sin\; (x+y).$

A positive integer appearing as a superscript after the symbol of the function denotes exponentiation, not function composition. For example $\backslash sin^2\; x$ and $\backslash sin^2\; (x)$ denote $(\backslash sin\; x)^2,$ not $\backslash sin(\backslash sin\; x).$ This differs from the (historically later) general functional notation in which $f^2(x)\; =\; (f\; \backslash circ\; f)(x)\; =\; f(f(x)).$

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

File:TrigonometryTriangle.svg

File:TrigFunctionDiagram.svg

If the acute angle {{mvar|θ}} is given, then any right triangles that have an angle of {{mvar|θ}} are 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.{{harvtxt|Protter|Morrey|1970|pp=APP-2, APP-3}}{{Cite web|title=Sine, Cosine, Tangent|url=https://www.mathsisfun.com/sine-cosine-tangent.html|access-date=29 August 2020|website=www.mathsisfun.com}}

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}} radians}}. Therefore $\backslash sin(\backslash theta)$ and $\backslash cos(90^\backslash circ\; -\; \backslash theta)$ 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

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 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 or complex numbers, 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,{{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}} or as solutions to differential equations given particular initial values{{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}} (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. Moreover, these definitions result in simple expressions for the derivatives and indefinite integrals for the trigonometric functions.{{Cite book|last=Spivak|first=Michael|title=Calculus|publisher=Addison-Wesley|year=1967|chapter=15|pages=256–257|lccn=67-20770}} Thus, in settings beyond elementary geometry, radians are regarded as the mathematically natural unit for describing angle measures.

When radians (rad) are employed, the angle is given as the length of the 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°),{{cite oeis|A072097|Decimal expansion of 180/Pi}} and a complete turn (360°) is an angle of 2{{pi}} (≈ 6.28) rad.{{cite oeis|A019692|Decimal expansion of 2*Pi}} For real number x, the notation {{math|sin x}}, {{math|cos x}}, etc. refers 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 ({{math|sin x°}}, {{math|cos x°}}, etc.). Using this standard notation, the argument x for the trigonometric functions satisfies the relationship x = (180x/{{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.{{cite oeis|A019685|Decimal expansion of Pi/180}}

Image:Circle-trig6.svg

File:Periodic sine.svg

File:Unit Circle Definitions of Six Trigonometric Functions.svg 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 like "all students take calculus" indicates when sine, cosine, and tangent are positive from quadrants I to IV.{{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}}]]

The six trigonometric functions can be defined as 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 allow for the definition of the trigonometric functions for angles between {{math|0}} and $\backslash frac\{\backslash pi\}\{2\}$ radians {{math|(90°),}} the unit circle definitions allow the domain of trigonometric functions to be extended to all positive and negative real numbers.

Let $\backslash mathcal\; L$ be the ray obtained by rotating by an angle {{mvar|θ}} the positive half of the {{math|x}}-axis (counterclockwise rotation for $\backslash theta\; >\; 0,$ and clockwise rotation for ). This ray intersects the unit circle at the point $\backslash mathrm\{A\}\; =\; (x\_\backslash mathrm\{A\},y\_\backslash mathrm\{A\}).$ The ray $\backslash mathcal\; L,$ extended to a line if necessary, intersects the line of equation $x=1$ at point $\backslash mathrm\{B\}\; =\; (1,y\_\backslash mathrm\{B\}),$ and the line of equation $y=1$ at point $\backslash mathrm\{C\}\; =\; (x\_\backslash mathrm\{C\},1).$ The tangent line to the unit circle at the point {{math|A}}, is perpendicular to $\backslash mathcal\; L,$ and intersects the {{math|y}}- and {{math|x}}-axes at points $\backslash mathrm\{D\}\; =\; (0,y\_\backslash mathrm\{D\})$ and $\backslash mathrm\{E\}\; =\; (x\_\backslash mathrm\{E\},0).$ The 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 x- and y-coordinate values of point {{math|A}}. That is,

:$\backslash cos\; \backslash theta\; =\; x\_\backslash mathrm\{A\}\; \backslash quad$ and $\backslash quad\; \backslash sin\; \backslash theta\; =\; y\_\backslash mathrm\{A\}.${{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}}

In the range $0\; \backslash le\; \backslash theta\; \backslash le\; \backslash pi/2$, 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 $x^2+y^2=1$ holds for all points $\backslash mathrm\{P\}\; =\; (x,y)$ on the unit circle, this definition of cosine and sine also satisfies the Pythagorean identity.

