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{{redirect|Trig}}{{pp-move-indef}}{{short description|In geometry, study of the relationship between angles and lengths}}{{Trigonometry}}Trigonometry (from Greek (wikt:τρίγωνον|trigōnon), "triangle" and (wikt:μέτρον|metron), "measure"WEB,weblink trigonometry, Online Etymology Dictionary, ) is a branch of mathematics that studies relationships between side lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies.R. Nagel (ed.), Encyclopedia of Science, 2nd Ed., The Gale Group (2002) The Greeks focused on the calculation of chords, while mathematicians in India created the earliest-known tables of values for trigonometric ratios (also called trigonometric functions) such as sine.BOOK, Carl Benjamin, Boyer, Carl Benjamin Boyer, A History of Mathematics, 2nd, John Wiley & Sons, Inc., 1991, 978-0-471-54397-8,weblink Throughout history, trigonometry has been applied in areas such as geodesy, surveying, celestial mechanics, and navigation.BOOK, Charles William Hackley, A treatise on trigonometry, plane and spherical: with its application to navigation and surveying, nautical and practical astronomy and geodesy, with logarithmic, trigonometrical, and nautical tables,weblink 1853, G. P. Putnam, Trigonometry is known for its many identities,BOOK, Mary Jane Sterling, Trigonometry For Dummies,weblink 24 February 2014, John Wiley & Sons, 978-1-118-82741-3, 185, BOOK, P.R. Halmos, I Want to be a Mathematician: An Automathography,weblink 1 December 2013, Springer Science & Business Media, 978-1-4612-1084-9, which are equations used for rewriting trigonometrical expressions to solve equations, to find a more useful expression, or to discover new relationships.BOOK, Ron Larson, Robert P. Hostetler, Trigonometry,weblink 10 March 2006, Cengage Learning, 0-618-64332-X, 230,


