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In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. They are commonly defined as ratios of two sides of a right triangle containing the angle, and can equivalently be defined as the lengths of various line segments from a unit circle. More modern definitions express them as infinite series or as solutions of certain differential equations, allowing their extension to positive and negative values and even to complex numbers. All of these approaches will be presented below. Mathematics is often defined as the study of topics such as quantity, structure, space, and change. ... Partial plot of a function f. ... This article is about angles in geometry. ... For alternate meanings, such as the musical instrument, see triangle (disambiguation). ... In algebra, a ratio is the relationship between two quantities. ... Illustration of a unit circle. ... In mathematics, a series is a sum of a sequence of terms. ... In mathematics, a differential equation is an equation in which the derivatives of a function appear as variables. ... Wikibooks Algebra has more about this subject: Complex numbers In mathematics, a complex number is an expression of the form a + bi, where a and b are real numbers, and i stands for the square root of negative one (−1), which cannot be represented by any real number. ...


In modern usage, there are six basic trigonometric functions, which are tabulated below along with equations relating them to one another. (Especially in the case of the last four, these relations are often taken as the definitions of those functions, but one can equally define them geometrically or by other means and derive the relations.)

Function Abbreviation Relation
Sine sin sin theta = cos left(frac{pi}{2} - theta right) ,
Cosine cos cos theta = sin left(frac{pi}{2} - theta right),
Tangent tan tan theta = frac{1}{cot theta} = frac{sin theta}{cos theta} = cot left(frac{pi}{2} - theta right) ,
Cotangent cot cot theta = frac{1}{tan theta} = frac{cos theta}{sin theta} = tan left(frac{pi}{2} - theta right) ,
Secant sec sec theta = frac{1}{cos theta} = csc left(frac{pi}{2} - theta right) ,
Cosecant csc
(or cosec)
csc theta =frac{1}{sin theta} = sec left(frac{pi}{2} - theta right) ,

A few other functions were common historically (and appeared in the earliest tables), but are now little-used, such as:

Many more relations between these functions are listed in the article about trigonometric identities. The versed sine, also called the versine and, in Latin, the sinus versus (flipped sine) or the sagitta (arrow), is a trigonometric function versin(θ) (sometimes further abbreviated vers) defined by the equation: versin(θ) = 1 − cos(θ) = 2 sin2(θ / 2) There are also three corresponding functions: the coversed... The trigonometric functions, including the exsecant, can be constructed geometrically in terms of a unit circle centered at O. The exsecant is the portion DE of the secant exterior to (ex) the circle. ... In mathematics, trigonometric identities are equations involving trigonometric functions that are true for all values of the occurring variables. ...

Contents


History

The earliest systematic study of trigonometric functions and tabulation of their values was performed by Hipparchus of Nicaea (180-125 BC), who tabulated the lengths of circle arcs (angle A times radius r) with the lengths of the subtending chords (2r sin(A/2)). Later, Ptolemy (2nd century) expanded upon this work in his Almagest, deriving addition/subtraction formulas for the equivalent of sin(A + B) and cos(A + B). Ptolemy also derived the equivalent of the half-angle formula sin2(A/2) = (1 − cos(A))/2, allowing him to create tables with any desired accuracy. Neither the tables of Hipparchus nor of Ptolemy have survived to the present day. Hipparchus (Greek Ἳππαρχος) (ca. ... Iznik (formerly Nicaea) is a city in Anatolia (now part of Turkey) which is known primarily as the site of two major meetings (or Ecumenical councils) in the early history of the Christian church. ... Centuries: 3rd century BC - 2nd century BC - 1st century BC Decades: 230s BC 220s BC 210s BC 200s BC 190s BC - 180s BC - 150s BC 140s BC 130s BC 120s BC 110s BC Years: 185 BC 184 BC 183 BC 182 BC 181 BC - 180 BC - 179 BC 178 BC... Centuries: 3rd century BC - 2nd century BC - 1st century BC Decades: 170s BC 160s BC 150s BC 140s BC 130s BC - 120s BC - 110s BC 100s BC 90s BC 80s BC 70s BC Years: 130 BC 129 BC 128 BC 127 BC 126 BC - 125 BC - 124 BC 123 BC... Claudius Ptolemaeus, given contemporary German styling, in a 16th century engraved book frontispiece. ... // Events Roman Empire governed by the Five Good Emperors (96–180) – Nerva, Trajan, Hadrian, Antoninus Pius, Marcus Aurelius. ... Almagest is the Latin form of the Arabic name (al-kitabu-l-mijisti, i. ...


The next significant development of trigonometry was in India, in the works known as the Siddhantas (4th5th century), which first defined the sine as the modern relationship between half an angle and half a chord. The Siddhantas also contained the earliest surviving tables of sine values (along with 1 − cos values), in 3.75-degree intervals from 0 to 90 degrees. As a means of recording the passage of time, the 4th century was that century which lasted from 301 to 400. ... // Overview Events Romulus Augustus, Last Western Roman Emperor 410: Rome sacked by Visigoths 452: Pope Leo I allegedly meets personally with Attila the Hun and convinces him not to sack Rome 439: Vandals conquer Carthage At some point after 440, the Anglo-Saxons settle in Britain. ...


The Hindu works were later translated and expanded by the Arabs, who by the 10th century (in the work of Abu'l-Wefa) were using all six trigonometric functions, and had sine tables in 0.25-degree increments, to 8 decimal places of accuracy, as well as tables of tangent values. A Hindu (archaic Hindoo) is an adherent of philosophies and scriptures of Hinduism, the predominant religious, philosophical and cultural system of India (Bharat), Nepal, and the island of Bali. ... The Arabs (Arabic: عرب ʻarab) are a large and heterogeneous ethnic group found throughout the Middle East and North Africa, originating in the Arabian Peninsula of southwest Asia. ... As a means of recording the passage of time, the 10th century was that century which lasted from 901 to 1000. ...