:$\backslash cos^2\backslash theta+\backslash sin^2\backslash theta=1.$

The other trigonometric functions can be found along the unit circle as

:$\backslash tan\; \backslash theta\; =\; y\_\backslash mathrm\{B\}\; \backslash quad$ and $\backslash quad\backslash cot\; \backslash theta\; =\; x\_\backslash mathrm\{C\},$

:$\backslash csc\; \backslash theta\backslash \; =\; y\_\backslash mathrm\{D\}\; \backslash quad$ and $\backslash quad\backslash sec\; \backslash theta\; =\; x\_\backslash mathrm\{E\}.$

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

: $\backslash tan\; \backslash theta\; =\backslash frac\{\backslash sin\; \backslash theta\}\{\backslash cos\backslash theta\},\backslash quad\; \backslash cot\backslash theta=\backslash frac\{\backslash cos\backslash theta\}\{\backslash sin\backslash theta\},\backslash quad\; \backslash sec\backslash theta=\backslash frac\{1\}\{\backslash cos\backslash theta\},\backslash quad\; \backslash csc\backslash theta=\backslash frac\{1\}\{\backslash sin\backslash theta\}.$

[[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 $\backslash pm2\backslash pi$ 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 $2\backslash pi$. Thus trigonometric functions are periodic functions with period $2\backslash pi$. That is, the equalities

: $\backslash sin\backslash theta\; =\; \backslash sin\backslash left(\backslash theta\; +\; 2\; k\; \backslash pi\; \backslash right)\backslash quad$ and $\backslash quad\; \backslash cos\backslash theta\; =\; \backslash cos\backslash left(\backslash theta\; +\; 2\; k\; \backslash pi\; \backslash right)$

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 $2\backslash pi$ is the smallest value for which they are periodic (i.e., $2\backslash pi$ is the fundamental period of these functions). However, after a rotation by an angle $\backslash pi$, 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 $\backslash pi$. That is, the equalities

: $\backslash tan\backslash theta\; =\; \backslash tan(\backslash theta\; +\; k\backslash pi)\; \backslash quad$ and $\backslash quad\; \backslash cot\backslash theta\; =\; \backslash cot(\backslash theta\; +\; k\backslash pi)$

hold for any angle {{mvar|θ}} and any integer {{mvar|k}}.

File:Unit circle angles color.svg, with some points labeled with their cosine and sine (in this order), and the corresponding angles in radians and degrees.]]

The algebraic expressions for the most important angles are as follows:

:$\backslash sin\; 0\; =\; \backslash sin\; 0^\backslash circ\; \backslash quad=\; \backslash frac\{\backslash sqrt0\}2\; =\; 0$ (zero angle)

:$\backslash sin\; \backslash frac\backslash pi6\; =\; \backslash sin\; 30^\backslash circ\; =\; \backslash frac\{\backslash sqrt1\}2\; =\; \backslash frac\{1\}\{2\}$

:$\backslash sin\; \backslash frac\backslash pi4\; =\; \backslash sin\; 45^\backslash circ\; =\; \backslash frac\{\backslash sqrt\{2\}\}\{2\}\; =\; \backslash frac\{1\}\{\backslash sqrt\{2\}\}$

:$\backslash sin\; \backslash frac\backslash pi3\; =\; \backslash sin\; 60^\backslash circ\; =\; \backslash frac\{\backslash sqrt\{3\}\}\{2\}$

:$\backslash sin\; \backslash frac\backslash pi2\; =\; \backslash sin\; 90^\backslash circ\; =\; \backslash frac\{\backslash sqrt4\}2\; =\; 1$ (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.

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 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 numbers, which may be expressed in terms of nth root. This results from the fact that the Galois groups of the cyclotomic polynomials are 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 numbers. 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.

{{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.

file:Trigonometrija-graf.svg of sine, cosine and tangent]]

File:Taylorsine.svg of degree 7 (pink) for a full cycle centered on the origin.]]

File:Taylor cos.gif

File:Taylorreihenentwicklung des Kosinus.svg

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.{{citation|first=G.H.|last=Hardy|title=A course of pure mathematics|year=1950|edition=8th|pages=432–438}} 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.
• By a power series, which is particularly well-suited to complex variables.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.
• By using an infinite product expansion.
• By inverting the inverse trigonometric functions, which can be defined as integrals of algebraic or rational functions.
• As solutions of a differential equation.Bartle, R. G., & Sherbert, D. R. (2000). Introduction to real analysis (3rd ed). Wiley.

