Movatterモバイル変換

[0]ホーム

Jump to content

Trigonometric functions

Edit links

From Wikipedia, the free encyclopedia

(Redirected fromTrigonometric function)

Functions of an angle

"Logarithmic sine" and "Logarithmic cosine" redirect here. For the Clausen-related functions, seelog cosine function andlog sine function.

Basis of trigonometry: if tworight triangles have equalacute angles, they aresimilar, so their corresponding side lengths areproportional.

Inmathematics, thetrigonometric functions (also calledcircular functions,angle functions orgoniometric functions)^[1] arereal functions which relate an angle of aright-angled triangle to ratios of two side lengths. They are widely used in all sciences that are related togeometry, such asnavigation,solid mechanics,celestial mechanics,geodesy, and many others. They are among the simplestperiodic functions, and as such are also widely used for studying periodic phenomena throughFourier analysis.

Trigonometry

Outline History Usage Functions (sin,cos,tan,inverse) Generalized trigonometry
Reference
Identities Exact constants Tables Unit circle
Laws and theorems
Sines Cosines Tangents Cotangents Pythagorean theorem
Calculus
Trigonometric substitution Integrals (inverse functions) Derivatives Trigonometric series
Mathematicians
Hipparchus Ptolemy Brahmagupta al-Hasib al-Battani Regiomontanus Viète de Moivre Euler Fourier
v t e

The trigonometric functions most widely used in modern mathematics are thesine, thecosine, and thetangent functions. Theirreciprocals are respectively thecosecant, thesecant, and thecotangent functions, which are less used. Each of these six trigonometric functions has a correspondinginverse function, and an analog among thehyperbolic functions.

The oldest definitions of trigonometric functions, related to right-angle triangles, define them only foracute angles. To extend the sine and cosine functions to functions whosedomain is the wholereal line, geometrical definitions using the standardunit circle (i.e., a circle withradius 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 asinfinite series or as solutions ofdifferential equations. This allows extending the domain of sine and cosine functions to the wholecomplex plane, and the domain of the other trigonometric functions to the complex plane with some isolated points removed.

Notation

[edit]

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 "sin" for sine, "cos" for cosine, "tan" or "tg" for tangent, "sec" for secant, "csc" or "cosec" for cosecant, and "cot" or "ctg" for cotangent. Historically, these abbreviations were first used in prose sentences to indicate particularline segments or their lengths related to anarc of an arbitrary circle, and later to indicate ratios of lengths, but as thefunction concept developed in the 17th–18th century, they began to be considered as functions of real-number-valued angle measures, and written withfunctional notation, for examplesin(x). Parentheses are still often omitted to reduce clutter, but are sometimes necessary; for example the expression $\sin x+y$ would typically be interpreted to mean $(\sin x)+y,$ so parentheses are required to express $\sin(x+y).$

Apositive integer appearing as a superscript after the symbol of the function denotesexponentiation, notfunction composition. For example $\sin ^{2}x$ and $\sin ^{2}(x)$ denote $(\sin x)^{2},$ not $\sin(\sin x).$ This differs from the (historically later) general functional notation in which $f^{2}(x)=(f\circ f)(x)=f(f(x)).$

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

Right-angled triangle definitions

[edit]

In this right triangle, denoting the measure of angle BAC as A:sinA =⁠a/c⁠;cosA =⁠b/c⁠;tanA =⁠a/b⁠.

Plot of the six trigonometric functions, the unit circle, and a line for the angleθ = 0.7 radians. The points labeled1,Sec(θ),Csc(θ) represent the length of the line segment from the origin to that point.Sin(θ),Tan(θ), and1 are the heights to the line starting from thex-axis, whileCos(θ),1, andCot(θ) are lengths along thex-axis starting from the origin.

If the acute angleθ is given, then any right triangles that have an angle ofθ aresimilar to each other. This means that the ratio of any two side lengths depends only onθ. Thus these six ratios define six functions ofθ, which are the trigonometric functions. In the following definitions, thehypotenuse is the length of the side opposite the right angle,opposite represents the side opposite the given angleθ, andadjacent represents the side between the angleθ and the right angle.^[2]^[3]

sine $\sin \theta ={\frac {\mathrm {opposite} }{\mathrm {hypotenuse} }}$	cosecant $\csc \theta ={\frac {\mathrm {hypotenuse} }{\mathrm {opposite} }}$
cosine $\cos \theta ={\frac {\mathrm {adjacent} }{\mathrm {hypotenuse} }}$	secant $\sec \theta ={\frac {\mathrm {hypotenuse} }{\mathrm {adjacent} }}$
tangent $\tan \theta ={\frac {\mathrm {opposite} }{\mathrm {adjacent} }}$	cotangent $\cot \theta ={\frac {\mathrm {adjacent} }{\mathrm {opposite} }}$

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,90° or⁠π/2⁠radians. Therefore $\sin(\theta )$ and $\cos(90^{\circ }-\theta )$ represent the same ratio, and thus are equal. This identity and analogous relationships between the other trigonometric functions are summarized in the following table.

**Top:** Trigonometric functionsinθ for selected anglesθ,π −θ,π +θ, and2π −θ in the four quadrants.
**Bottom:** Graph of sine versus angle. Angles from the top panel are identified.

Summary of relationships between trigonometric functions^[4]
Function	Description	Relationship
Function	Description	usingradians	usingdegrees
sine	⁠opposite/hypotenuse⁠	$\sin \theta =\cos \left({\frac {\pi }{2}}-\theta \right)={\frac {1}{\csc \theta }}$	$\sin x=\cos \left(90^{\circ }-x\right)={\frac {1}{\csc x}}$
cosine	⁠adjacent/hypotenuse⁠	$\cos \theta =\sin \left({\frac {\pi }{2}}-\theta \right)={\frac {1}{\sec \theta }}\,$	$\cos x=\sin \left(90^{\circ }-x\right)={\frac {1}{\sec x}}\,$
tangent	⁠opposite/adjacent⁠	$\tan \theta ={\frac {\sin \theta }{\cos \theta }}=\cot \left({\frac {\pi }{2}}-\theta \right)={\frac {1}{\cot \theta }}$	$\tan x={\frac {\sin x}{\cos x}}=\cot \left(90^{\circ }-x\right)={\frac {1}{\cot x}}$
cotangent	⁠adjacent/opposite⁠	$\cot \theta ={\frac {\cos \theta }{\sin \theta }}=\tan \left({\frac {\pi }{2}}-\theta \right)={\frac {1}{\tan \theta }}$	$\cot x={\frac {\cos x}{\sin x}}=\tan \left(90^{\circ }-x\right)={\frac {1}{\tan x}}$
secant	⁠hypotenuse/adjacent⁠	$\sec \theta =\csc \left({\frac {\pi }{2}}-\theta \right)={\frac {1}{\cos \theta }}$	$\sec x=\csc \left(90^{\circ }-x\right)={\frac {1}{\cos x}}$
cosecant	⁠hypotenuse/opposite⁠	$\csc \theta =\sec \left({\frac {\pi }{2}}-\theta \right)={\frac {1}{\sin \theta }}$	$\csc x=\sec \left(90^{\circ }-x\right)={\frac {1}{\sin x}}$

Radians versus degrees

[edit]

In geometric applications, the argument of a trigonometric function is generally the measure of anangle. For this purpose, anyangular unit is convenient. One common unit isdegrees, in which a right angle is 90° and a complete turn is 360° (particularly inelementary mathematics).