Our modern word sine comes, via sinus ("bay" or "fold") in Latin, from a mistranslation of the Sanskrit jiva (or jya). jiva (originally called ardha-jiva, "half-chord", in the 6th century Aryabhata) was transliterated by the Arabs as jiba (جب), but was confused for another word, jaib (جب) ("bay"), by European translators such as Robert of Chester and Gherardo of Cremona in Toledo in the 12th century, probably because jiba (جب) and jaib (جب) are written the same in Arabic (many vowels are excluded from words written in the Arabic alphabet). Latin is an ancient Indo-European language originally spoken in the region around Rome called Latium. ... Sanskrit ( संस्कृतम्) is an Indo-European classical language of India and a liturgical language of Hinduism, Buddhism, and Jainism. ... This Buddhist stela from China, Northern Wei period, was built in the early 6th century. ... Aryabhata (आर्यभट) Ä€ryabhaá¹­a) (476 - 550) is the first of the great astronomers of the classical age of India. ... This article is about the city in Spain named Toledo. ... (11th century - 12th century - 13th century - other centuries) As a means of recording the passage of time, the 12th century was that century which lasted from 1101 to 1200. ... The Arabic alphabet is the script used for writing in the Arabic language. ...


All of these earlier works on trigonometry treated it mainly as an adjunct to astronomy; perhaps the first treatment as a subject in its own right was by the De triangulis omnimodus (1464) of Regiomontanus (14361476), as well as his later Tabulae directionum (which included the tangent function, unnamed). Events February - Christian I of Denmark and Norway who was also serving as King of Sweden is declared deposed from the later throne. ... Johannes Müller von Königsberg (June 6, 1436 – July 6, 1476), known by his Latin pseudonym Regiomontanus, was an important German mathematician, astronomer and astrologer. ... Events April - Paris is recaptured by the French End of the Hussite Wars in Bohemia. ... Events March 2 - Battle of Grandson. ...


The Opus palatinum de triangulis of Rheticus, a student of Copernicus, was the first to define trigonometric functions directly in terms of right triangles instead of circles, with tables for all six trigonometric functions; this work was finished by Rheticus' student Valentin Otho in 1596. Georg Joachim von Lauchen Rheticus was born in 1514 at Feldkirch, Austria and died in 1574 at Kosice, Hungary. ... Nicolaus Copernicus (in Latin; Polish Mikołaj Kopernik, German Nikolaus Kopernikus - February 19, 1473 – May 24, 1543) was a Polish astronomer, mathematician and economist who developed a heliocentric (Sun-centered) theory of the solar system in a form detailed enough to make it scientifically useful. ... Events February 5 - 26 catholics crucified in Nagasaki, Japan. ...


The Introductio in analysin infinitorum (1748) of Euler was primarily responsible for establishing the analytic treatment of trigonometric functions, defining them as infinite series and presenting "Euler's formula" eix = cos(x) + i sin(x). Euler used the near-modern abbreviations sin., cos., tang., cot., sec., and cosec.. Events April 24 - A congress assembles at Aix-la-Chapelle with the intent to conclude the struggle known as the War of Austrian Succession - at October 18 - The Treaty of Aix-la-Chapelle is signed to end the war Adam Smith begins to deliver public lectures in Edinburgh Building of... Leonhard Euler aged 49 (oil painting by Emanuel Handmann, 1756) Leonhard Euler (April 15, 1707 - September 18, 1783) (pronounced oiler) was a Swiss mathematician and physicist. ... Eulers formula, named after Leonhard Euler (pronounced oiler), is a mathematical formula in the subfield of complex analysis that shows a deep relationship between the trigonometric functions and the complex exponential function. ...


Right triangle definitions

A right triangle always includes a 90° (π/2 radians) angle, here labeled C. Angles A and B may vary. Trigonometric functions specify the relationships between side lengths and interior angles of a right triangle.
A right triangle always includes a 90° (π/2 radians) angle, here labeled C. Angles A and B may vary. Trigonometric functions specify the relationships between side lengths and interior angles of a right triangle.

In order to define the trigonometric functions for the angle A, start with an arbitrary right triangle that contains the angle A: Triangle with basic trigonometry labels. ... Triangle with basic trigonometry labels. ... For alternate meanings, such as the musical instrument, see triangle (disambiguation). ... For alternate meanings, such as the musical instrument, see triangle (disambiguation). ...


We use the following names for the sides of the triangle:

  • The hypotenuse is the side opposite the right angle, or defined as the longest side of a right-angled triangle, in this case h.
  • The opposite side is the side opposite to the angle we are interested in, in this case a.
  • The adjacent side is the side that is in contact with the angle we are interested in and the right angle, hence its name. In this case the adjacent side is b.

All triangles are taken to exist in the Euclidean plane so that the inside angles of each triangle sum to π radians (or 180°); therefore, for a right triangle the two non-right angles are between zero and π/2 radians. The reader should note that the following definitions, strictly speaking, only define the trigonometric functions for angles in this range. We extend them to the full set of real arguments by requiring that they be periodic functions. For alternate meanings, such as the musical instrument, see triangle (disambiguation). ... In mathematics and astronomy, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ... The radian (symbol: rad) is the SI unit of plane angle. ... A degree (in full, a degree of arc, arc degree, or arcdegree), usually symbolized °, is a measurement of plane angle, representing 1/360 of a full rotation. ... In mathematics, a periodic function is a function that repeats its values after some definite period has been added to its independent variable. ...


1) The sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse. In our case

sin A = frac {textrm{opposite}} {textrm{hypotenuse}} = frac {a} {h}.

Note that this ratio does not depend on the particular right triangle chosen, as long as it contains the angle A, since all those triangles are similar. Several equivalence relations in mathematics are called similarity. ...


The set of zeroes of sine is left{npibig| nisinmathbb{Z}right}.