Sine and cosine can be defined as the unique solution to the initial value problem:{{sfn|Bartle|Sherbert|1999|p=247}}

:$\backslash frac\{d\}\{dx\}\backslash sin\; x=\; \backslash cos\; x,\backslash \; \backslash frac\{d\}\{dx\}\backslash cos\; x=\; -\backslash sin\; x,\backslash \; \backslash sin(0)=0,\backslash \; \backslash cos(0)=1.$

Differentiating again, $\backslash frac\{d^2\}\{dx^2\}\backslash sin\; x\; =\; \backslash frac\{d\}\{dx\}\backslash cos\; x\; =\; -\backslash sin\; x$ and $\backslash frac\{d^2\}\{dx^2\}\backslash cos\; x\; =\; -\backslash frac\{d\}\{dx\}\backslash sin\; x\; =\; -\backslash cos\; x$, so both sine and cosine are solutions of the same ordinary differential equation

:$y\text{'}\text{'}+y=0\backslash ,.$

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 $\backslash cos,\backslash sin$ are periodic, having the same period. Writing this period as $2\backslash pi$ is then a definition of the real number $\backslash pi$ which is independent of geometry.

Applying the quotient rule to the tangent $\backslash tan\; x\; =\; \backslash sin\; x\; /\; \backslash cos\; x$,

:$\backslash frac\{d\}\{dx\}\backslash tan\; x\; =\; \backslash frac\{\backslash cos^2\; x\; +\; \backslash sin^2\; x\}\{\backslash cos^2\; x\}\; =\; 1+\backslash tan^2\; x\backslash ,,$

so the tangent function satisfies the ordinary differential equation

:$y\text{'}\; =\; 1\; +\; y^2\backslash ,.$

It is the unique solution with {{math|y(0) {{=}} 0}}.

The basic trigonometric functions can be defined by the following power series expansions.Whitaker and Watson, p 584 These series are also known as the Taylor series or Maclaurin series of these trigonometric functions:

:$$

\begin{align}

\sin x & = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots \\[6mu]

& = \sum_{n=0}^\infty \frac{(-1)^n}{(2n+1)!}x^{2n+1} \\[8pt]

\cos x & = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots \\[6mu]

& = \sum_{n=0}^\infty \frac{(-1)^n}{(2n)!}x^{2n}.

\end{align}

The radius of convergence of these series is infinite. Therefore, the sine and the cosine can be extended to entire functions (also called "sine" and "cosine"), which are (by definition) complex-valued functions 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 functions, that is functions that are holomorphic in the whole complex plane, except some isolated points called poles. Here, the poles are the numbers of the form $(2k+1)\backslash frac\; \backslash pi\; 2$ for the tangent and the secant, or $k\backslash pi$ 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 combinatorial interpretation: they enumerate alternating permutations of finite sets.Stanley, Enumerative Combinatorics, Vol I., p. 149

More precisely, defining

: {{mvar|U_n}}, the {{mvar|n}}th up/down number,

: {{mvar|B_n}}, the {{mvar|n}}th Bernoulli number, and

: {{mvar|E_n}}, is the {{mvar|n}}th Euler number,

one has the following series expansions:Abramowitz; Weisstein.

: $$

\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}

: $$

\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}

: $$

\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}

: $$

\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}

The following continued fractions are valid in the whole complex plane:

:$\backslash sin\; x\; =$

\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}}}}

:$\backslash cos\; x\; =\; \backslash cfrac\{1\}\{1\; +\; \backslash cfrac\{x^2\}\{1\; \backslash cdot\; 2\; -\; x^2\; +\; \backslash cfrac\{1\; \backslash cdot\; 2x^2\}\{3\; \backslash cdot\; 4\; -\; x^2\; +\; \backslash cfrac\{3\; \backslash cdot\; 4x^2\}\{5\; \backslash cdot\; 6\; -\; x^2\; +\; \backslash ddots\}\}\}\}$

:$\backslash tan\; x\; =\; \backslash cfrac\{x\}\{1\; -\; \backslash cfrac\{x^2\}\{3\; -\; \backslash cfrac\{x^2\}\{5\; -\; \backslash cfrac\{x^2\}\{7\; -\; \backslash ddots\}\}\}\}=\backslash cfrac\{1\}\{\backslash cfrac\{1\}\{x\}\; -\; \backslash cfrac\{1\}\{\backslash cfrac\{3\}\{x\}\; -\; \backslash cfrac\{1\}\{\backslash cfrac\{5\}\{x\}\; -\; \backslash cfrac\{1\}\{\backslash cfrac\{7\}\{x\}\; -\; \backslash ddots\}\}\}\}$

The last one was used in the historically first proof that π is irrational.{{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-Verlag |year = 2004|edition = 3rd|pages = 129–140|isbn = 0-387-20571-3}}

There is a series representation as partial fraction expansion where just translated reciprocal functions are summed up, such that the poles of the cotangent function and the reciprocal functions match:

: $$

\pi \cot \pi x = \lim_{N\to\infty}\sum_{n=-N}^N \frac{1}{x+n}.