However, incalculus andmathematical analysis, the trigonometric functions are generally regarded more abstractly as functions ofreal orcomplex numbers, rather than angles. In fact, the functionssin andcos can be defined for all complex numbers in terms of theexponential function, via power series,^[5] or as solutions todifferential equations given particular initial values^[6] (see below), without reference to any geometric notions. The other four trigonometric functions (tan,cot,sec,csc) can be defined as quotients and reciprocals ofsin andcos, 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.^[5] Moreover, these definitions result in simple expressions for thederivatives andindefinite integrals for the trigonometric functions.^[7] Thus, in settings beyond elementary geometry, radians are regarded as the mathematically natural unit for describing angle measures.

Whenradians (rad) are employed, the angle is given as the length of thearc of theunit circle subtended by it: the angle that subtends an arc of length 1 on the unit circle is 1 rad (≈ 57.3°),^[8] and a completeturn (360°) is an angle of 2π (≈ 6.28) rad.^[9] For real numberx, the notationsinx,cosx, etc. refers to the value of the trigonometric functions evaluated at an angle ofx rad. If units of degrees are intended, the degree sign must be explicitly shown (sinx°,cosx°, etc.). Using this standard notation, the argumentx for the trigonometric functions satisfies the relationshipx = (180x/π)°, so that, for example,sinπ = sin 180° when we takex =π. In this way, the degree symbol can be regarded as a mathematical constant such that 1° =π/180 ≈ 0.0175.^[10]

Unit-circle definitions

[edit]

All of the trigonometric functions of the angleθ (theta) can be constructed geometrically in terms of a unit circle centered atO.

Sine function on unit circle (top) and its graph (bottom)

In this illustration, the six trigonometric functions of an arbitrary angleθ are represented asCartesian coordinates of points related to theunit circle. They-axis ordinates ofA,B andD aresinθ,tanθ andcscθ, respectively, while thex-axis abscissas ofA,C andE arecosθ,cotθ andsecθ, respectively.

Signs of trigonometric functions in each quadrant.Mnemonics like "**all**studentstakecalculus" indicates whensine,cosine, andtangent are positive from quadrants I to IV.^[11]

The six trigonometric functions can be defined ascoordinate values of points on theEuclidean plane that are related to theunit circle, which is thecircle of radius one centered at the originO of this coordinate system. Whileright-angled triangle definitions allow for the definition of the trigonometric functions for angles between0 and ${\textstyle {\frac {\pi }{2}}}$ radians(90°), the unit circle definitions allow the domain of trigonometric functions to be extended to all positive and negative real numbers.

Let ${\mathcal {L}}$ be theray obtained by rotating by an angleθ the positive half of thex-axis (counterclockwise rotation for $\theta >0,$ and clockwise rotation for $\theta <0$ ). This ray intersects the unit circle at the point $\mathrm {A} =(x_{\mathrm {A} },y_{\mathrm {A} }).$ The ray ${\mathcal {L}},$ extended to aline if necessary, intersects the line of equation $x=1$ at point $\mathrm {B} =(1,y_{\mathrm {B} }),$ and the line of equation $y=1$ at point $\mathrm {C} =(x_{\mathrm {C} },1).$ Thetangent line to the unit circle at the pointA, isperpendicular to ${\mathcal {L}},$ and intersects they- andx-axes at points $\mathrm {D} =(0,y_{\mathrm {D} })$ and $\mathrm {E} =(x_{\mathrm {E} },0).$ Thecoordinates of these points give the values of all trigonometric functions for any arbitrary real value ofθ in the following manner.

The trigonometric functionscos andsin are defined, respectively, as thex- andy-coordinate values of pointA. That is,

\cos \theta =x_{\mathrm {A} }\quad

and

\quad \sin \theta =y_{\mathrm {A} }.

^[12]

In the range $0\leq \theta \leq \pi /2$ , this definition coincides with the right-angled triangle definition, by taking the right-angled triangle to have the unit radiusOA ashypotenuse. And since the equation $x^{2}+y^{2}=1$ holds for all points $\mathrm {P} =(x,y)$ on the unit circle, this definition of cosine and sine also satisfies thePythagorean identity.

\cos ^{2}\theta +\sin ^{2}\theta =1.

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

\tan \theta =y_{\mathrm {B} }\quad

and

\quad \cot \theta =x_{\mathrm {C} },

\csc \theta \ =y_{\mathrm {D} }\quad

and

\quad \sec \theta =x_{\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

\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 }}.

Trigonometric functions:Sine,Cosine,Tangent,Cosecant (dotted),Secant (dotted),Cotangent (dotted) –animation

Since a rotation of an angle of $\pm 2\pi$ does not change the position or size of a shape, the pointsA,B,C,D, andE are the same for two angles whose difference is an integer multiple of $2\pi$ . Thus trigonometric functions areperiodic functions with period $2\pi$ . That is, the equalities

\sin \theta =\sin \left(\theta +2k\pi \right)\quad

and

\quad \cos \theta =\cos \left(\theta +2k\pi \right)

hold for any angleθ and anyintegerk. 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 $2\pi$ is the smallest value for which they are periodic (i.e., $2\pi$ is thefundamental period of these functions). However, after a rotation by an angle $\pi$ , the pointsB andC already return to their original position, so that the tangent function and the cotangent function have a fundamental period of $\pi$ . That is, the equalities

\tan \theta =\tan(\theta +k\pi )\quad

and

\quad \cot \theta =\cot(\theta +k\pi )

hold for any angleθ and any integerk.

Algebraic values

[edit]

Theunit circle, with some points labeled with their cosine and sine (in this order), and the corresponding angles in radians and degrees.