2) The cosine of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse. In our case

cos A = frac {textrm{adjacent}} {textrm{hypotenuse}} = frac {b} {h}.

The set of zeroes of cosine is left{left(frac{2n+1}{2}right)pibigg| nisinmathbb{Z}right}.


3) The tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side. In our case

tan A = frac {textrm{opposite}} {textrm{adjacent}} = frac {a} {b}.

The set of zeroes of tangent is left{npibig| nisinmathbb{Z}right}.


The remaining three functions are best defined using the above three functions.


4) The cosecant csc(A) is the multiplicative inverse of sin(A), i.e. the ratio of the length of the hypotenuse to the length of the opposite side: In mathematics, the reciprocal, or multiplicative inverse, of a number x is the number which, when multiplied by x, yields 1. ...

csc A = frac {textrm{hypotenuse}} {textrm{opposite}} = frac {h} {a}.

5) The secant sec(A) is the multiplicative inverse of cos(A), i.e. the ratio of the length of the hypotenuse to the length of the adjacent side:

sec A = frac {textrm{hypotenuse}} {textrm{adjacent}} = frac {h} {b}.

6) The cotangent cot(A) is the multiplicative inverse of tan(A), i.e. the ratio of the length of the adjacent side to the length of the opposite side:

cot A = frac {textrm{adjacent}} {textrm{opposite}} = frac {b} {a}.

Mnemonics

There are a number of mnemonics for the above definitions, for example SOHCAHTOA (sounds like "soak a toe-a" or "sock-a toe-a" depending upon which side of the Atlantic you hail from. Can also be read as "soccer tour"). It means: A mnemonic (pronounced in American English, in British English) is a memory aid. ...

  • SOH ... sin = opposite/hypotenuse
  • CAH ... cos = adjacent/hypotenuse
  • TOA ... tan = opposite/adjacent.

Many other such words and phrases have been contrived. For more see: trigonometry mnemonics. A number of mnemonics have been invented by educators to help students remember the rules defining the various trigonometric functions. ...


Slope definitions

Equivalent to the right-triangle definitions, the trigonometric functions can be defined in terms of the rise, run, and slope of a line segment relative to some horizontal line. The slope is commonly taught as "rise over run" or rise/run. The three main trigonometric functions are commonly taught in the order sine, cosine, tangent. With a unit circle, this gives rise to the following matchings: Look up Slope in Wiktionary, the free dictionary In mathematics, the slope or the gradient of a straight line (within a Cartesian coordinate system) is a measure for the steepness of the line relative to the horizontal axis. ... Illustration of a unit circle. ...

  1. Sine is first, rise is first. Sine takes an angle and tells the rise.
  2. Cosine is second, run is second. Cosine takes an angle and tells the run.
  3. Tangent is the slope formula that combines the rise and run. Tangent takes an angle and tells the slope.

This shows the main use of tangent and arctangent, which is converting between the two ways of telling how slanted a line is: angles and slopes.


While the radius of the circle makes no difference for the slope (the slope doesn't depend on the length of the slanted line), it does affect rise and run. To adjust and find the actual rise and run, just multiply the sine and cosine by the radius. For instance, if the circle has radius 5, the run at an angle of 1 is 5 cos(1).


Unit-circle definitions

The six trigonometric functions can also be defined in terms of the unit circle, the circle of radius one centered at the origin. The unit circle definition provides little in the way of practical calculation; indeed it relies on right triangles for most angles. The unit circle definition does, however, permit the definition of the trig functions for all positive and negative arguments, not just for angles between 0 and π/2 radians. It also provides a single visual picture that encapsulates at once all the important triangles used so far. The equation for the unit circle is: Image File history File links Download high resolution version (1050x1050, 98 KB) // Summary Angles and values on the Unit circle. ... Image File history File links Download high resolution version (1050x1050, 98 KB) // Summary Angles and values on the Unit circle. ... Illustration of a unit circle. ... Illustration of a unit circle. ... A circle, in Euclidean geometry, is the set of all points at a fixed distance, called the radius, from a fixed point, the centre. ...

x^2 + y^2 = 1 ,

In the picture, some common angles, measured in radians, are given. Measurements in the counter clockwise direction are positive angles and measurements in the clockwise direction are negative angles. Let a line making an angle of θ with the positive half of the x-axis intersect the unit circle. The x- and y-coordinates of this point of intersection are equal to cos θ and sin θ, respectively. The triangle in the graphic enforces the formula; the radius is equal to the hypotenuse and has length 1, so we have sin θ = y/1 and cos θ = x/1. The unit circle can be thought of as a way of looking at an infinite number of triangles by varying the lengths of their legs but keeping the lengths of their hypotenuses equal to 1.

The f(x) = sin(x) and f(x) = cos(x) functions graphed on the cartesian plane.
The f(x) = sin(x) and f(x) = cos(x) functions graphed on the cartesian plane.

For angles greater than 2π or less than −2π, simply continue to rotate around the circle. In this way, sine and cosine become periodic functions with period 2π: A Plot I made with Gnuplot and manually improved the labeling on. ... A Plot I made with Gnuplot and manually improved the labeling on. ... In mathematics, a periodic function is a function that repeats its values after some definite period has been added to its independent variable. ...

sintheta = sinleft(theta + 2pi k right)
costheta = cosleft(theta + 2pi k right)

for any angle θ and any integer k. The integers consist of the positive natural numbers (1, 2, 3, …), their negatives (−1, −2, −3, ...) and the number zero. ...


The smallest positive period of a periodic function is called the primitive period of the function. The primitive period of the sine, cosine, secant, or cosecant is a full circle, i.e. 2π radians or 360 degrees; the primitive period of the tangent or cotangent is only a half-circle, i.e. π radians or 180 degrees.