This identity can be proved with the Herglotz trick.

Combining the {{math|(–n)}}th with the {{math|n}}th term lead to absolutely convergent series:

:$$

\pi \cot \pi x = \frac{1}{x} + 2x\sum_{n=1}^\infty \frac{1}{x^2-n^2}.

Similarly, one can find a partial fraction expansion for the secant, cosecant and tangent functions:

:$$

\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},

:$\backslash pi^2\backslash csc^2\backslash pi\; x=\backslash sum\_\{n=-\backslash infty\}^\backslash infty\; \backslash frac\{1\}\{(x+n)^2\},$

:$$

\pi\sec\pi x = \sum_{n=0}^\infty (-1)^n \frac{(2n+1)}{(n+\tfrac12)^2 - x^2},

:$$

\pi \tan \pi x = 2x\sum_{n=0}^\infty \frac{1}{(n+\tfrac12)^2 - x^2}.

The following infinite product for the sine is due to Leonhard Euler, and is of great importance in complex analysis:Whittaker and Watson, p 137

:$\backslash sin\; z\; =\; z\; \backslash prod\_\{n=1\}^\backslash infty\; \backslash left(1-\backslash frac\{z^2\}\{n^2\; \backslash pi^2\}\backslash right),\; \backslash quad\; z\backslash in\backslash mathbb\; C.$

This may be obtained from the partial fraction decomposition of $\backslash cot\; z$ given above, which is the logarithmic derivative of $\backslash sin\; z$.Ahlfors, p 197 From this, it can be deduced also that

:$\backslash cos\; z\; =\; \backslash prod\_\{n=1\}^\backslash infty\; \backslash left(1-\backslash frac\{z^2\}\{(n-1/2)^2\; \backslash pi^2\}\backslash right),\; \backslash quad\; z\backslash in\backslash mathbb\; C.$

File:Sinus und Kosinus am Einheitskreis 3.svg

Euler's formula relates sine and cosine to the exponential function:

:$e^\{ix\}\; =\; \backslash cos\; x\; +\; i\backslash sin\; x.$

This formula is commonly considered for real values of {{mvar|x}}, but it remains true for all complex values.

Proof: Let $f\_1(x)=\backslash cos\; x\; +\; i\backslash sin\; x,$ and $f\_2(x)=e^\{ix\}.$ One has $df\_j(x)/dx=\; if\_j(x)$ for {{math|1=j = 1, 2}}. The quotient rule implies thus that $d/dx\backslash ,\; (f\_1(x)/f\_2(x))=0$. Therefore, $f\_1(x)/f\_2(x)$ is a constant function, which equals {{val|1}}, as $f\_1(0)=f\_2(0)=1.$ This proves the formula.

One has

:$\backslash begin\{align\}$

e^{ix} &= \cos x + i\sin x\\[5pt]

e^{-ix} &= \cos x - i\sin x.

\end{align}

Solving this linear system in sine and cosine, one can express them in terms of the exponential function:

:

\cos x &= \frac{e^{i x} + e^{-i x}}{2}.

\end{align}

When {{mvar|x}} is real, this may be rewritten as

: $\backslash cos\; x\; =\; \backslash operatorname\{Re\}\backslash left(e^\{i\; x\}\backslash right),\; \backslash qquad\; \backslash sin\; x\; =\; \backslash operatorname\{Im\}\backslash left(e^\{i\; x\}\backslash right).$

Most 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 $e^\{a+b\}=e^ae^b$ 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 groups.{{cite book |last=Bourbaki |first=Nicolas |author-link=Nicolas Bourbaki |title=Topologie generale |publisher=Springer |year=1981|at=§VIII.2}} The set $U$ 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 $\backslash mathbb\; R/\backslash mathbb\; Z$, via an isomorphism

$$e:\backslash mathbb\; R/\backslash mathbb\; Z\backslash to\; U.$$

In pedestrian terms $e(t)\; =\; \backslash exp(2\backslash pi\; i\; t)$, and this isomorphism is unique up to taking complex conjugates.