Thealgebraic expressions for the most important angles are as follows:

\sin 0=\sin 0^{\circ }\quad ={\frac {\sqrt {0}}{2}}=0

(zero angle)

\sin {\frac {\pi }{6}}=\sin 30^{\circ }={\frac {\sqrt {1}}{2}}={\frac {1}{2}}

\sin {\frac {\pi }{4}}=\sin 45^{\circ }={\frac {\sqrt {2}}{2}}={\frac {1}{\sqrt {2}}}

\sin {\frac {\pi }{3}}=\sin 60^{\circ }={\frac {\sqrt {3}}{2}}

\sin {\frac {\pi }{2}}=\sin 90^{\circ }={\frac {\sqrt {4}}{2}}=1

(right angle)

Writing the numerators assquare roots of consecutive non-negative integers, with a denominator of 2, provides an easy way to remember the values.^[13]

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, theexact 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 byruler 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 thecube root of a non-realcomplex 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 arational number, the sine and the cosine arealgebraic numbers, which may be expressed in terms ofnth roots. This results from the fact that theGalois groups of thecyclotomic polynomials arecyclic.
For an angle which, expressed in degrees, is not a rational number, then either the angle or both the sine and the cosine aretranscendental numbers. This is a corollary ofBaker'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

[edit]

Main article:Exact trigonometric values § Common angles

The following table lists the sines, cosines, and tangents of multiples of 15 degrees from 0 to 90 degrees.

Angle,θ, in		$\sin(\theta )$	$\cos(\theta )$	$\tan(\theta )$
radians	degrees	$\sin(\theta )$	$\cos(\theta )$	$\tan(\theta )$
$0 {\displaystyle 0}$	$0^{\circ }$	$0 {\displaystyle 0}$	$1 {\displaystyle 1}$	$0 {\displaystyle 0}$
${\frac {\pi }{12}}$	$15^{\circ }$	${\frac {{\sqrt {6}}-{\sqrt {2}}}{4}}$	${\frac {{\sqrt {6}}+{\sqrt {2}}}{4}}$	$2-{\sqrt {3}}$
${\frac {\pi }{6}}$	$30^{\circ }$	${\frac {1}{2}}$	${\frac {\sqrt {3}}{2}}$	${\frac {\sqrt {3}}{3}}$
${\frac {\pi }{4}}$	$45^{\circ }$	${\frac {\sqrt {2}}{2}}$	${\frac {\sqrt {2}}{2}}$	$1 {\displaystyle 1}$
${\frac {\pi }{3}}$	$60^{\circ }$	${\frac {\sqrt {3}}{2}}$	${\frac {1}{2}}$	${\sqrt {3}}$
${\frac {5\pi }{12}}$	$75^{\circ }$	${\frac {{\sqrt {6}}+{\sqrt {2}}}{4}}$	${\frac {{\sqrt {6}}-{\sqrt {2}}}{4}}$	$2+{\sqrt {3}}$
${\frac {\pi }{2}}$	$90^{\circ }$	$1 {\displaystyle 1}$	$0 {\displaystyle 0}$	Undefined

Definitions in analysis

[edit]

The sine function (blue) is closely approximated by itsTaylor polynomial of degree 7 (pink) for a full cycle centered on the origin.

Animation for the approximation of cosine via Taylor polynomials.

$\cos(x)$ together with the first Taylor polynomials $p_{n}(x)=\sum _{k=0}^{n}(-1)^{k}{\frac {x^{2k}}{(2k)!}}$

{\displaystyle \cos(x)} — $\cos(x)$ together with the first Taylor polynomials $p_{n}(x)=\sum _{k=0}^{n}(-1)^{k}{\frac {x^{2k}}{(2k)!}}$

G. H. Hardy noted in his 1908 workA 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.^[14] 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.^[14]
By a power series, which is particularly well-suited to complex variables.^[14]^[15]
By using an infinite product expansion.^[14]
By inverting the inverse trigonometric functions, which can be defined as integrals of algebraic or rational functions.^[14]
As solutions of a differential equation.^[16]

Definition by differential equations

[edit]

Sine and cosine can be defined as the unique solution to theinitial value problem:^[17]

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

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

y''+y=0\,.

Sine is the unique solution withy(0) = 0 andy′(0) = 1; cosine is the unique solution withy(0) = 1 andy′(0) = 0.

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

Applying thequotient rule to the tangent $\tan x=\sin x/\cos x$ ,

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

so the tangent function satisfies the ordinary differential equation

y'=1+y^{2}\,.

It is the unique solution withy(0) = 0.

Power series expansion

[edit]

The basic trigonometric functions can be defined by the following power series expansions.^[18] These series are also known as theTaylor series orMaclaurin series of these trigonometric functions:

{\begin{aligned}\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{aligned}}

Theradius of convergence of these series is infinite. Therefore, the sine and the cosine can be extended toentire functions (also called "sine" and "cosine"), which are (by definition)complex-valued functions that are defined andholomorphic on the wholecomplex 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 tomeromorphic functions, that is functions that are holomorphic in the whole complex plane, except some isolated points calledpoles. Here, the poles are the numbers of the form ${\textstyle (2k+1){\frac {\pi }{2}}}$ for the tangent and the secant, or $k\pi$ for the cotangent and the cosecant, wherek is an arbitrary integer.

Recurrences relations may also be computed for the coefficients of theTaylor series of the other trigonometric functions. These series have a finiteradius of convergence. Their coefficients have acombinatorial interpretation: they enumeratealternating permutations of finite sets.^[19]

More precisely, defining

U_n, thenthup/down number,

B_n, thenthBernoulli number, and

E_n, is thenthEuler number,

one has the following series expansions:^[20]

{\begin{aligned}\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{aligned}}

{\begin{aligned}\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{aligned}}

{\begin{aligned}\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{aligned}}

{\begin{aligned}\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{aligned}}

Continued fraction expansion

[edit]

The followingcontinued fractions are valid in the whole complex plane:

\sin x={\cfrac {x}{1+{\cfrac {x^{2}}{2\cdot 3-x^{2}+{\cfrac {2\cdot 3x^{2}}{4\cdot 5-x^{2}+{\cfrac {4\cdot 5x^{2}}{6\cdot 7-x^{2}+\ddots }}}}}}}}

\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 }}}}}}}}

\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 }}}}}}}}

The last one was used in the historically firstproof that π is irrational.^[21]

Partial fraction expansion

[edit]

There is a series representation aspartial fraction expansion where just translatedreciprocal functions are summed up, such that thepoles of the cotangent function and the reciprocal functions match:^[22]

\pi \cot \pi x=\lim _{N\to \infty }\sum _{n=-N}^{N}{\frac {1}{x+n}}.