Above, only sine and cosine were defined directly by the unit circle, but the other four trig functions can be defined by:

tantheta = frac{sintheta}{costheta} quad sectheta = frac{1}{costheta}
csctheta = frac{1}{sintheta} quad cottheta = frac{costheta}{sintheta}
All of the trigonometric functions can be constructed geometrically in terms of a unit circle centered at O.
Enlarge
All of the trigonometric functions can be constructed geometrically in terms of a unit circle centered at O.

Alternatively, all of the basic trigonometric functions can be defined in terms of a unit circle centered at O (shown at right), and similar such geometric definitions were used historically. In particular, for a chord AB of the circle, where θ is half of the subtended angle, sin(θ) is AC (half of the chord), a definition introduced in India (see below). cos(θ) is the horizontal distance OC, and versin(θ) = 1 − cos(θ) is CD. tan(θ) is the length of the segment AE of the tangent line through A, hence the word tangent for this function. cot(θ) is another tangent segment, AF. sec(θ) = OE and csc(θ) = OF are segments of secant lines (intersecting the circle at two points), and can also be viewed as projections of OA along the tangent at A to the horizontal and vertical axes, respectively. DE is exsec(θ) = sec(θ) − 1 (the portion of the secant outside, or ex, the circle). From these constructions, it is easy to see that the secant and tangent functions diverge as θ approaches π/2 (90 degrees) and that the cosecant and cotangent diverge as θ approaches zero. (Many similar constructions are possible, and the basic trigonometric identities can also be proven graphically.) Define several trig functions from unit circle. ... Define several trig functions from unit circle. ... The versed sine, also called the versine and, in Latin, the sinus versus (flipped sine) or the sagitta (arrow), is a trigonometric function versin(θ) (sometimes further abbreviated vers) defined by the equation: versin(θ) = 1 − cos(θ) = 2 sin2(θ / 2) There are also three corresponding functions: the coversed... In mathematics, the word tangent has two distinct, but etymologically related meanings: one in geometry, and one in trigonometry. ... It has been suggested that Chord (geometry) be merged into this article or section. ... The trigonometric functions, including the exsecant, can be constructed geometrically in terms of a unit circle centered at O. The exsecant is the portion DE of the secant exterior to (ex) the circle. ...


Series definitions

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

Please note: Here, and generally in calculus, all angles are measured in radians. (See also below). The sine function (blue) along with its taylors polyn of degree 7 (pink) File links The following pages link to this file: Mathematics Talk:Mathematics Trigonometric function Wikipedia:List of images Wikipedia:List of images/Mathematics Categories: GFDL images ... The sine function (blue) along with its taylors polyn of degree 7 (pink) File links The following pages link to this file: Mathematics Talk:Mathematics Trigonometric function Wikipedia:List of images Wikipedia:List of images/Mathematics Categories: GFDL images ... Integral and differential calculus is a central branch of mathematics, developed from algebra and geometry. ... The radian (symbol: rad) is the SI unit of plane angle. ...


Using only geometry and properties of limits, it can be shown that the derivative of sine is cosine and the derivative of cosine is the negative of sine. One can then use the theory of Taylor series to show that the following identities hold for all real numbers x: In mathematics, the limit of a function is a fundamental concept in mathematical analysis. ... In mathematics, the derivative is one of the two central concepts of calculus. ... As the degree of the Taylor series rises, it approaches the correct function. ... In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ...

sin x = x - frac{x^3}{3!} + frac{x^5}{5!} - frac{x^7}{7!} + cdots = sum_{n=0}^infty frac{(-1)^nx^{2n+1}}{(2n+1)!}
cos x = 1 - frac{x^2}{2!} + frac{x^4}{4!} - frac{x^6}{6!} + cdots = sum_{n=0}^infty frac{(-1)^nx^{2n}}{(2n)!}

These identities are often taken as the definitions of the sine and cosine function. They are often used as the starting point in a rigorous treatment of trigonometric functions and their applications (e.g. in Fourier series), since the theory of infinite series can be developed from the foundations of the real number system, independent of any geometric considerations. The differentiability and continuity of these functions are then established from the series definitions alone. The Fourier series, named in honor of Joseph Fourier (1768-1830), is an extremely useful mathematical tool. ... In mathematics, a series is a sum of a sequence of terms. ... In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ... In mathematics, the derivative of a function is one of the two central concepts of calculus. ... In mathematics, a continuous function is a function in which arbitrarily small changes in the input produce arbitrarily small changes in the output. ...


Other series can be found (Abramowitz and Stegun 1964, Weinstein 2006):

tan x , {} = sum_{n=0}^infty frac{U_{2n+1} x^{2n+1}}{(2n+1)!}
{} = sum_{n=1}^infty frac{(-1)^{n-1} 2^{2n} (2^{2n}-1) B_{2n} x^{2n-1}}{(2n)!}
{} = x + frac{x^3}{3} + frac{2 x^5}{15} + frac{17 x^7}{315} + cdots, qquad mbox{for } |x| < frac {pi} {2}
csc x , {} = sum_{n=0}^infty frac{(-1)^{n+1} 2 (2^{2n-1}-1) B_{2n} x^{2n-1}}{(2n)!}
{} = frac {1} {x} + frac {x} {6} + frac {7 x^3} {360} + frac {31 x^5} {15120} + cdots, qquad mbox{for } 0 < |x| < pi
sec x , {} = sum_{n=0}^infty frac{U_{2n} x^{2n}}{(2n)!} = sum_{n=0}^infty frac{(-1)^n E_n x^{2n}}{(2n)!}
{} = 1 + frac {x^2} {2} + frac {5 x^4} {24} + frac {61 x^6} {720} + cdots, qquad mbox{for } |x| < frac {pi} {2}
cot x , {} = sum_{n=0}^infty frac{(-1)^n 2^{2n} B_{2n} x^{2n-1}}{(2n)!}
{} = frac {1} {x} - frac {x}{3} - frac {x^3} {45} - frac {2 x^5} {945} - cdots, qquad mbox{for } 0 < |x| < pi

where

B_n , is the nth Bernoulli number,
E_n , is the nth Euler number, and
U_n , is the nth up/down number.