For a nonzero real number $a$ (the base), the function $t\backslash mapsto\; e(t/a)$ defines an isomorphism of the group $\backslash mathbb\; R/a\backslash mathbb\; Z\backslash to\; U$. The real and imaginary parts of $e(t/a)$ are the cosine and sine, where $a$ is used as the base for measuring angles. For example, when $a=2\backslash pi$, we get the measure in radians, and the usual trigonometric functions. When $a=360$, we get the sine and cosine of angles measured in degrees.

Note that $a=2\backslash pi$ is the unique value at which the derivative

$$\backslash frac\{d\}\{dt\}\; e(t/a)$$

becomes a unit vector with positive imaginary part at $t=0$. This fact can, in turn, be used to define the constant $2\backslash pi$.

Another way to define the trigonometric functions in analysis is using integration.{{citation|last=Bartle|year=1964|title=Elements of real analysis|publisher=|pages=315–316}} For a real number $t$, put

$$\backslash theta(t)\; =\; \backslash int\_0^t\; \backslash frac\{d\backslash tau\}\{1+\backslash tau^2\}=\backslash arctan\; t$$

where this defines this inverse tangent function. Also, $\backslash pi$ is defined by

$$\backslash frac12\backslash pi\; =\; \backslash int\_0^\backslash infty\; \backslash frac\{d\backslash tau\}\{1+\backslash tau^2\}$$

a definition that goes back to Karl Weierstrass.{{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/ }}

On the interval , the trigonometric functions are defined by inverting the relation $\backslash theta\; =\; \backslash arctan\; t$. Thus we define the trigonometric functions by

$$\backslash tan\backslash theta\; =\; t,\backslash quad\; \backslash cos\backslash theta\; =\; (1+t^2)^\{-1/2\},\backslash quad\; \backslash sin\backslash theta\; =\; t(1+t^2)^\{-1/2\}$$

where the point $(t,\backslash theta)$ is on the graph of $\backslash theta=\backslash arctan\; t$ and the positive square root is taken.

This defines the trigonometric functions on $(-\backslash pi/2,\backslash pi/2)$. The definition can be extended to all real numbers by first observing that, as $\backslash theta\backslash to\backslash pi/2$, $t\backslash to\backslash infty$, and so $\backslash cos\backslash theta\; =\; (1+t^2)^\{-1/2\}\backslash to\; 0$ and $\backslash sin\backslash theta\; =\; t(1+t^2)^\{-1/2\}\backslash to\; 1$. Thus $\backslash cos\backslash theta$ and $\backslash sin\backslash theta$ are extended continuously so that $\backslash cos(\backslash pi/2)=0,\backslash sin(\backslash pi/2)=1$. Now the conditions $\backslash cos(\backslash theta+\backslash pi)=-\backslash cos(\backslash theta)$ and $\backslash sin(\backslash theta+\backslash pi)=-\backslash sin(\backslash theta)$ define the sine and cosine as periodic functions with period $2\backslash pi$, 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,

$$\backslash arctan\; s\; +\; \backslash arctan\; t\; =\; \backslash arctan\; \backslash frac\{s+t\}\{1-st\}$$

holds, provided $\backslash arctan\; s+\backslash arctan\; t\backslash in(-\backslash pi/2,\backslash pi/2)$, since

$$\backslash arctan\; s\; +\; \backslash arctan\; t=\; \backslash int\_\{-s\}^t\backslash frac\{d\backslash tau\}\{1+\backslash tau^2\}=\backslash int\_0^\{\backslash frac\{s+t\}\{1-st\}\}\backslash frac\{d\backslash tau\}\{1+\backslash tau^2\}$$

after the substitution $\backslash tau\; \backslash to\; \backslash frac\{s+\backslash tau\}\{1-s\backslash tau\}$. In particular, the limiting case as $s\backslash to\backslash infty$ gives

$$\backslash arctan\; t\; +\; \backslash frac\{\backslash pi\}\{2\}\; =\; \backslash arctan(-1/t),\backslash quad\; t\backslash in\; (-\backslash infty,0).$$

Thus we have

$$\backslash sin\backslash left(\backslash theta\; +\; \backslash frac\{\backslash pi\}\{2\}\backslash right)\; =\; \backslash frac\{-1\}\{t\backslash sqrt\{1+(-1/t)^2\}\}\; =\; \backslash frac\{-1\}\{\backslash sqrt\{1+t^2\}\}\; =\; -\backslash cos(\backslash theta)$$

and

$$\backslash cos\backslash left(\backslash theta\; +\; \backslash frac\{\backslash pi\}\{2\}\backslash right)\; =\; \backslash frac\{1\}\{\backslash sqrt\{1+(-1/t)^2\}\}\; =\; \backslash frac\{t\}\{\backslash sqrt\{1+t^2\}\}\; =\; \backslash sin(\backslash theta).$$

So the sine and cosine functions are related by translation over a quarter period $\backslash pi/2$.