This identity can be proved with theHerglotz trick.^[23]Combining the(–n)th with thenth term lead toabsolutely 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}}},

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

\pi \sec \pi x=\sum _{n=0}^{\infty }(-1)^{n}{\frac {(2n+1)}{(n+{\tfrac {1}{2}})^{2}-x^{2}}},

\pi \tan \pi x=2x\sum _{n=0}^{\infty }{\frac {1}{(n+{\tfrac {1}{2}})^{2}-x^{2}}}.

Infinite product expansion

[edit]

The following infinite product for the sine is due toLeonhard Euler, and is of great importance in complex analysis:^[24]

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

This may be obtained from the partial fraction decomposition of $\cot z$ given above, which is the logarithmic derivative of $\sin z$ .^[25] From this, it can be deduced also that

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

Euler's formula and the exponential function

[edit]

$\cos(\theta )$ and $\sin(\theta )$ are the real and imaginary part of $e^{i\theta }$ respectively.

{\displaystyle \cos(\theta )} — $\cos(\theta )$ and $\sin(\theta )$ are the real and imaginary part of $e^{i\theta }$ respectively.

Euler's formula relates sine and cosine to theexponential function:

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

This formula is commonly considered for real values ofx, but it remains true for all complex values.

Proof: Let $f_{1}(x)=\cos x+i\sin x,$ and $f_{2}(x)=e^{ix}.$ One has $df_{j}(x)/dx=if_{j}(x)$ forj = 1, 2. Thequotient rule implies thus that $d/dx\,(f_{1}(x)/f_{2}(x))=0$ . Therefore, $f_{1}(x)/f_{2}(x)$ is a constant function, which equals1, as $f_{1}(0)=f_{2}(0)=1.$ This proves the formula.

One has

{\begin{aligned}e^{ix}&=\cos x+i\sin x\\[5pt]e^{-ix}&=\cos x-i\sin x.\end{aligned}}

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

{\begin{aligned}\sin x&={\frac {e^{ix}-e^{-ix}}{2i}}\\[5pt]\cos x&={\frac {e^{ix}+e^{-ix}}{2}}.\end{aligned}}

Whenx is real, this may be rewritten as

\cos x=\operatorname {Re} \left(e^{ix}\right),\qquad \sin x=\operatorname {Im} \left(e^{ix}\right).

Mosttrigonometric identities can be proved by expressing trigonometric functions in terms of the complex exponential function by using above formulas, and then using the identity $e^{a+b}=e^{a}e^{b}$ for simplifying the result.

Euler's formula can also be used to define the basic trigonometric function directly, as follows, using the language oftopological groups.^[26] The set $U {\displaystyle U}$ 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 $\mathbb {R} /\mathbb {Z}$ , via an isomorphism $e:\mathbb {R} /\mathbb {Z} \to U.$ In pedestrian terms $e(t)=\exp(2\pi it)$ , and this isomorphism is unique up to taking complex conjugates.

For a nonzero real number $a {\displaystyle a}$ (thebase), the function $t\mapsto e(t/a)$ defines an isomorphism of the group $\mathbb {R} /a\mathbb {Z} \to U$ . The real and imaginary parts of $e(t/a)$ are the cosine and sine, where $a {\displaystyle a}$ is used as the base for measuring angles. For example, when $a=2\pi$ , 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 $a=2\pi$ is the unique value at which the derivative ${\frac {d}{dt}}e(t/a)$ becomes aunit vector with positive imaginary part at $t=0$ . This fact can, in turn, be used to define the constant $2\pi$ .

Definition via integration

[edit]

Another way to define the trigonometric functions in analysis is using integration.^[14]^[27] For a real number $t {\displaystyle t}$ , put $\theta (t)=\int _{0}^{t}{\frac {d\tau }{1+\tau ^{2}}}=\arctan t$ where this defines this inverse tangent function. Also, $\pi$ is defined by ${\frac {1}{2}}\pi =\int _{0}^{\infty }{\frac {d\tau }{1+\tau ^{2}}}$ a definition that goes back toKarl Weierstrass.^[28]

On the interval $-\pi /2<\theta <\pi /2$ , the trigonometric functions are defined by inverting the relation $\theta =\arctan t$ . Thus we define the trigonometric functions by $\tan \theta =t,\quad \cos \theta =(1+t^{2})^{-1/2},\quad \sin \theta =t(1+t^{2})^{-1/2}$ where the point $(t,\theta )$ is on the graph of $\theta =\arctan t$ and the positive square root is taken.

This defines the trigonometric functions on $(-\pi /2,\pi /2)$ . The definition can be extended to all real numbers by first observing that, as $\theta \to \pi /2$ , $t\to \infty$ , and so $\cos \theta =(1+t^{2})^{-1/2}\to 0$ and $\sin \theta =t(1+t^{2})^{-1/2}\to 1$ . Thus $\cos \theta$ and $\sin \theta$ are extended continuously so that $\cos(\pi /2)=0,\sin(\pi /2)=1$ . Now the conditions $\cos(\theta +\pi )=-\cos(\theta )$ and $\sin(\theta +\pi )=-\sin(\theta )$ define the sine and cosine as periodic functions with period $2\pi$ , 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, $\arctan s+\arctan t=\arctan {\frac {s+t}{1-st}}$ holds, provided $\arctan s+\arctan t\in (-\pi /2,\pi /2)$ , since $\arctan s+\arctan t=\int _{-s}^{t}{\frac {d\tau }{1+\tau ^{2}}}=\int _{0}^{\frac {s+t}{1-st}}{\frac {d\tau }{1+\tau ^{2}}}$ after the substitution $\tau \to {\frac {s+\tau }{1-s\tau }}$ . In particular, the limiting case as $s\to \infty$ gives $\arctan t+{\frac {\pi }{2}}=\arctan(-1/t),\quad t\in (-\infty ,0).$ Thus we have $\sin \left(\theta +{\frac {\pi }{2}}\right)={\frac {-1}{t{\sqrt {1+(-1/t)^{2}}}}}={\frac {-1}{\sqrt {1+t^{2}}}}=-\cos(\theta )$ and $\cos \left(\theta +{\frac {\pi }{2}}\right)={\frac {1}{\sqrt {1+(-1/t)^{2}}}}={\frac {t}{\sqrt {1+t^{2}}}}=\sin(\theta ).$ So the sine and cosine functions are related by translation over a quarter period $\pi /2$ .