File:Hipparchos 1.jpeg|thumb|upright|left|Hipparchus, credited with compiling the first trigonometric table, has been described as "the father of trigonometry".BOOK, Boyer, Carl Benjamin Boyer, A History of Mathematics, 1991, Greek Trigonometry and Mensuration, 162, ]]Sumerian astronomers studied angle measure, using a division of circles into 360 degrees.Aaboe, Asger (2001). Episodes from the Early History of Astronomy. New York: Springer. {{isbn|0-387-95136-9}} They, and later the Babylonians, studied the ratios of the sides of similar triangles and discovered some properties of these ratios but did not turn that into a systematic method for finding sides and angles of triangles. The ancient Nubians used a similar method.BOOK, Otto Neugebauer, A history of ancient mathematical astronomy. 1,weblink 1975, Springer-Verlag, 978-3-540-06995-9, 744, In the 3rd century BC, Hellenistic mathematicians such as Euclid and Archimedes studied the properties of chords and inscribed angles in circles, and they proved theorems that are equivalent to modern trigonometric formulae, although they presented them geometrically rather than algebraically. In 140 BC, Hipparchus (from Nicaea, Asia Minor) gave the first tables of chords, analogous to modern tables of sine values, and used them to solve problems in trigonometry and spherical trigonometry.Thurston, pp. 235–236. In the 2nd century AD, the Greco-Egyptian astronomer Ptolemy (from Alexandria, Egypt) constructed detailed trigonometric tables (Ptolemy's table of chords) in Book 1, chapter 11 of his Almagest.{{Citation|title=Ptolemy's Almagest|last1=Toomer|first1=G.|authorlink=Gerald J. Toomer|publisher=Princeton University Press|year= 1998|isbn =978-0-691-00260-6}} Ptolemy used chord length to define his trigonometric functions, a minor difference from the sine convention we use today.Thurston, pp. 239–243. (The value we call sin(θ) can be found by looking up the chord length for twice the angle of interest (2θ) in Ptolemy's table, and then dividing that value by two.) Centuries passed before more detailed tables were produced, and Ptolemy's treatise remained in use for performing trigonometric calculations in astronomy throughout the next 1200 years in the medieval Byzantine, Islamic, and, later, Western European worlds.The modern sine convention is first attested in the Surya Siddhanta, and its properties were further documented by the 5th century (AD) Indian mathematician and astronomer Aryabhata.Boyer p. 215 These Greek and Indian works were translated and expanded by medieval Islamic mathematicians. By the 10th century, Islamic mathematicians were using all six trigonometric functions, had tabulated their values, and were applying them to problems in spherical geometry.Gingerich, Owen. "Islamic astronomy." Scientific American 254.4 (1986): 74-83BOOK, Michael Willers, Armchair Algebra: Everything You Need to Know From Integers To Equations,weblink 13 February 2018, Book Sales, 978-0-7858-3595-0, 37, The Persian polymath Nasir al-Din al-Tusi has been described as the creator of trigonometry as a mathematical discipline in its own right.WEB,weblink Al-Tusi_Nasir biography, MacTutor History of Mathematics archive, 2018-08-05, One of al-Tusi's most important mathematical contributions was the creation of trigonometry as a mathematical discipline in its own right rather than as just a tool for astronomical applications. In Treatise on the quadrilateral al-Tusi gave the first extant exposition of the whole system of plane and spherical trigonometry. This work is really the first in history on trigonometry as an independent branch of pure mathematics and the first in which all six cases for a right-angled spherical triangle are set forth., WEB,weblink the cambridge history of science, October 2013, WEB,weblink ṬUSI, NAṢIR-AL-DIN i. Biography – Encyclopaedia Iranica,,, en, 2018-08-05, His major contribution in mathematics (Nasr, 1996, pp. 208-214) is said to be in trigonometry, which for the first time was compiled by him as a new discipline in its own right. Spherical trigonometry also owes its development to his efforts, and this includes the concept of the six fundamental formulas for the solution of spherical right-angled triangles., Nasīr al-Dīn al-Tūsī was the first to treat trigonometry as a mathematical discipline independent from astronomy, and he developed spherical trigonometry into its present form.WEB, trigonometry,weblink Encyclopædia Britannica, 2008-07-21, He listed the six distinct cases of a right-angled triangle in spherical trigonometry, and in his On the Sector Figure, he stated the law of sines for plane and spherical triangles, discovered the law of tangents for spherical triangles, and provided proofs for both these laws.BOOK, J. Lennart, Berggren, The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook, Mathematics in Medieval Islam, Princeton University Press, 2007, 978-0-691-11485-9, 518, Knowledge of trigonometric functions and methods reached Western Europe via Latin translations of Ptolemy's Greek Almagest as well as the works of Persian and Arab astronomers such as Al Battani and Nasir al-Din al-Tusi.Boyer pp. 237, 274 One of the earliest works on trigonometry by a northern European mathematician is De Triangulis by the 15th century German mathematician Regiomontanus, who was encouraged to write, and provided with a copy of the Almagest, by the Byzantine Greek scholar cardinal Basilios Bessarion with whom he lived for several years.WEB,weblink Regiomontanus biography,, 2017-03-08, At the same time, another translation of the Almagest from Greek into Latin was completed by the Cretan George of Trebizond.N.G. Wilson (1992). From Byzantium to Italy. Greek Studies in the Italian Renaissance, London. {{isbn|0-7156-2418-0}} Trigonometry was still so little known in 16th-century northern Europe that Nicolaus Copernicus devoted two chapters of De revolutionibus orbium coelestium to explain its basic concepts.Driven by the demands of navigation and the growing need for accurate maps of large geographic areas, trigonometry grew into a major branch of mathematics.BOOK, Grattan-Guinness, Ivor, 1997, The Rainbow of Mathematics: A History of the Mathematical Sciences, W.W. Norton, 978-0-393-32030-5, Bartholomaeus Pitiscus was the first to use the word, publishing his Trigonometria in 1595.BOOK, Robert E. Krebs, Groundbreaking Scientific Experiments, Inventions, and Discoveries of the Middle Ages and the Renaissance,weblink 2004, Greenwood Publishing Group, 978-0-313-32433-8, 153, Gemma Frisius described for the first time the method of triangulation still used today in surveying. It was Leonhard Euler who fully incorporated complex numbers into trigonometry. The works of the Scottish mathematicians James Gregory in the 17th century and Colin Maclaurin in the 18th century were influential in the development of trigonometric series.William Bragg Ewald (2007). From Kant to Hilbert: a source book in the foundations of mathematics. Oxford University Press US. p. 93. {{isbn|0-19-850535-3}} Also in the 18th century, Brook Taylor defined the general Taylor series.Kelly Dempski (2002). Focus on Curves and Surfaces. p. 29. {{isbn|1-59200-007-X}}