One can also define the trigonometric functions using various functional equations.

For example, the sine and the cosine form the unique pair of continuous functions that satisfy the difference formula

: $\backslash cos(x-\; y)\; =\; \backslash cos\; x\backslash cos\; y\; +\; \backslash sin\; x\backslash sin\; y\backslash ,$

and the added condition

:

The sine and cosine of a complex number $z=x+iy$ can be expressed in terms of real sines, cosines, and hyperbolic functions as follows:

:

\cos z &= \cos x \cosh y - i \sin x \sinh y\end{align}

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 $z$ becomes larger (since the color white represents infinity), and the fact that the functions contain simple 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.

The sine and cosine functions are periodic, with period $2\backslash pi$, which is the smallest positive period:

$$\backslash sin(z+2\backslash pi)\; =\; \backslash sin(z),\backslash quad\; \backslash cos(z+2\backslash pi)\; =\; \backslash cos(z).$$

Consequently, the cosecant and secant also have $2\backslash pi$ as their period.

The functions sine and cosine also have semiperiods $\backslash pi$, and

$$\backslash sin(z+\backslash pi)=-\backslash sin(z),\backslash quad\; \backslash cos(z+\backslash pi)=-\backslash cos(z)$$

and consequently

$$\backslash tan(z+\backslash pi)\; =\; \backslash tan(z),\backslash quad\; \backslash cot(z+\backslash pi)\; =\; \backslash cot(z).$$

Also,

$$\backslash sin(x+\backslash pi/2)=\backslash cos(x),\backslash quad\; \backslash cos(x+\backslash pi/2)\; =\; -\backslash sin(x)$$

(see Complementary angles).

The function $\backslash sin(z)$ has a unique zero (at $z=0$) in the strip . The function $\backslash cos(z)$ has the pair of zeros $z=\backslash pm\backslash pi/2$ in the same strip. Because of the periodicity, the zeros of sine are

$$\backslash pi\backslash mathbb\; Z\; =\; \backslash left\backslash \{\backslash dots,-2\backslash pi,-\backslash pi,0,\backslash pi,2\backslash pi,\backslash dots\backslash right\backslash \}\backslash subset\backslash mathbb\; C.$$

There zeros of cosine are

$$\backslash frac\{\backslash pi\}\{2\}\; +\; \backslash pi\backslash mathbb\; Z\; =\; \backslash left\backslash \{\backslash dots,-\backslash frac\{3\backslash pi\}\{2\},-\backslash frac\{\backslash pi\}\{2\},\backslash frac\{\backslash pi\}\{2\},\backslash frac\{3\backslash pi\}\{2\},\backslash dots\backslash right\backslash \}\backslash subset\backslash mathbb\; C.$$

All of the zeros are simple zeros, and both functions have derivative $\backslash pm\; 1$ at each of the zeros.

The tangent function $\backslash tan(z)=\backslash sin(z)/\backslash cos(z)$ has a simple zero at $z=0$ and vertical asymptotes at $z=\backslash pm\backslash pi/2$, where it has a simple pole of residue $-1$. Again, owing to the periodicity, the zeros are all the integer multiples of $\backslash pi$ and the poles are odd multiples of $\backslash pi/2$, all having the same residue. The poles correspond to vertical asymptotes

$$\backslash lim\_\{x\backslash to\backslash pi^-\}\backslash tan(x)\; =\; +\backslash infty,\backslash quad\; \backslash lim\_\{x\backslash to\backslash pi^+\}\backslash tan(x)\; =\; -\backslash infty.$$

The cotangent function $\backslash cot(z)=\backslash cos(z)/\backslash sin(z)$ has a simple pole of residue 1 at the integer multiples of $\backslash pi$ and simple zeros at odd multiples of $\backslash pi/2$. The poles correspond to vertical asymptotes

$$\backslash lim\_\{x\backslash to\; 0^-\}\backslash cot(x)\; =\; -\backslash infty,\backslash quad\; \backslash lim\_\{x\backslash to\; 0^+\}\backslash cot(x)\; =\; +\backslash infty.$$

Many 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 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.

The cosine and the secant are even functions; the other trigonometric functions are odd functions. That is:

:$\backslash begin\{align\}$

\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}

All trigonometric functions are periodic functions 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

:$\backslash begin\{array\}\{lrl\}$

\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}

See Periodicity and asymptotes.