Definitions using functional equations

[edit]

One can also define the trigonometric functions using variousfunctional equations.

For example,^[29] the sine and the cosine form the unique pair ofcontinuous functions that satisfy the difference formula

\cos(x-y)=\cos x\cos y+\sin x\sin y\,

and the added condition

0<x\cos x<\sin x<x\quad {\text{ for }}\quad 0<x<1.

In the complex plane

[edit]

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

{\begin{aligned}\sin z&=\sin x\cosh y+i\cos x\sinh y\\[5pt]\cos z&=\cos x\cosh y-i\sin x\sinh y\end{aligned}}

By taking advantage ofdomain 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 $z {\displaystyle z}$ becomes larger (since the color white represents infinity), and the fact that the functions contain simplezeros 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.

**Trigonometric functions in the complex plane**
$\sin z\,$ $\cos z\,$	$\tan z\,$ $\cot z\,$	$\sec z\,$ $\csc z\,$

Periodicity and asymptotes

[edit]

The sine and cosine functions areperiodic, with period $2\pi$ , which is the smallest positive period: $\sin(z+2\pi )=\sin(z),\quad \cos(z+2\pi )=\cos(z).$ Consequently, the cosecant and secant also have $2\pi$ as their period.

The functions sine and cosine also have semiperiods $\pi$ , and $\sin(z+\pi )=-\sin(z),\quad \cos(z+\pi )=-\cos(z)$ and consequently $\tan(z+\pi )=\tan(z),\quad \cot(z+\pi )=\cot(z).$ Also, $\sin(x+\pi /2)=\cos(x),\quad \cos(x+\pi /2)=-\sin(x)$ (seeComplementary angles).

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

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

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

Basic identities

[edit]

Manyidentities interrelate the trigonometric functions. This section contains the most basic ones; for more identities, seeList 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[0,π/2], seeProofs of trigonometric identities). For non-geometrical proofs using only tools ofcalculus, one may use directly the differential equations, in a way that is similar to that of theabove 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

[edit]

The cosine and the secant areeven functions; the other trigonometric functions areodd functions. That is:

{\begin{aligned}\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{aligned}}

Periods

[edit]

All trigonometric functions areperiodic functions of period2π. This is the smallest period, except for the tangent and the cotangent, which haveπ as smallest period. This means that, for every integerk, one has

{\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}}

SeePeriodicity and asymptotes.

Pythagorean identity

[edit]

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

\sin ^{2}x+\cos ^{2}x=1

Dividing through by either $\cos ^{2}x$ or $\sin ^{2}x$ gives

\tan ^{2}x+1=\sec ^{2}x

1+\cot ^{2}x=\csc ^{2}x

and

\sec ^{2}x+\csc ^{2}x=\sec ^{2}x\csc ^{2}x

Sum and difference formulas

[edit]

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 toPtolemy. One can also produce them algebraically usingEuler's formula.

Sum: ${\begin{aligned}\sin \left(x+y\right)&=\sin x\cos y+\cos x\sin y,\\[5mu]\cos \left(x+y\right)&=\cos x\cos y-\sin x\sin y,\\[5mu]\tan(x+y)&={\frac {\tan x+\tan y}{1-\tan x\tan y}}.\end{aligned}}$
Difference: ${\begin{aligned}\sin \left(x-y\right)&=\sin x\cos y-\cos x\sin y,\\[5mu]\cos \left(x-y\right)&=\cos x\cos y+\sin x\sin y,\\[5mu]\tan(x-y)&={\frac {\tan x-\tan y}{1+\tan x\tan y}}.\end{aligned}}$

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

{\begin{aligned}\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{aligned}}

These identities can be used to derive theproduct-to-sum identities.

By setting $t=\tan {\tfrac {1}{2}}\theta ,$ all trigonometric functions of $\theta$ can be expressed asrational fractions of $t {\displaystyle t}$ :

{\begin{aligned}\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{aligned}}

Together with

d\theta ={\frac {2}{1+t^{2}}}\,dt,

this is thetangent half-angle substitution, which reduces the computation ofintegrals andantiderivatives of trigonometric functions to that of rational fractions.

Derivatives and antiderivatives

[edit]

Thederivatives of trigonometric functions result from those of sine and cosine by applying thequotient rule. The values given for theantiderivatives in the following table can be verified by differentiating them. The number C is aconstant of integration.

$f(x)$	$f'(x)$	${\textstyle \int f(x)\,dx}$
$\sin x$	$\cos x$	$-\cos x+C$
$\cos x$	$-\sin x$	$\sin x+C$
$\tan x$	$\sec ^{2}x$	$\ln \left\|\sec x\right\|+C$
$\csc x$	$-\csc x\cot x$	$\ln \left\|\csc x-\cot x\right\|+C$
$\sec x$	$\sec x\tan x$	$\ln \left\|\sec x+\tan x\right\|+C$
$\cot x$	$-\csc ^{2}x$	$-\ln \left\|\csc x\right\|+C$

Note: For $0<x<\pi$ the integral of $\csc x$ can also be written as $-\operatorname {arsinh} (\cot x),$ and for the integral of $\sec x$ for $-\pi /2<x<\pi /2$ as $\operatorname {arsinh} (\tan x),$ where $\operatorname {arsinh}$ is theinverse hyperbolic sine.

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

{\begin{aligned}{\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{aligned}}

Inverse functions

[edit]

Main article:Inverse trigonometric functions

The trigonometric functions are periodic, and hence notinjective, so strictly speaking, they do not have aninverse function. However, on each interval on which a trigonometric function ismonotonic, one can define an inverse function, and this defines inverse trigonometric functions asmultivalued functions. To define a true inverse function, one must restrict the domain to an interval where the function is monotonic, and is thusbijective from this interval to its image by the function. The common choice for this interval, called the set ofprincipal 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.

Function	Definition	Domain	Set of principal values
$y=\arcsin x$	$\sin y=x$	$-1\leq x\leq 1$	${\textstyle -{\frac {\pi }{2}}\leq y\leq {\frac {\pi }{2}}}$
$y=\arccos x$	$\cos y=x$	$-1\leq x\leq 1$	${\textstyle 0\leq y\leq \pi }$
$y=\arctan x$	$\tan y=x$	$-\infty <x<\infty$	${\textstyle -{\frac {\pi }{2}}<y<{\frac {\pi }{2}}}$
$y=\operatorname {arccot} x$	$\cot y=x$	$-\infty <x<\infty$	${\textstyle 0<y<\pi }$
$y=\operatorname {arcsec} x$	$\sec y=x$	$x<-1{\text{ or }}x>1$	${\textstyle 0\leq y\leq \pi ,\;y\neq {\frac {\pi }{2}}}$
$y=\operatorname {arccsc} x$	$\csc y=x$	$x<-1{\text{ or }}x>1$	${\textstyle -{\frac {\pi }{2}}\leq y\leq {\frac {\pi }{2}},\;y\neq 0}$

The notationssin⁻¹,cos⁻¹, etc. are often used forarcsin andarccos, 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 ofcomplex logarithms.