In mathematics, the Bernoulli numbers Bn were first discovered in connection with the closed forms of the sums for various fixed values of n. ... The Euler numbers are a sequence En of integers defined by the following Taylor series expansion: (Note that e, the base of the natural logarithm, is also occasionally called Eulers number, as is the Euler characteristic. ... In mathematics, the boustrophedon transform is a procedure which maps one sequence to another. ...

Relationship to exponential function

It can be shown from the series definitions that the sine and cosine functions are the imaginary and real parts, respectively, of the complex exponential function when its argument is purely imaginary: Wikibooks Algebra has more about this subject: Complex numbers In mathematics, a complex number is an expression of the form a + bi, where a and b are real numbers, and i stands for the square root of negative one (−1), which cannot be represented by any real number. ... The exponential function is one of the most important functions in mathematics. ...

e^{i theta} = costheta + isintheta ,.

This relationship was first noted by Euler and the identity is called Euler's formula. In this way, trigonometric functions become essential in the geometric interpretation of complex analysis. For example, with the above identity, if one considers the unit circle in the complex plane, defined by eix, and as above, we can parametrize this circle in terms of cosines and sines, the relationship between the complex exponential and the trigonometric functions becomes more apparent. Leonhard Euler aged 49 (oil painting by Emanuel Handmann, 1756) Leonhard Euler (April 15, 1707 - September 18, 1783) (pronounced oiler) was a Swiss mathematician and physicist. ... This article is about the Eulers formula in complex analysis. ... Complex analysis is the branch of mathematics investigating functions of complex numbers. ... In mathematics, the complex plane is a way of visualising the space of the complex numbers. ...


Furthermore, this allows for the definition of the trigonometric functions for complex arguments z:

sin z , = , sum_{n=0}^{infty}frac{(-1)^{n}}{(2n+1)!}z^{2n+1} , = , {e^{imath z} - e^{-imath z} over 2imath} = -imath sinh left( imath zright)
cos z , = , sum_{n=0}^{infty}frac{(-1)^{n}}{(2n)!}z^{2n} , = , {e^{imath z} + e^{-imath z} over 2} = cosh left(imath zright)

where i2 = −1. Also, for purely real x,

cos x , = , mbox{Re } (e^{imath x})
sin x , = , mbox{Im } (e^{imath x})

It is also shown that exponential processes are intimately linked to periodic behavior.


Definitions via differential equations

Both the sine and cosine functions satisfy the differential equation In mathematics, a differential equation is an equation in which the derivatives of a function appear as variables. ...

y,''=-y

i.e. each is the additive inverse of its own second derivative. Within the 2-dimensional vector space V consisting of all solutions of this equation, the sine function is the unique solution satisfying the initial conditions y(0) = 0 and y′(0) = 1, and the cosine function is the unique solution satisfying the initial conditions y(0) = 1 and y′(0) = 0. Since the sine and cosine functions are linearly independent, together they form a basis of V. This method of defining the sine and cosine functions is essentially equivalent to using Euler's formula. (See linear differential equation.) It turns out that this differential equation can be used not only to define the sine and cosine functions but also to prove the trigonometric identities for the sine and cosine functions. A vector space (or linear space) is the basic object of study in the branch of mathematics called linear algebra. ... In linear algebra, a basis is a minimum set of vectors that, when combined, can address every vector in a given space. ... In mathematics, a linear differential equation is a differential equation Lf = g, where the differential operator L is a linear operator. ... In mathematics, trigonometric identities are equations involving trigonometric functions that are true for all values of the occurring variables. ...


The tangent function is the unique solution of the nonlinear differential equation

y,'=1+y^2

satisfying the initial condition y(0) = 0. There is a very interesting visual proof that the tangent function satisfies this differential equation; see [1].


The significance of radians

Radians specify an angle by measuring the length around the path of the circle and constitute a special argument to the sine and cosine functions. In particular, only those sines and cosines which map radians to ratios satisfy the differential equations which classically describe them. If an argument to sine or cosine in radians is scaled by frequency,

f(x) = sin(kx); k ne 0, k ne 1 ,

then the derivatives will scale by amplitude.

f'(x) = kcos(kx) ,.

Here, k is a constant that represents a mapping between units. If x is in degrees, then

k = xfrac{pi}{180^circ}.

This means that the second derivative of a sine in degrees satisfies not the differential equation

y'' = -y ,,

but

y'' = -k^2y ,;

similarly for cosine.


This means that these sines and cosines are different functions, and that the fourth derivative of sine will be sine again only if the argument is in radians.


Other definitions

Theorem: There exists exactly one pair of real functions s, c with the following properties:


For any x, y inmathbb{R}:

s(x)^2 + c(x)^2 = 1,,
s(x+y) = s(x)c(y) + c(x)s(y),,
c(x+y) = c(x)c(y) - s(x)s(y),,
0 < xc(x) < s(x) < x mathrm{for} 0 < x < 1.

Computation

The computation of trigonometric functions is a complicated subject, which can today be avoided by most people because of the widespread availability of computers and scientific calculators that provide built-in trigonometric functions for any angle. In this section, however, we describe more details of their computation in three important contexts: the historical use of trigonometric tables, the modern techniques used by computers, and a few "important" angles where simple exact values are easily found. (Below, it suffices to consider a small range of angles, say 0 to π/2, since all other angles can be reduced to this range by the periodicity and symmetries of the trigonometric functions.) A computer is a machine capable of undergoing complex calculations. ... A basic arithmetic calculator. ...