Trigonometric ratios

(File:TrigonometryTriangle.svg|thumb|245px|In this right triangle: {{math|1= sin A = a/c;}} {{math|1= cos A = b/c;}} {{math|1= tan A = a/b.}})Trigonometric ratios are the ratios between edges of a right triangle. These ratios are given by the following trigonometric functions of the known angle A, where a, b and c refer to the lengths of the sides in the accompanying figure:
  • Sine function (sin), defined as the ratio of the side opposite the angle to the hypotenuse.

sin A=frac{textrm{opposite}}{textrm{hypotenuse}}=frac{a}{c}.
  • Cosine function (cos), defined as the ratio of the adjacent leg (the side of the triangle joining the angle to the right angle) to the hypotenuse.

cos A=frac{textrm{adjacent}}{textrm{hypotenuse}}=frac{b}{c}.
  • Tangent function (tan), defined as the ratio of the opposite leg to the adjacent leg.

tan A=frac{textrm{opposite}}{textrm{adjacent}}=frac{a}{b}=frac{a/c}{b/c}=frac{sin A}{cos A}.
The hypotenuse is the side opposite to the 90 degree angle in a right triangle; it is the longest side of the triangle and one of the two sides adjacent to angle A. The adjacent leg is the other side that is adjacent to angle A. The opposite side is the side that is opposite to angle A. The terms perpendicular and base are sometimes used for the opposite and adjacent sides respectively. See below under Mnemonics.Since any two right triangles with the same acute angle A are similarBOOK, James Stewart, Lothar Redlin, Saleem Watson, Algebra and Trigonometry,weblink 16 January 2015, Cengage Learning, 978-1-305-53703-3, 448, , the value of a trigonometric ratio depends only on the angle A.The reciprocals of these functions are named the cosecant (csc), secant (sec), and cotangent (cot), respectively:
csc A=frac{1}{sin A}=frac{textrm{hypotenuse}}{textrm{opposite}}=frac{c}{a} ,
sec A=frac{1}{cos A}=frac{textrm{hypotenuse}}{textrm{adjacent}}=frac{c}{b} ,
cot A=frac{1}{tan A}=frac{textrm{adjacent}}{textrm{opposite}}=frac{cos A}{sin A}=frac{b}{a} .
The cosine, cotangent, and cosecant are so named because they are respectively the sine, tangent, and secant of the complementary angle abbreviated to "co-".BOOK, Dick Jardine, Amy Shell-Gellasch, Mathematical Time Capsules: Historical Modules for the Mathematics Classroom,weblink 2011, MAA, 978-0-88385-984-1, 182, With these functions, one can answer virtually all questions about arbitrary triangles by using the law of sines and the law of cosines.BOOK, Krystle Rose Forseth, Christopher Burger, Michelle Rose Gilman, Deborah J. Rumsey, Pre-Calculus For Dummies,weblink 7 April 2008, John Wiley & Sons, 978-0-470-16984-1, 218, These laws can be used to compute the remaining angles and sides of any triangle as soon as two sides and their included angle or two angles and a side or three sides are known.


A common use of mnemonics is to remember facts and relationships in trigonometry. For example, the sine, cosine, and tangent ratios in a right triangle can be remembered by representing them and their corresponding sides as strings of letters. For instance, a mnemonic is SOH-CAH-TOA:{{MathWorld|title=SOHCAHTOA|urlname=SOHCAHTOA}}
Sine = Opposite ÷ Hypotenuse Cosine = Adjacent ÷ Hypotenuse Tangent = Opposite ÷ Adjacent
One way to remember the letters is to sound them out phonetically (i.e., SOH-CAH-TOA, which is pronounced 'so-ka-toe-uh' {{IPAc-en|s|oʊ|k|æ|ˈ|t|oʊ|ə}}). Another method is to expand the letters into a sentence, such as "Some Old Hippie Caught Another Hippie Trippin' On Acid".A sentence more appropriate for high schools is "Some Old Horse Came A'Hopping Through Our Alley". BOOK, Memory: A Very Short Introduction, Jonathan K., Foster, Oxford, 2008, 978-0-19-280675-8, 128,