Applications

[edit]

Main article:Uses of trigonometry

Angles and sides of a triangle

[edit]

In this sectionA,B,C denote the three (interior) angles of a triangle, anda,b,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

[edit]

Main article:Law of sines

The law of sines states that for an arbitrary triangle with sidesa,b, andc and angles opposite those sidesA,B andC: ${\frac {\sin A}{a}}={\frac {\sin B}{b}}={\frac {\sin C}{c}}={\frac {2\Delta }{abc}},$ whereΔ is the area of the triangle,or, equivalently, ${\frac {a}{\sin A}}={\frac {b}{\sin B}}={\frac {c}{\sin C}}=2R,$ whereR is the triangle'scircumradius.

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 intriangulation, a technique to determine unknown distances by measuring two angles and an accessible enclosed distance.

Law of cosines

[edit]

Main article:Law of cosines

The law of cosines (also known as the cosine formula or cosine rule) is an extension of thePythagorean theorem: $c^{2}=a^{2}+b^{2}-2ab\cos C,$ or equivalently, $\cos C={\frac {a^{2}+b^{2}-c^{2}}{2ab}}.$

In this formula the angle atC is opposite to the side c. This theorem can be proved by dividing the triangle into two right ones and using thePythagorean 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

[edit]

Main article:Law of tangents

The law of tangents says that:

{\frac {\tan {\frac {A-B}{2}}}{\tan {\frac {A+B}{2}}}}={\frac {a-b}{a+b}}

Law of cotangents

[edit]

Main article:Law of cotangents

Ifs is the triangle's semiperimeter, (a +b +c)/2, andr is the radius of the triangle'sincircle, thenrs is the triangle's area. ThereforeHeron's formula implies that:

r={\sqrt {{\frac {1}{s}}(s-a)(s-b)(s-c)}}

The law of cotangents says that:^[30]

\cot {\frac {A}{2}}={\frac {s-a}{r}}

It follows that

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

Periodic functions

[edit]

ALissajous curve, a figure formed with a trigonometry-based function.

An animation of theadditive synthesis of asquare wave with an increasing number of harmonics

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 (k = 1) have additional nodes. The oscillation seen about the sawtooth whenk is large is called theGibbs phenomenon.

The trigonometric functions are also important in physics. The sine and the cosine functions, for example, are used to describesimple 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 ofuniform circular motion.

Trigonometric functions also prove to be useful in the study of generalperiodic functions. The characteristic wave patterns of periodic functions are useful for modeling recurring phenomena such as sound or lightwaves.^[31]

Under rather general conditions, a periodic functionf (x) can be expressed as a sum of sine waves or cosine waves in aFourier series.^[32] Denoting the sine or cosinebasis functions byφ_k, the expansion of the periodic functionf (t) takes the form: $f(t)=\sum _{k=1}^{\infty }c_{k}\varphi _{k}(t).$

For example, thesquare wave can be written as theFourier series $f_{\text{square}}(t)={\frac {4}{\pi }}\sum _{k=1}^{\infty }{\sin {\big (}(2k-1)t{\big )} \over 2k-1}.$

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 asawtooth wave are shown underneath.

History

[edit]

Main article: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. Thechord function was defined byHipparchus ofNicaea (180–125 BCE) andPtolemy ofRoman Egypt (90–165 CE). The functions of sine andversine (1 – cosine) are closely related to thejyā andkoti-jyā functions used inGupta period Indian astronomy (Aryabhatiya,Surya Siddhanta), via translation from Sanskrit to Arabic and then from Arabic to Latin.^[33] (SeeAryabhata's sine table.)

All six trigonometric functions in current use were known inIslamic mathematics by the 9th century, as was thelaw of sines, used insolving triangles.^[34]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.^[35]^[36]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°.^[36] The trigonometric functions were later studied by mathematicians includingOmar Khayyám,Bhāskara II,Nasir al-Din al-Tusi,Jamshīd al-Kāshī (14th century),Ulugh Beg (14th century),Regiomontanus (1464),Rheticus, and Rheticus' studentValentinus Otho.

Madhava of Sangamagrama (c. 1400) made early strides in theanalysis of trigonometric functions in terms ofinfinite series.^[37] (SeeMadhava series andMadhava's sine table.)

The tangent function was brought to Europe byGiovanni Bianchini in 1467 in trigonometry tables he created to support the calculation of stellar coordinates.^[38]

The termstangent andsecant were first introduced by the Danish mathematicianThomas Fincke in his bookGeometria rotundi (1583).^[39]

The 17th century French mathematicianAlbert Girard made the first published use of the abbreviationssin,cos, andtan in his bookTrigonométrie.^[40]

In a paper published in 1682,Gottfried Leibniz proved thatsinx is not analgebraic function ofx.^[41] Though defined as ratios of sides of aright triangle, and thus appearing to berational functions, Leibnitz result established that they are actuallytranscendental functions of their argument. The task of assimilating circular functions into algebraic expressions was accomplished by Euler in hisIntroduction to the Analysis of the Infinite (1748). His method was to show that the sine and cosine functions arealternating series formed from the even and odd terms respectively of theexponential series. He presented "Euler's formula", as well as near-modern abbreviations (sin.,cos.,tang.,cot.,sec., andcosec.).^[33]

A few functions were common historically, but are now seldom used, such as thechord,versine (which appeared in the earliest tables^[33]),haversine,coversine,^[42] half-tangent (tangent of half an angle), andexsecant.List of trigonometric identities shows more relations between these functions.