Prior to computers, people typically evaluated trigonometric functions by interpolating from a detailed table of their values, calculated to many significant figures. Such tables have been available for as long as trigonometric functions have been described (see History, above), and were typically generated by repeated application of the half-angle and angle-addition identities starting from a known value (such as sin(π/2)=1). See also: Generating trigonometric tables. In the mathematical subfield of numerical analysis, interpolation is a method of constructing new data points from a discrete set of known data points. ... The hypothetical idea of significant figures (sig figs or sf), also called significant digits (sig digs) is a method of expressing error in measurement. ... In mathematics, trigonometric identities are equations involving trigonometric functions that are true for all values of the occurring variables. ... Tables of trigonometric functions are useful in a number of areas. ...


Modern computers use a variety of techniques (Kantabutra, 1996). One common method, especially on higher-end processors with floating point units, is to combine a polynomial approximation (such as a Taylor series or a rational function) with a table lookup — they first look up the closest angle in a small table, and then use the polynomial to compute the correction. On simpler devices that lack hardware multipliers, there is an algorithm called CORDIC (as well as related techniques) that is more efficient, since it uses only shifts and additions. All of these methods are commonly implemented in hardware for performance reasons. A floating-point number is a digital representation for a number in a certain subset of the rational numbers, and is often used to approximate an arbitrary real number on a computer. ... In mathematics, polynomial functions, or polynomials, are an important class of simple and smooth functions. ... As the degree of the Taylor series rises, it approaches the correct function. ... In mathematics, a rational function is a ratio of polynomials. ... ALU redirects here. ... CORDIC (for COordinate Rotation DIgital Computer) is a simple and efficient algorithm to calculate hyperbolic and trigonometric functions. ... In mathematics, and in particular functional analysis, the shift operators are examples of linear operators, important for their simplicity and natural occurrence. ... Computer hardware is the physical parts of a computer, as distinguished from the computer software or computer programs and data that operate within the hardware. ...


Finally, for some simple angles, the values can be easily computed by hand using the Pythagorean theorem, as in the following examples. In fact, the sine, cosine and tangent of any integer multiple of π/60 radians (three degrees) can be found exactly by hand. The Pythagorean theorem: The sum of the areas of the two squares on the legs (blue and red) equals the area of the square on the hypotenuse (purple). ... Exact constant expressions for trigonometric expressions are sometimes useful, mainly for simplifying solutions into radical forms which allow further simplification. ...


Consider a right triangle where the two other angles are equal, and therefore are both π/4 radians (45 degrees). Then the length of side b and the length of side a are equal; we can choose a = b = 1. The values of sine, cosine and tangent of an angle of π/4 radians (45 degrees) can then be found using the Pythagorean theorem: The radian (symbol: rad) is the SI unit of plane angle. ... The radian (symbol: rad) is the SI unit of plane angle. ...

c = sqrt { a^2+b^2 } = sqrt2

Therefore:

sin left(pi / 4 right) = sin left(45^circright) = cos left(pi / 4 right) = cos left(45^circright) = {1 over sqrt2}
tan left(pi / 4 right) = tan left(45^circright) = {sqrt2 over sqrt2} = 1

To determine the trigonometric functions for angles of π/3 radians (60 degrees) and π/6 radians (30 degrees), we start with an equilateral triangle of side length 1. All its angles are π/3 radians (60 degrees). By dividing it into two, we obtain a right triangle with π/6 radians (30 degrees) and π/3 radians (60 degrees) angles. For this triangle, the shortest side = 1/2, the next largest side =(√3)/2 and the hypotenuse = 1. This yields:

sin left(pi / 6 right) = sin left(30^circright) = cos left(pi / 3 right) = cos left(60^circright) = {1 over 2}
cos left(pi / 6 right) = cos left(30^circright) = sin left(pi / 3 right) = sin left(60^circright) = {sqrt3 over 2}
tan left(pi / 6 right) = tan left(30^circright) = cot left(pi / 3 right) = cot left(60^circright) = {1 over sqrt3}

See also: Exact trigonometric constants Exact constant expressions for trigonometric expressions are sometimes useful, mainly for simplifying solutions into radical forms which allow further simplification. ...


Inverse functions

The trigonometric functions are periodic, so we must restrict their domains before we are able to define a unique inverse. In the following, the functions on the left are defined by the equation on the right; these are not proved identities. The principal inverses are usually defined as:

begin{matrix} mbox{for} & -frac{pi}{2} le y le frac{pi}{2}, & y = arcsin(x) & mbox{if and only if} & x = sin(y)  mbox{for} & 0 le y le pi, & y = arccos(x) & mbox{if and only if} & x = cos(y)  mbox{for} & -frac{pi}{2} < y < frac{pi}{2}, & y = arctan(x) & mbox{if and only if} & x = tan(y)  mbox{for} & -frac{pi}{2} le y le frac{pi}{2}, y ne 0, & y = arccsc(x) & mbox{if and only if} & x = csc(y)  mbox{for} & 0 le y le pi, y ne frac{pi}{2}, & y = arcsec(x) & mbox{if and only if} & x = sec(y)  mbox{for} & -frac{pi}{2} < y < frac{pi}{2}, y ne 0, & y = arccot(x) & mbox{if and only if} & x = cot(y)  end{matrix}

For inverse trigonometric functions, the notations sin−1 and cos−1 are often used for arcsin and arccos, etc. When this notation is used, the inverse functions are sometimes confused with the multiplicative inverses of the functions. Our notation avoids such confusion.