The Pythagorean identity, is the expression of the Pythagorean theorem in terms of trigonometric functions. It is

:$\backslash sin^2\; x\; +\; \backslash cos^2\; x\; =\; 1$.

Dividing through by either $\backslash cos^2\; x$ or $\backslash sin^2\; x$ gives

:$\backslash tan^2\; x\; +\; 1\; =\; \backslash sec^2\; x$

:$1\; +\; \backslash cot^2\; x\; =\; \backslash csc^2\; x$

and

:$\backslash sec^2\; x\; +\; \backslash csc^2\; x\; =\; \backslash sec^2\; x\; \backslash csc^2\; x$.

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 Angle sum and difference identities). One can also produce them algebraically using Euler's formula.

; Sum

:$\backslash begin\{align\}$

\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}

; Difference

:$\backslash begin\{align\}$

\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}

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

:$\backslash begin\{align\}$

\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}

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

By setting $t=\backslash tan\; \backslash tfrac12\; \backslash theta,$ all trigonometric functions of $\backslash theta$ can be expressed as rational fractions of $t$:

:$\backslash begin\{align\}$

\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}

Together with

:$d\backslash theta\; =\; \backslash frac\{2\}\{1+t^2\}\; \backslash ,\; dt,$

this is the tangent half-angle substitution, which reduces the computation of integrals and antiderivatives of trigonometric functions to that of rational fractions.

The derivatives of trigonometric functions result from those of sine and cosine by applying the quotient rule. The values given for the antiderivatives in the following table can be verified by differentiating them. The number {{mvar|C}} is a constant of integration.

Note: For

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

:$$

\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}

{{Main|Inverse trigonometric functions}}

The trigonometric functions are periodic, and hence not 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 functions. To define a true inverse function, one must restrict the domain to an interval where the function is monotonic, and is thus bijective from this interval to its image by the function. The common choice for this interval, called the set of principal values, 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.

The notations {{math|sin⁻¹}}, {{math|cos⁻¹}}, 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 logarithms.

{{Main|Uses of trigonometry}}

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.

{{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}}:

$$\backslash frac\{\backslash sin\; A\}\{a\}\; =\; \backslash frac\{\backslash sin\; B\}\{b\}\; =\; \backslash frac\{\backslash sin\; C\}\{c\}\; =\; \backslash frac\{2\backslash Delta\}\{abc\},$$

where {{math|Δ}} is the area of the triangle,

or, equivalently,

$$\backslash frac\{a\}\{\backslash sin\; A\}\; =\; \backslash frac\{b\}\{\backslash sin\; B\}\; =\; \backslash frac\{c\}\{\backslash sin\; C\}\; =\; 2R,$$

where {{mvar|R}} is the triangle's 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.

{{Main|Law of cosines}}

The law of cosines (also known as the cosine formula or cosine rule) is an extension of the Pythagorean theorem:

$$c^2=a^2+b^2-2ab\backslash cos\; C,$$

or equivalently,

$$\backslash cos\; C=\backslash frac\{a^2+b^2-c^2\}\{2ab\}.$$

In this formula the angle at {{mvar|C}} is opposite to the side {{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.

{{main|Law of tangents}}

The law of tangents says that:

:$\backslash frac\{\backslash tan\; \backslash frac\{A-B\}\{2\; \}\}\{\backslash tan\; \backslash frac\{A+B\}\{2\; \}\; \}\; =\; \backslash frac\{a-b\}\{a+b\}$.

{{main|Law of cotangents}}

If s is the triangle's semiperimeter, (a + b + c)/2, and r is the radius of the triangle's incircle, then rs is the triangle's area. Therefore Heron's formula implies that:

:$r\; =\; \backslash sqrt\{\backslash frac\{1\}\{s\}\; (s-a)(s-b)(s-c)\}$.

The law of cotangents says that:

:$\backslash cot\{\; \backslash frac\{A\}\{2\}\}\; =\; \backslash frac\{s-a\}\{r\}$

It follows that

:$\backslash frac\{\backslash cot\; \backslash dfrac\{A\}\{2\}\}\{s-a\}=\backslash frac\{\backslash cot\; \backslash dfrac\{B\}\{2\}\}\{s-b\}=\backslash frac\{\backslash cot\; \backslash dfrac\{C\}\{2\}\}\{s-c\}=\backslash frac\{1\}\{r\}.$

File:Lissajous curve 5by4.svg, a figure formed with a trigonometry-based function.]]

File:Synthesis square.gif of a square wave with an increasing number of harmonics]]

File:Sawtooth Fourier Animation.gif.]]