The unit circle and common trigonometric values

(File:Sin-cos-defn-I.png|right|thumb|Fig. 1a – Sine and cosine of an angle θ defined using the unit circle.)Trigonometric ratios can also be represented using the unit circle, which is the circle of radius 1 centered at the origin in the plane.BOOK, David Cohen, Lee B. Theodore, David Sklar, Precalculus: A Problems-Oriented Approach, Enhanced Edition,weblink 17 July 2009, Cengage Learning, 1-4390-4460-0, In this setting, the terminal side of an angle A placed in standard position will intersect the unit circle in a point (x,y), where x = cos A and y = sin A . This representation allows for the calculation of commonly found trigonometric values, such as those in the following table:BOOK, W. Michael Kelley, The Complete Idiot's Guide to Calculus,weblink 2002, Alpha Books, 978-0-02-864365-6, 45, {| class="wikitable"! Function! 0! pi/6! pi/4! pi/3! pi/2! 2pi/3! 3pi/4! 5pi/6! pi
! sine| 0| 1/2| sqrt{2}/2| sqrt{3}/2| 1| sqrt{3}/2| sqrt{2}/2| 1/2| 0
! cosine| 1| sqrt{3}/2| sqrt{2}/2| 1/2| 0| -1/2| -sqrt{2}/2| -sqrt{3}/2| -1
! tangent| 0| sqrt{3}/3| 1| sqrt{3}| undefined| -sqrt{3}| -1| -sqrt{3}/3| 0
! secant| 1| 2sqrt{3}/3| sqrt{2}| 2| undefined| -2| -sqrt{2}| -2sqrt{3}/3| -1
! cosecant| undefined| 2| sqrt{2}| 2sqrt{3}/3| 1| 2sqrt{3}/3| sqrt{2}| 2| undefined
! cotangent| undefined| sqrt{3}| 1| sqrt{3}/3| 0| -sqrt{3}/3| -1| -sqrt{3}| undefined

Trigonometric functions of real or complex variables

Using the unit circle, one can extend the definitions of trigonometric ratios to all positive and negative argumentsBOOK, Jenny Olive, Maths: A Student's Survival Guide: A Self-Help Workbook for Science and Engineering Students,weblink 18 September 2003, Cambridge University Press, 978-0-521-01707-7, 175, (see trigonometric function).

Graphs of trigonometric functions

The following table summarizes the properties of the graphs of the six main trigonometric functions:BOOK, Mary P Attenborough, Mathematics for Electrical Engineering and Computing,weblink 30 June 2003, Elsevier, 978-0-08-047340-6, 418, BOOK, Ron Larson, Bruce H. Edwards, Calculus of a Single Variable,weblink 10 November 2008, Cengage Learning, 0-547-20998-3, 21, {| class="wikitable"! Function! Period! Domain! Range! Graph
! sine| 2pi| (-infty,infty)| [-1,1]
Image:Sine one period.svg>200 px
! cosine| 2pi| (-infty,infty)| [-1,1]
Image:Cosine one period.svg>200 px
! tangent| pi| x neq pi/2+npi| (-infty,infty)
Image:Tangent-plot.svg>200 px
! secant| 2pi| x neq pi/2 + npi| (-infty,-1] cup [1,infty)
Image:Secant.svg>200 px
! cosecant| 2pi| x neq npi| (-infty,-1] cup [1,infty)
Image:Cosecant.svg>200 px
! cotangent| pi| x neq npi| (-infty,infty)
Image:Cotangent.svg>200 px

Inverse trigonometric functions

Because the six main trigonometric functions are periodic, they are not injective (or, 1 to 1), and thus are not invertible. By restricting the domain of a trigonometric function, however, they can be made invertible.BOOK, Elizabeth G. Bremigan, Ralph J. Bremigan, John D. Lorch, Mathematics for Secondary School Teachers,weblink 2011, MAA, 978-0-88385-773-1, {{rp|48ff}}The names of the inverse trigonometric functions, together with their domains and range, can be found in the following table:{{rp|48ff}}BOOK, Martin Brokate, Pammy Manchanda, Abul Hasan Siddiqi, Calculus for Scientists and Engineers,weblink 3 August 2019, Springer, 9789811384646, {{rp|521ff}}{| class="wikitable" style="text-align:center" !Name!Usual notation!Definition!Domain of x for real result!Range of usual principal value (radians)!Range of usual principal value (degrees)
arcsine >y = {{math>arcsin(x)}} x = {{mathsine>sin(y)}} −1 ≤ x ≤ 1 −{{sfrac2}} ≤ y ≤ {{sfrac2}} −90° ≤ y ≤ 90°
arccosine >y = {{math>arccos(x)}} x = {{mathcosine>cos(y)}} −1 ≤ x ≤ 1 0 ≤ y ≤ {{pi}} 0° ≤ y ≤ 180°
< y < {{sfrac< y < 90°
arctangent >y = {{math>arctan(x)}} x = {{mathTangent (trigonometry)>tan(y)}} all real numbers −{{sfrac2}} 2}} −90°
< y < 180°
arccotangent >y = {{math>arccot(x)}} x = {{mathcotangent>cot(y)}} all real numbersy < {{pi}} >| 0°
< {{sfrac< y ≤ {{pi}} < 90° or 90° < y ≤ 180°
arcsecant >y = {{math>arcsec(x)}} x = {{mathSecant (trigonometry)>sec(y)}} x ≤ −1 or 1 ≤ x 0 ≤ y 2}} or {{sfrac2}} 0° ≤ y
< 0 or 0 < y ≤ {{sfrac< 0° or 0° < y ≤ 90°
arccosecant >y = {{math>arccsc(x)}} x = {{mathcosecant>csc(y)}} x ≤ −1 or 1 ≤ x −{{sfrac2}} ≤ y 2}} −90° ≤ y