The following series definition may be obtained:

begin{matrix} arcsin z & = & z + left( frac {1} {2} right) frac {z^3} {3} + left( frac {1 cdot 3} {2 cdot 4} right) frac {z^5} {5} + left( frac{1 cdot 3 cdot 5} {2 cdot 4 cdot 6 } right) frac{z^7} {7} + cdots & = & sum_{n=0}^infty left( frac {(2n)!} {2^{2n}(n!)^2} right) frac {z^{2n+1}} {(2n+1)} end{matrix}  , quad left| z right| < 1
begin{matrix} arccos z & = & frac {pi} {2} - arcsin z  & = & frac {pi} {2} - (z + left( frac {1} {2} right) frac {z^3} {3} + left( frac {1 cdot 3} {2 cdot 4} right) frac {z^5} {5} + left( frac{1 cdot 3 cdot 5} {2 cdot 4 cdot 6 } right) frac{z^7} {7} + cdots )  & = & frac {pi} {2} - sum_{n=0}^infty left( frac {(2n)!} {2^{2n}(n!)^2} right) frac {z^{2n+1}} {(2n+1)} end{matrix}  , quad left| z right| < 1
begin{matrix} arctan z & = & z - frac {z^3} {3} +frac {z^5} {5} -frac {z^7} {7} +cdots  & = & sum_{n=0}^infty frac {(-1)^n z^{2n+1}} {2n+1} end{matrix}  , quad left| z right| < 1
begin{matrix} arccsc z & = & arcsinleft(z^{-1}right)  & = & z^{-1} + left( frac {1} {2} right) frac {z^{-3}} {3} + left( frac {1 cdot 3} {2 cdot 4 } right) frac {z^{-5}} {5} + left( frac {1 cdot 3 cdot 5} {2 cdot 4 cdot 6} right) frac {z^{-7}} {7} +cdots  & = & sum_{n=0}^infty left( frac {(2n)!} {2^{2n}(n!)^2} right) frac {z^{-(2n+1)}} {2n+1}end{matrix}  , quad left| z right| > 1
begin{matrix} arcsec z & = & arccosleft(z^{-1}right)  & = & frac {pi} {2} - (z^{-1} + left( frac {1} {2} right) frac {z^{-3}} {3} + left( frac {1 cdot 3} {2 cdot 4} right) frac {z^{-5}} {5} + left( frac{1 cdot 3 cdot 5} {2 cdot 4 cdot 6 } right) frac{z^{-7}} {7} + cdots )  & = & frac {pi} {2} - sum_{n=0}^infty left( frac {(2n)!} {2^{2n}(n!)^2} right) frac {z^{-(2n+1)}} {(2n+1)} end{matrix}  , quad left| z right| > 1
begin{matrix} arccot z & = & frac {pi} {2} - arctan z  & = & frac {pi} {2} - ( z - frac {z^3} {3} +frac {z^5} {5} -frac {z^7} {7} +cdots )  & = & frac {pi} {2} - sum_{n=0}^infty frac {(-1)^n z^{2n+1}} {2n+1} end{matrix}  , quad left| z right| < 1


These functions may also be defined by proving that they are antiderivatives of other functions.

arcsinleft(xright) = int_0^x frac 1 {sqrt{1 - z^2}},mathrm{d}z, quad |x| < 1
arccosleft(xright) = int_x^1 frac {1} {sqrt{1 - z^2}},mathrm{d}z,quad |x| < 1
arctanleft(xright) = int_0^x frac 1 {1 + z^2},mathrm{d}z, quad forall x in mathbb{R}
arccotleft(xright) = int_x^infty frac {1} {z^2 + 1},mathrm{d}z, quad z > 0
arcsecleft(xright) = int_x^1 frac 1 {|z| sqrt{z^2 - 1}},mathrm{d}z, quad x > 1
arccscleft(xright) = int_x^infty frac {-1} {|z| sqrt{z^2 - 1}},mathrm{d}z, quad x > 1


Inverse trigonometric functions can be generalized to complex arguments using the complex logarithm. Logarithms to various bases: is to base e, is to base 10, and is to base 1. ...

arcsin (z) = -i log left( i left( z + sqrt{1 - z^2}right) right)
arccos (z) = -i log left( z + sqrt{z^2 - 1}right)
arctan (z) = frac{i}{2} logleft(frac{1-iz}{1+iz}right)

Note: arcsec can also mean arcsecond. A second of arc or arcsecond is a unit of angular measurement which comprises one-sixtieth of an arcminute, or 1/3600 of a degree of arc or 1/1296000 ≈ 7. ...


Identities

sin left(x+yright)=sin x cos y + cos x sin y
sin left(x-yright)=sin x cos y - cos x sin y
cos left(x+yright)=cos x cos y - sin x sin y
cos left(x-yright)=cos x cos y + sin x sin y
sin x+sin y=2sin left( frac{x+y}{2} right) cos left( frac{x-y}{2} right)
sin x-sin y=2cos left( frac{x+y}{2} right) sin left( frac{x-y}{2} right)
cos x+cos y=2cos left( frac{x+y}{2} right) cos left( frac{x-y}{2} right)
cos x-cos y=-2sin left( frac{x+y}{2} right)sin left( frac{x-y}{2} right)
tan x+tan y=frac{sin left( x+yright) }{cos xcos y}
tan x-tan y=frac{sin left( x-yright) }{cos xcos y}
cot x+cot y=frac{sin left( x+yright) }{sin xsin y}
cot x-cot y=frac{-sin left( x-yright) }{sin xsin y}

See also trigonometric identity. For integrals and derivatives of trigonometric functions, see the relevant sections of table of derivatives, table of integrals, and list of integrals of trigonometric functions. In mathematics, trigonometric identities are equations involving trigonometric functions that are true for all values of the occurring variables. ... In calculus, the integral of a function is a generalization of area, mass, volume, sum, and total. ... In mathematics, the derivative is one of the two central concepts of calculus. ... The primary operation in differential calculus is finding a derivative. ... Integration is one of the two basic operations in calculus and since it, unlike differentiation, is non-trivial, tables of known integrals are often useful. ... The following is a list of integrals (antiderivative functions) of trigonometric functions. ...


Properties and applications

The trigonometric functions, as the name suggests, are of crucial importance in trigonometry, mainly because of the following two results: Trigonometry (from the Greek trigonon = three angles and metro = measure) is a branch of mathematics dealing with angles, triangles and trigonometric functions such as sine, cosine and tangent. ...