{\begin{aligned}\operatorname {crd} \theta &=2\sin {\tfrac {1}{2}}\theta ,\\[5mu]\operatorname {vers} \theta &=1-\cos \theta =2\sin ^{2}{\tfrac {1}{2}}\theta ,\\[5mu]\operatorname {hav} \theta &={\tfrac {1}{2}}\operatorname {vers} \theta =\sin ^{2}{\tfrac {1}{2}}\theta ,\\[5mu]\operatorname {covers} \theta &=1-\sin \theta =\operatorname {vers} {\bigl (}{\tfrac {1}{2}}\pi -\theta {\bigr )},\\[5mu]\operatorname {exsec} \theta &=\sec \theta -1.\end{aligned}}

Historically, trigonometric functions were often combined withlogarithms in compound functions like the logarithmic sine, logarithmic cosine, logarithmic secant, logarithmic cosecant, logarithmic tangent and logarithmic cotangent.^[43]^[44]^[45]^[46]

Etymology

[edit]

Main article:History of trigonometry § Etymology

The wordsine derives^[47] fromLatinsinus, meaning "bend; bay", and more specifically "the hanging fold of the upper part of atoga", "the bosom of a garment", which was chosen as the translation of what was interpreted as the Arabic wordjaib, meaning "pocket" or "fold" in the twelfth-century translations of works byAl-Battani andal-Khwārizmī intoMedieval Latin.^[48]The choice was based on a misreading of the Arabic written formj-y-b (جيب), which itself originated as atransliteration from Sanskritjīvā, which along with its synonymjyā (the standard Sanskrit term for the sine) translates to "bowstring", being in turn adopted fromAncient Greekχορδή "string".^[49]

The wordtangent comes from Latintangens meaning "touching", since the linetouches the circle of unit radius, whereassecant stems from Latinsecans—"cutting"—since the linecuts the circle.^[50]

The prefix "co-" (in "cosine", "cotangent", "cosecant") is found inEdmund Gunter'sCanon triangulorum (1620), which defines thecosinus as an abbreviation of thesinus complementi (sine of thecomplementary angle) and proceeds to define thecotangens similarly.^[51]^[52]

Notes

[edit]

^Klein, Felix (1924) [1902]."Die goniometrischen Funktionen".Elementarmathematik vom höheren Standpunkt aus: Arithmetik, Algebra, Analysis (in German). Vol. 1 (3rd ed.). Berlin: J. Springer.Ch. 3.2, p. 175 ff. Translated as"The Goniometric Functions".Elementary Mathematics from an Advanced Standpoint: Arithmetic, Algebra, Analysis. Translated by Hedrick, E. R.; Noble, C. A. Macmillan. 1932. Ch. 3.2, p. 162 ff.
^Protter & Morrey (1970, pp. APP-2, APP-3)
^"Sine, Cosine, Tangent".www.mathsisfun.com. Retrieved2020-08-29.
^Protter & Morrey (1970, p. APP-7)
^^a ^bRudin, Walter, 1921–2010.Principles of mathematical analysis (Third ed.). New York.ISBN 0-07-054235-X.OCLC 1502474.{{cite book}}: CS1 maint: multiple names: authors list (link) CS1 maint: numeric names: authors list (link)
^Diamond, Harvey (2014)."Defining Exponential and Trigonometric Functions Using Differential Equations".Mathematics Magazine.87 (1):37–42.doi:10.4169/math.mag.87.1.37.ISSN 0025-570X.S2CID 126217060.
^Spivak, Michael (1967). "15".Calculus. Addison-Wesley. pp. 256–257.LCCN 67-20770.
^Sloane, N. J. A. (ed.)."Sequence A072097 (Decimal expansion of 180/Pi)".TheOn-Line Encyclopedia of Integer Sequences. OEIS Foundation.
^Sloane, N. J. A. (ed.)."Sequence A019692 (Decimal expansion of 2*Pi)".TheOn-Line Encyclopedia of Integer Sequences. OEIS Foundation.
^Sloane, N. J. A. (ed.)."Sequence A019685 (Decimal expansion of Pi/180)".TheOn-Line Encyclopedia of Integer Sequences. OEIS Foundation.
^Stueben, Michael; Sandford, Diane (1998).Twenty years before the blackboard: the lessons and humor of a mathematics teacher. Spectrum series. Washington, DC: Mathematical Association of America. p. 119.ISBN 978-0-88385-525-6.
^Bityutskov, V.I. (2011-02-07)."Trigonometric Functions".Encyclopedia of Mathematics.Archived from the original on 2017-12-29. Retrieved2017-12-29.
^Larson, Ron (2013).Trigonometry (9th ed.). Cengage Learning. p. 153.ISBN 978-1-285-60718-4.Archived from the original on 2018-02-15.Extract of page 153 Archived 15 February 2018 at theWayback Machine
^^a ^b ^c ^d ^e ^fHardy, G.H. (1950),A course of pure mathematics (8th ed.), pp. 432–438
^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.
^Bartle, R. G., & Sherbert, D. R. (2000). Introduction to real analysis (3rd ed). Wiley.
^Bartle & Sherbert 1999, p. 247.
^Whitaker and Watson, p 584
^Stanley, Enumerative Combinatorics, Vol I., p. 149
^Abramowitz; Weisstein.
^Lambert, Johann Heinrich (2004) [1768], "Mémoire sur quelques propriétés remarquables des quantités transcendantes circulaires et logarithmiques", in Berggren, Lennart;Borwein, Jonathan M.;Borwein, Peter B. (eds.),Pi, a source book (3rd ed.), New York:Springer-Verlag, pp. 129–140,ISBN 0-387-20571-3
^Aigner, Martin;Ziegler, Günter M. (2000).Proofs from THE BOOK (Second ed.).Springer-Verlag. p. 149.ISBN 978-3-642-00855-9.Archived from the original on 2014-03-08.
^Remmert, Reinhold (1991).Theory of complex functions. Springer. p. 327.ISBN 978-0-387-97195-7.Archived from the original on 2015-03-20.Extract of page 327 Archived 20 March 2015 at theWayback Machine
^Whittaker and Watson, p 137
^Ahlfors, p 197
^Bourbaki, Nicolas (1981).Topologie generale. Springer. §VIII.2.
^Bartle (1964),Elements of real analysis, pp. 315–316
^Weierstrass, Karl (1841)."Darstellung einer analytischen Function einer complexen Veränderlichen, deren absoluter Betrag zwischen zwei gegebenen Grenzen liegt" [Representation of an analytical function of a complex variable, whose absolute value lies between two given limits].Mathematische Werke (in German). Vol. 1. Berlin: Mayer & Müller (published 1894). pp. 51–66.
^Kannappan, Palaniappan (2009).Functional Equations and Inequalities with Applications. 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.
^Farlow, Stanley J. (1993).Partial differential equations for scientists and engineers (Reprint of Wiley 1982 ed.). Courier Dover Publications. p. 82.ISBN 978-0-486-67620-3.Archived from the original on 2015-03-20.
^See for example,Folland, Gerald B. (2009)."Convergence and completeness".Fourier Analysis and its Applications (Reprint of Wadsworth & Brooks/Cole 1992 ed.). American Mathematical Society. pp. 77ff.ISBN 978-0-8218-4790-9.Archived from the original on 2015-03-19.
^^a ^b ^cBoyer, Carl B. (1991). A History of Mathematics (Second ed.). John Wiley & Sons, Inc.ISBN 0-471-54397-7, p. 210.
^Gingerich, Owen (1986)."Islamic Astronomy".Scientific American. Vol. 254. p. 74. Archived fromthe original on 2013-10-19. Retrieved2010-07-13.
^Jacques Sesiano, "Islamic mathematics", p. 157, inSelin, Helaine;D'Ambrosio, Ubiratan, eds. (2000).Mathematics Across Cultures: The History of Non-western Mathematics.Springer Science+Business Media.ISBN 978-1-4020-0260-1.
^^a ^b"trigonometry". Encyclopedia Britannica. 2023-11-17.
^O'Connor, J. J.; Robertson, E. F."Madhava of Sangamagrama". School of Mathematics and Statistics University of St Andrews, Scotland. Archived fromthe original on 2006-05-14. Retrieved2007-09-08.
^Van Brummelen, Glen (2018)."The end of an error: Bianchini, Regiomontanus, and the tabulation of stellar coordinates".Archive for History of Exact Sciences.72 (5):547–563.doi:10.1007/s00407-018-0214-2.JSTOR 45211959.S2CID 240294796.
^"Fincke biography".Archived from the original on 2017-01-07. Retrieved2017-03-15.
^O'Connor, John J.;Robertson, Edmund F.,"Trigonometric functions",MacTutor History of Mathematics Archive,University of St Andrews
^Bourbaki, Nicolás (1994).Elements of the History of Mathematics. Springer.ISBN 9783540647676.
^Nielsen (1966, pp. xxiii–xxiv)
^von Hammer, Ernst Hermann Heinrich[in German], ed. (1897).Lehrbuch der ebenen und sphärischen Trigonometrie. Zum Gebrauch bei Selbstunterricht und in Schulen, besonders als Vorbereitung auf Geodäsie und sphärische Astronomie (in German) (2 ed.). Stuttgart, Germany:J. B. Metzlerscher Verlag. Retrieved2024-02-06.
^Heß, Adolf (1926) [1916].Trigonometrie für Maschinenbauer und Elektrotechniker - Ein Lehr- und Aufgabenbuch für den Unterricht und zum Selbststudium (in German) (6 ed.). Winterthur, Switzerland: Springer.doi:10.1007/978-3-662-36585-4.ISBN 978-3-662-35755-2.
^Lötzbeyer, Philipp (1950)."§ 14. Erläuterungen u. Beispiele zu T. 13: lg sin X; lg cos X und T. 14: lg tg x; lg ctg X".Erläuterungen und Beispiele für den Gebrauch der vierstelligen Tafeln zum praktischen Rechnen (in German) (1 ed.). Berlin, Germany:Walter de Gruyter & Co.doi:10.1515/9783111507545.ISBN 978-3-11114038-4. Archive ID 541650. Retrieved2024-02-06.
^Roegel, Denis, ed. (2016-08-30).A reconstruction of Peters's table of 7-place logarithms (volume 2, 1940). Vandoeuvre-lès-Nancy, France:Université de Lorraine. hal-01357842.Archived from the original on 2024-02-06. Retrieved2024-02-06.
^The anglicized form is first recorded in 1593 inThomas Fale'sHorologiographia, the Art of Dialling.
^
Various sources credit the first use ofsinus to either
- Plato Tiburtinus's 1116 translation of theAstronomy ofAl-Battani
- Gerard of Cremona's translation of theAlgebra ofal-Khwārizmī
- Robert of Chester's 1145 translation of the tables of al-Khwārizmī
See Merlet,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.
SeeKatx, Victor (July 2008).A history of mathematics (3rd ed.). Boston:Pearson. p. 210 (sidebar).ISBN 978-0321387004.
^See Plofker,Mathematics in India, Princeton University Press, 2009, p. 257
See"Clark University".Archived from the original on 2008-06-15.
See Maor (1998), chapter 3, regarding the etymology.
^Oxford English Dictionary
^Gunter, Edmund (1620).Canon triangulorum.
^Roegel, Denis, ed. (2010-12-06)."A reconstruction of Gunter's Canon triangulorum (1620)" (Research report). HAL. inria-00543938.Archived from the original on 2017-07-28. Retrieved2017-07-28.