Power series representations

When considered as functions of a real variable, the trigonometric ratios can be represented by an infinite series. For instance, sine and cosine have the following representations:BOOK, Serge Lang, Complex Analysis,weblink 14 March 2013, Springer, 978-3-642-59273-7, 63,
begin{align}sin x & = x - frac{x^3}{3!} + frac{x^5}{5!} - frac{x^7}{7!} + cdots & = sum_{n=0}^infty frac{(-1)^n x^{2n+1}}{(2n+1)!} end{align}
begin{align}cos x & = 1 - frac{x^2}{2!} + frac{x^4}{4!} - frac{x^6}{6!} + cdots & = sum_{n=0}^infty frac{(-1)^n x^{2n}}{(2n)!}.end{align}With these definitions the trigonometric functions can be defined for complex numbers.BOOK, Silvia Maria Alessio, Digital Signal Processing and Spectral Analysis for Scientists: Concepts and Applications,weblink 9 December 2015, Springer, 978-3-319-25468-5, 339, When extended as functions of real or complex variables, the following formula holds for the complex exponential:
e^{x+iy} = e^x(cos y + i sin y).
This complex exponential function, written in terms of trigonometric functions, is particularly useful.BOOK, K. RAJA RAJESWARI, B. VISVESVARA RAO, SIGNALS AND SYSTEMS,weblink 24 March 2014, PHI Learning, 978-81-203-4941-4, 263, BOOK, John Stillwell, Mathematics and Its History,weblink 23 July 2010, Springer Science & Business Media, 978-1-4419-6053-5, 313,

Calculating trigonometric functions

Trigonometric functions were among the earliest uses for mathematical tables.BOOK, Martin Campbell-Kelly, Professor Emeritus of Computer Science Martin Campbell-Kelly, Visiting Fellow Department of Computer Science Mary Croarken, Raymond Flood, Eleanor Robson, The History of Mathematical Tables: From Sumer to Spreadsheets,weblink 2 October 2003, OUP Oxford, 978-0-19-850841-0, Such tables were incorporated into mathematics textbooks and students were taught to look up values and how to interpolate between the values listed to get higher accuracy.BOOK, George S. Donovan, Beverly Beyreuther Gimmestad, Trigonometry with calculators,weblink 1980, Prindle, Weber & Schmidt, 978-0-87150-284-1, Slide rules had special scales for trigonometric functions.BOOK, Ross Raymond Middlemiss, Instructions for Post-trig and Mannheim-trig Slide Rules,weblink 1945, Frederick Post Company, Scientific calculators have buttons for calculating the main trigonometric functions (sin, cos, tan, and sometimes cis and their inverses).BOOK, Bonnier Corporation, Popular Science,weblink April 1974, Bonnier Corporation, 125, Most allow a choice of angle measurement methods: degrees, radians, and sometimes gradians. Most computer programming languages provide function libraries that include the trigonometric functions.BOOK, Steven S Skiena, Miguel A. Revilla, Programming Challenges: The Programming Contest Training Manual,weblink 18 April 2006, Springer Science & Business Media, 978-0-387-22081-9, 302, The floating point unit hardware incorporated into the microprocessor chips used in most personal computers has built-in instructions for calculating trigonometric functions.BOOK, Intel® 64 and IA-32 Architectures Software Developer's Manual Combined Volumes: 1, 2A, 2B, 2C, 3A, 3B and 3C, 2013, Intel,weblink