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 functions. The characteristic wave patterns of periodic functions are useful for modeling recurring phenomena such as sound or light waves.

Under rather general conditions, a periodic function {{math|1=f (x)}} can be expressed as a sum of sine waves or cosine waves in a Fourier series. Denoting the sine or cosine basis functions by {{mvar|φ_k}}, the expansion of the periodic function {{math|1=f (t)}} takes the form:

$$f(t)\; =\; \backslash sum\; \_\{k=1\}^\backslash infty\; c\_k\; \backslash varphi\_k(t).$$

For example, the square wave can be written as the Fourier series

$$f\_\backslash text\{square\}(t)\; =\; \backslash frac\{4\}\{\backslash pi\}\; \backslash sum\_\{k=1\}^\backslash infty\; \{\backslash sin\; \backslash big(\; (2k-1)t\; \backslash big)\; \backslash over\; 2k-1\}.$$

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.

{{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 function was defined by Hipparchus of Nicaea (180–125 BCE) and Ptolemy of Roman Egypt (90–165 CE). The functions of sine and versine (1 – cosine) are closely related to the 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. (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. 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.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}}{{cite web |title=trigonometry |date=17 November 2023 |url=http://www.britannica.com/EBchecked/topic/605281/trigonometry |publisher=Encyclopedia Britannica}} 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°. 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), Rheticus, and Rheticus' student Valentinus Otho.

Madhava of Sangamagrama (c. 1400) made early strides in the analysis of trigonometric functions in terms of infinite series. (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.{{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 }}

The terms tangent and secant were first introduced by the Danish mathematician Thomas Fincke in his book Geometria rotundi (1583).

The 17th century French mathematician Albert Girard made the first published use of the abbreviations sin, cos, and tan in his book Trigonométrie.{{MacTutor|id=Girard_Albert}}

In a paper published in 1682, Gottfried Leibniz proved that {{math|sin x}} is not an algebraic function of {{mvar|x}}. Though defined as ratios of sides of a right triangle, and thus appearing to be rational functions, Leibnitz result established that they are actually transcendental functions 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 series. He presented "Euler's formula", as well as near-modern abbreviations (sin., cos., tang., cot., sec., and cosec.).

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

: $\backslash begin\{align\}$

\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}

{{anchor|Logarithmic sine|Logarithmic cosine|Logarithmic secant|Logarithmic cosecant|Logarithmic tangent|Logarithmic cotangent}}Historically, trigonometric functions were often combined with logarithms in compound functions like the logarithmic sine, logarithmic cosine, logarithmic secant, logarithmic cosecant, logarithmic tangent and logarithmic cotangent.{{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}}{{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}}{{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}}{{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}}

{{main|History of trigonometry#Etymology}}

The word {{Lang|la-x-medieval|sine}} derivesThe anglicized form is first recorded in 1593 in Thomas Fale's Horologiographia, the Art of Dialling. from Latin 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 al-Khwārizmī into Medieval Latin.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 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
See Maor (1998), chapter 3, for an earlier etymology crediting Gerard.
See {{cite book |last=Katx |first=Victor |date=July 2008 |title=A history of mathematics |edition=3rd |location=Boston |publisher=Pearson |page=210 (sidebar) |isbn= 978-0321387004 |language=en }}

The choice was based on a misreading of the Arabic written form j-y-b ({{lang|ar|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 {{lang|grc|χορδή}} "string".

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.Oxford English Dictionary

The 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.

{{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}}

{{notelist}}

{{reflist|refs=

{{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.}} }}

{{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 }}

{{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 }}

{{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 }}

{{cite book |author-last=Kannappan |author-first=Palaniappan |title=Functional Equations and Inequalities with Applications |date=2009 |publisher=Springer |isbn=978-0387894911}}

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.

{{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 }}

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 }}

Boyer, Carl B. (1991). A History of Mathematics (Second ed.). John Wiley & Sons, Inc. {{isbn|0-471-54397-7}}, p. 210.

{{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}}

{{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 }}

{{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 }}

{{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 }}

{{cite book |author-first=Edmund |author-last=Gunter |author-link=Edmund Gunter |title=Canon triangulorum |date=1620}}

{{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}}

See Plofker, Mathematics in India, Princeton University Press, 2009, p. 257
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 }}
See Maor (1998), chapter 3, regarding the etymology.

}}

{{refbegin}}

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{{refend}}

{{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}}

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{{DEFAULTSORT:Trigonometric Functions}}

Category:Analytic functions

Category:Angle

Category:Ratios