Law of sines

The law of sines for an arbitrary triangle states: In trigonometry, the law of sines (or sine law) is a statement about arbitrary triangles in the plane. ...

frac{sin A}{a} = frac{sin B}{b} = frac{sin C}{c}
A Lissajous curve, a figure formed with a trigonometry-based function.
A Lissajous curve, a figure formed with a trigonometry-based function.

It can be proven by dividing the triangle into two right ones and using the above definition of sine. The common number (sinA)/a occurring in the theorem is the reciprocal of the diameter of the circle through the three points A, B and C. 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. Image File history File links Download high resolution version (469x630, 39 KB) Summary This image was created by a trigonometric function plotting program I wrote. ... Image File history File links Download high resolution version (469x630, 39 KB) Summary This image was created by a trigonometric function plotting program I wrote. ... Lissajous figure on an Oscilloscope Lissajous figure in three dimensions In mathematics, a Lissajous curve (Lissajous figure or Bowditch curve) is the graph of the system of parametric equations which describes complex harmonic motion. ... Triangulation can be used to find the distance from the shore to the ship. ...


Law of cosines

The law of cosines (also known as the cosine formula) is an extension of the Pythagorean theorem: In trigonometry, the law of cosines (also known as the cosine formula) is a statement about arbitrary triangles which generalizes the Pythagorean theorem by correcting it with a term proportional to the cosine of the opposing angle. ... The Pythagorean theorem: The sum of the areas of the two squares on the legs (blue and red) equals the area of the square on the hypotenuse (purple). ...

c^2=a^2+b^2-2abcos C ,

Again, this theorem can be proven by dividing the triangle into two right ones. The law of cosines is useful to determine the unknown data of a triangle if two sides and an angle are known.


If the angle is not contained between the two sides, the triangle may not be unique. Be aware of this ambiguous case of the Cosine law.


Law of tangents

There is also a law of tangents: In trigonometry, the law of tangents is a statement about arbitrary triangles in the plane. ...

frac{a+b}{a-b} = frac{tan[frac{1}{2}(A+B)]}{tan[frac{1}{2}(A-B)]}
Functions based on sine and cosine can make appealing pictures.
Functions based on sine and cosine can make appealing pictures.

The trigonometric functions are also important outside of the study of triangles. They are periodic functions with characteristic wave patterns as graphs, useful for modelling recurring phenomena such as sound or light waves. Every signal can be written as a (typically infinite) sum of sine and cosine functions of different frequencies; this is the basic idea of Fourier analysis, where trigonometric series are used to solve a variety of boundary-value problems in partial differential equations. Graph of sum of sines and cosines (made by me) File links The following pages link to this file: Trigonometric function Categories: GFDL images ... Graph of sum of sines and cosines (made by me) File links The following pages link to this file: Trigonometric function Categories: GFDL images ... In mathematics, a periodic function is a function that repeats its values after some definite period has been added to its independent variable. ... A wave is a disturbance that propagates in a periodically repeating fashion, often transferring energy. ... Harmonic analysis is the branch of mathematics which studies the representation of functions or signals as the superposition of basic waves. ...


The image on the right displays a two-dimensional graph based on such a summation of sines and cosines, illustrating the fact that arbitrarily complicated closed curves can be described by a Fourier series. Its equation is: In mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object. ... The Fourier series, named in honor of Joseph Fourier (1768-1830), is an extremely useful mathematical tool. ...

(x(theta),,y(theta)) = sum_{n=1}^infty frac {1}{F(n+1)} (sin(thetacdot F(n)),, cos(thetacdot F(n)))

where F(n) is the nth Fibonacci number. In mathematics, the Fibonacci numbers form a sequence defined recursively by: In other words: one starts with 0 and 1, and then produces the next Fibonacci number by adding the two previous Fibonacci numbers. ...


For a compilation of many relations between the trigonometric functions, see trigonometric identities. In mathematics, trigonometric identities are equations involving trigonometric functions that are true for all values of the occurring variables. ...


References

Abramowitz and Stegun is the informal moniker of a mathematical reference work edited by Milton Abramowitz and Irene Stegun of the U.S. National Bureau of Standards. ... The MacTutor history of mathematics archive is a website hosted by University of St Andrews in Scotland. ... MathWorld is an online mathematics reference work, sponsored by Wolfram Research Inc. ... January 21 is the 21st day of the year in the Gregorian calendar. ... 2006 (MMVI) is a common year starting on Sunday of the Gregorian calendar. ...

See also

Tables of trigonometric functions are useful in a number of areas. ... In mathematics, the hyperbolic functions are analogs of the ordinary trigonometric, or circular, functions. ... This topic is considered to be an essential subject on Wikipedia. ... The Pythagorean theorem: The sum of the areas of the two squares on the legs (blue and red) equals the area of the square on the hypotenuse (purple). ... In mathematics, trigonometric identities are equations involving trigonometric functions that are true for all values of the occurring variables. ... Knowledge of nearly nothing beyond trigonometry itself is enough to make clear the nature of some of the applications of trigonometry to such endeavors as navigation, land surveying, building, and the like, but that impression is misleading in that it fails to indicate the nature and enormous variety of the... In mathematics, a unit vector in a normed vector space is a vector (most commonly a spatial vector) whose length is 1. ... In mathematics, a Newtonian series is a sum over a sequence written in the form where is the binomial coefficient and is the rising factorial. ...

External links


  Results from FactBites:
 
Trigonometric function - Wikipedia, the free encyclopedia (3381 words)
1) The sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse.
The sine function (blue) is closely approximated by its Taylor polynomial of degree 7 (pink) for a full cycle centered on the origin.
Sine and cosine function with an implementation in Rexx.
sine (633 words)
In the antiquity the sine was used as the length of a chord.
Imagine a triangle, the sine formula states that the ratio of the sine of an angle and the opposite side is equal for all three angles.
The inverse functions of the sine and the cosine are called the arc sine arcsin(x) and the arc cosine arccos(x), respectively.
  More results at FactBites »

 
 

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