Other Trigonometric Functions

In addition to the six ratios listed earlier, there are additional trigonometric functions that were historically important, though seldom used today. These include the chord ({{math|1=crd(θ) = 2 sin({{sfrac|θ|2}})}}), the versine ({{math|1=versin(θ) = 1 − cos(θ) = 2 sin2({{sfrac|θ|2}})}}) (which appeared in the earliest tables{{harvtxt|Boyer|1991|pp=xxiii–xxiv}}), the coversine ({{math|1=coversin(θ) = 1 − sin(θ) = versin({{sfrac|{{pi}}|2}} − θ)}}), the haversine ({{math|1=haversin(θ) = {{sfrac|1|2}}versin(θ) = sin2({{sfrac|θ|2}})}}),{{harvtxt|Nielsen|1966|pp=xxiii–xxiv}} the exsecant ({{math|1=exsec(θ) = sec(θ) − 1}}), and the excosecant ({{math|1=excsc(θ) = exsec({{sfrac|{{pi}}|2}} − θ) = csc(θ) − 1}}). See List of trigonometric identities for more relations between these functions.



For centuries, spherical trigonometry has been used for locating solar, lunar, and stellar positions,BOOK, Olinthus Gregory, Elements of Plane and Spherical Trigonometry: With Their Applications to Heights and Distances Projections of the Sphere, Dialling, Astronomy, the Solution of Equations, and Geodesic Operations,weblink 1816, Baldwin, Cradock, and Joy, predicting eclipses, and describing the orbits of the planets.Neugebauer, Otto. "Mathematical methods in ancient astronomy." Bulletin of the American Mathematical Society 54.11 (1948): 1013-1041.In modern times, the technique of triangulation is used in astronomy to measure the distance to nearby stars,BOOK, Michael Seeds, Dana Backman, Astronomy: The Solar System and Beyond,weblink 5 January 2009, Cengage Learning, 0-495-56203-3, 254, as well as in satellite navigation systems.


File:Frieberger drum marine sextant.jpg|thumb|200px|Sextants are used to measure the angle of the sun or stars with respect to the horizon. Using trigonometry and a marine chronometermarine chronometerHistorically, trigonometry has been used for locating latitudes and longitudes of sailing vessels, plotting courses, and calculating distances during navigation.BOOK, John Sabine, The Practical Mathematician, Containing Logarithms, Geometry, Trigonometry, Mensuration, Algebra, Navigation, Spherics and Natural Philosophy, Etc,weblink 1800, 1, Trigonometry is still used in navigation through such means as the Global Positioning System and artificial intelligence for autonomous vehicles.BOOK, Mordechai Ben-Ari, Francesco Mondada, Elements of Robotics,weblink 25 October 2017, Springer, 978-3-319-62533-1, 16,


In land surveying, trigonometry is used in the calculation of lengths, areas, and relative angles between objects.BOOK, George Roberts Perkins, Plane Trigonometry and Its Application to Mensuration and Land Surveying: Accompanied with All the Necessary Logarithmic and Trigonometric Tables,weblink 1853, D. Appleton & Company, On a larger scale, trigonometry is used in geography to measure distances between landmarks,BOOK, Charles W. J. Withers, Hayden Lorimer, Geographers: Biobibliographical Studies,weblink 14 December 2015, A&C Black, 978-1-4411-0785-5, 6,

Periodic functions

(File:Fourier series and transform.gif|frame|right|Function s(x) (in red) is a sum of six sine functions of different amplitudes and harmonically related frequencies. Their summation is called a Fourier series. The Fourier transform, S(f) (in blue), which depicts amplitude vs frequency, reveals the 6 frequencies (at odd harmonics) and their amplitudes (1/odd number).)The sine and cosine functions are fundamental to the theory of periodic functions,BOOK, H. G. ter Morsche, J. C. van den Berg, E. M. van de Vrie, Fourier and Laplace Transforms,weblink 7 August 2003, Cambridge University Press, 978-0-521-53441-3, 61, such as those that describe sound and light waves. Fourier discovered that every continuous, periodic function could be described as an infinite sum of trigonometric functions.Even non-periodic functions can be represented as an integral of sines and cosines through the Fourier transform. This has applications to quantum mechanicsBOOK, Bernd Thaller, Visual Quantum Mechanics: Selected Topics with Computer-Generated Animations of Quantum-Mechanical Phenomena,weblink 8 May 2007, Springer Science & Business Media, 978-0-387-22770-2, 15, and communicationsBOOK, M. Rahman, Applications of Fourier Transforms to Generalized Functions,weblink 2011, WIT Press, 978-1-84564-564-9, , among other fields.