References

[edit]

Abramowitz, Milton;Stegun, Irene Ann, eds. (1983) [June 1964].Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Applied Mathematics Series. Vol. 55 (Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.). Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications.ISBN 978-0-486-61272-0.LCCN 64-60036.MR 0167642.LCCN 65-12253.
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.
Bartle, Robert G.; Sherbert, Donald R. (1999).Introduction to Real Analysis (3rd ed.). Wiley.ISBN 9780471321484.
Boyer, Carl B.,A History of Mathematics, John Wiley & Sons, Inc., 2nd edition. (1991).ISBN 0-471-54397-7.
Cajori, Florian (1929)."§2.2.1. Trigonometric Notations".A History of Mathematical Notations. Vol. 2. Open Court. pp. 142–179 (¶511–537).
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. Computers45 (3), 328–339 (1996).
Maor, Eli,Trigonometric Delights, Princeton Univ. Press. (1998). Reprint edition (2002):ISBN 0-691-09541-8.
Needham, Tristan,"Preface"" toVisual Complex Analysis. Oxford University Press, (1999).ISBN 0-19-853446-9.
Nielsen, Kaj L. (1966),Logarithmic and Trigonometric Tables to Five Places (2nd ed.), New York:Barnes & Noble,LCCN 61-9103
O'Connor, J. J., and E. F. Robertson,"Trigonometric functions",MacTutor History of Mathematics archive. (1996).
O'Connor, J. J., and E. F. Robertson,"Madhava of Sangamagramma",MacTutor History of Mathematics archive. (2000).
Pearce, Ian G.,"Madhava of Sangamagramma"Archived 2006-05-05 at theWayback Machine,MacTutor History of Mathematics archive. (2002).
Protter, Murray H.; Morrey, Charles B. Jr. (1970),College Calculus with Analytic Geometry (2nd ed.), Reading:Addison-Wesley,LCCN 76087042
Weisstein, Eric W.,"Tangent" fromMathWorld, accessed 21 January 2006.