Optics and Acoustics

Trigonometry is useful in many physical sciences,BOOK, Lawrence Bornstein, Basic Systems, Inc, Trigonometry for the Physical Sciences,weblink 1966, Appleton-Century-Crofts, including acoustics,BOOK, John J. Schiller, Marie A. Wurster, College Algebra and Trigonometry: Basics Through Precalculus,weblink 1988, Scott, Foresman, 978-0-673-18393-4, and optics. In these areas, they are used to describe sound and light waves, and to solve boundary- and transmission-related problems.BOOK, Dudley H. Towne, Wave Phenomena,weblink 5 May 2014, Dover Publications, 978-0-486-14515-0,

Other applications

Other fields that use trigonometry or trigonometric functions include music theory,BOOK, E. Richard Heineman, J. Dalton Tarwater, Plane Trigonometry,weblink 1 November 1992, McGraw-Hill, 978-0-07-028187-5, geodesy, audio synthesis,BOOK, Mark Kahrs, Karlheinz Brandenburg, Applications of Digital Signal Processing to Audio and Acoustics,weblink 18 April 2006, Springer Science & Business Media, 978-0-306-47042-4, 404, architecture,BOOK, Kim Williams, Michael J. Ostwald, Architecture and Mathematics from Antiquity to the Future: Volume I: Antiquity to the 1500s,weblink 9 February 2015, Birkhäuser, 978-3-319-00137-1, 260, electronics, biology,BOOK, Dan Foulder, Essential Skills for GCSE Biology,weblink 15 July 2019, Hodder Education, 978-1-5104-6003-4, 78, medical imaging (CT scans and ultrasound),BOOK, Luciano Beolchi, Michael H. Kuhn, Medical Imaging: Analysis of Multimodality 2D/3D Images,weblink 1995, IOS Press, 978-90-5199-210-6, 122, chemistry,BOOK, Marcus Frederick Charles Ladd, Symmetry of Crystals and Molecules,weblink 2014, Oxford University Press, 978-0-19-967088-8, 13, number theory (and hence cryptology),BOOK, Gennady I. Arkhipov, Vladimir N. Chubarikov, Anatoly A. Karatsuba, Trigonometric Sums in Number Theory and Analysis,weblink 22 August 2008, Walter de Gruyter, 978-3-11-019798-3, seismology, meteorology,BOOK, Study Guide for the Course in Meteorological Mathematics: Latest Revision, Feb. 1, 1943,weblink 1943, oceanography,BOOK, Mary Sears, Daniel Merriman, Woods Hole Oceanographic Institution, Oceanography, the past,weblink 1980, Springer-Verlag, 978-0-387-90497-9, image compression,WEB,weblink JPEG Standard (JPEG ISO/IEC 10918-1 ITU-T Recommendation T.81), 1993, International Telecommunications Union, 6 April 2019, phonetics,BOOK, Kirsten Malmkjaer, The Routledge Linguistics Encyclopedia,weblink 4 December 2009, Routledge, 978-1-134-10371-3, 1, economics,BOOK, Kamran Dadkhah, Foundations of Mathematical and Computational Economics,weblink 11 January 2011, Springer Science & Business Media, 978-3-642-13748-8, 46, electrical engineering, mechanical engineering, civil engineering, computer graphics, cartography, crystallographyBOOK, John Joseph Griffin, A System of Crystallography, with Its Application to Mineralogy,weblink 1841, R. Griffin, 119, and game development.BOOK, Christopher Griffith, Real-World Flash Game Development: How to Follow Best Practices AND Keep Your Sanity,weblink 12 November 2012, CRC Press, 978-1-136-13702-0, 153,


(File:Triangle ABC with Sides a b c 2.png|thumb|240px|right|Triangle with sides a,b,c and respectively opposite angles A,B,C)Trigonometry has been noted for its many identities, that is, equations that are true for all possible inputs.BOOK, Dugopolski, Trigonometry I/E Sup,weblink July 2002, Addison Wesley, 978-0-201-78666-8, Identities involving only angles are known as trigonometric identities. Other equations, known as triangle identities,BOOK, V&S EDITORIAL BOARD, CONCISE DICTIONARY OF MATHEMATICS,weblink 6 January 2015, V&S Publishers, 978-93-5057-414-0, 288, relate both the sides and angles of a given triangle.

Triangle identities

{{Anchor|Triangle identities|Common formulas}}In the following identities, A, B and C are the angles of a triangle and a, b and c are the lengths of sides of the triangle opposite the respective angles (as shown in the diagram).Lecture 3 | Quantum Entanglements, Part 1 (Stanford), Leonard Susskind, trigonometry in five minutes, law of sin, cos, euler formula 2006-10-09.

Law of sines

The law of sines (also known as the "sine rule") for an arbitrary triangle states:
frac{a}{sin A} = frac{b}{sin B} = frac{c}{sin C} = 2R = frac{abc}{2Delta},
where Delta is the area of the triangle and R is the radius of the circumscribed circle of the triangle:
R = frac{abc}{sqrt{(a+b+c)(a-b+c)(a+b-c)(b+c-a)}}.

Law of cosines

The law of cosines (known as the cosine formula, or the "cos rule") is an extension of the Pythagorean theorem to arbitrary triangles:
c^2=a^2+b^2-2abcos C ,
or equivalently:
cos C=frac{a^2+b^2-c^2}{2ab}.

Law of tangents

The law of tangents, developed by François Viète, is an alternative to the Law of Cosines when solving for the unknown edges of a triangle, providing simpler computations when using trigonometric tables.BOOK, Ron Larson, Trigonometry,weblink 29 January 2010, Cengage Learning, 1-4390-4907-6, 331, It is given by:


Given two sides a and b and the angle between the sides C, the area of the triangle is given by half the product of the lengths of two sides and the sine of the angle between the two sides:BOOK, Cynthia Y. Young, Precalculus,weblink 19 January 2010, John Wiley & Sons, 978-0-471-75684-2, 435, Heron's formula is another method that may be used to calculate the area of a triangle. This formula states that if a triangle has sides of lengths a, b, and c, and if the semiperimeter is
then the area of the triangle is:BOOK, Richard N. Aufmann, Vernon C. Barker, Richard D. Nation, College Trigonometry,weblink 5 February 2007, Cengage Learning, 0-618-82507-X, 306,
mbox{Area} = Delta = sqrt{s(s-a)(s-b)(s-c)} = frac{abc}{4R},
where R is the radius of the circumcircle of the triangle.
mbox{Area} = Delta = frac{1}{2}a bsin C.

Trigonometric identities

Pythagorean identities

The following trigonometric identities are related to the Pythagorean theorem and hold for any value:BOOK, Technical Mathematics with Calculus, illustrated, John C., Peterson, Cengage Learning, 2004, 978-0-7668-6189-3, 856,weblink Extract of page 856
sin^2 A + cos^2 A = 1
tan^2 A + 1 = sec^2 A
cot^2 A + 1 = csc^2 A

Euler's formula

Euler's formula, which states that e^{ix} = cos x + i sin x, produces the following analytical identities for sine, cosine, and tangent in terms of e and the imaginary unit i:
sin x = frac{e^{ix} - e^{-ix}}{2i}, qquad cos x = frac{e^{ix} + e^{-ix}}{2}, qquad tan x = frac{i(e^{-ix} - e^{ix})}{e^{ix} + e^{-ix}}.

Other trigonometric identities

Other commonly used trigonometric identities include the half-angle identities, the angle sum and difference identities, and the product-to-sum identities.

See also

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  • BOOK, Carl B., Boyer, Carl Benjamin Boyer, A History of Mathematics, Second, John Wiley & Sons, Inc., 1991, 978-0-471-54397-8, registration,weblink
  • {{Springer |title=Trigonometric functions |id=p/t094210}}
  • Christopher M. Linton (2004). From Eudoxus to Einstein: A History of Mathematical Astronomy. Cambridge University Press.
  • {{citation |last1=Nielsen |first1=Kaj L. |title=Logarithmic and Trigonometric Tables to Five Places |edition=2nd |location=New York, USA |publisher=Barnes & Noble |date=1966 |lccn=61-9103}}
  • {{MathWorld|title=Trigonometric Addition Formulas|urlname=TrigonometricAdditionFormulas|author=Weisstein, Eric W.}}

External links

{{Sister project links|Trigonometry}}{{Library resources box |by=no |onlinebooks=no |others=no |about=yes |label=Trigonometry}} {{Areas of mathematics | state=collapsed}}{{Authority control}}

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