Trigonometry: Difference between revisions

{{shortShort description|InArea of geometry, study of the relationship betweenabout angles and lengths}}

'''Trigonometry''' ({{etymology|grc|''{{wikt-lang|grc|τρίγωνον}}'' ({{grc-transl|τρίγωνον}})|triangle||''{{wikt-lang|grc|μέτρον}}'' ({{grc-transl|μέτρον}})|measure}})<ref>{{OEtymD|trigonometry |access-date=2022-03-18}}</ref> is a branch of [[mathematics]] thatconcerned studieswith relationships between [[angle]]s and side lengths andof triangles. In particular, the [[angletrigonometric functions]]s relate the angles of a [[right triangle]] with [[ratio]]s of its side lengths. The field emerged in the [[Hellenistic period|Hellenistic world]] during the 3rd century BC from applications of [[geometry]] to [[Astronomy|astronomical studies]].<ref>R. Nagel (ed.), ''Encyclopedia of Science'', 2nd Ed., The Gale Group (2002)</ref> The Greeks focused on the [[Ptolemy's table of chords|calculation of chords]], while mathematicians in India created the earliest-known tables of values for trigonometric ratios (also called [[trigonometric functions]]) such as [[sine]].{{sfnp|Boyer|1991|p={{page needed|date=January 2021}}}}

[[List of trigonometric identities|trigonometric identities]]<ref name="Sterling2014">{{cite book|author=Mary Jane Sterling|title=Trigonometry For Dummies|url=https://books.google.com/books?id=cb7RAgAAQBAJ&pg=PA185|date=24 February 2014|publisher=John Wiley & Sons|isbn=978-1-118-82741-3|page=185}}</ref><ref name="Halmos2013">{{cite book|author=P.R. Halmos|title=I Want to be a Mathematician: An Automathography|url=https://books.google.com/books?id=7VblBwAAQBAJ&pg=PA24|date=1 December 2013|publisher=Springer Science & Business Media|isbn=978-1-4612-1084-9}}</ref> are commonly used for rewriting trigonometrical [[expression (mathematics)|expression]]s with the aim to simplify an expression, to find a more useful form of an expression, or to [[equation solving|solve an equation]].<ref name="LarsonHostetler2006">{{cite book|author1=Ron Larson|author2=Robert P. Hostetler|title=Trigonometry|url=https://books.google.com/books?id=RI-t-w0AXVAC&pg=PA230|date=10 March 2006|publisher=Cengage Learning|isbn=0-618-64332-X|page=230}}</ref>

[[File:HipparchosHead 1of Hipparchus (cropped).jpegjpg|thumb|upright=0.8|right|[[Hipparchus]], credited with compiling the first [[Trigonometric tables|trigonometric table]], has been described as "the father of trigonometry".{{sfnp|Boyer|1991 |loc="Greek Trigonometry and Mensuration" |p=[https://archive.org/details/historyofmathema00boye/page/162 162]}}]]

In the 3rd century BC, [[Greek mathematics|Hellenistic mathematicians]] such as [[Euclid]] and [[Archimedes]] studied the properties of [[chord (geometry)|chords]] and [[inscribed angle]]s 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 [[trigonometric tables|tables of sine values]], and used them to solve problems in trigonometry and [[spherical trigonometry]].{{sfnp|Thurston|1996|pp=[https://books.google.com/books?id=rNpHjqxQQ9oC&pg=PA235#v=onepage&q&f=false 235–236]|loc="Appendix 1: Hipparchus's Table of Chords"}} 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]]''.<ref name=toomer>{{Citation|title=Ptolemy's Almagest|last1=Toomer |first1=G.|author-link=Gerald J. Toomer|publisher=Princeton University Press|year=1998|isbn=978-0-691-00260-6}}</ref> Ptolemy used [[chord (geometry)|chord]] length to define his trigonometric functions, a minor difference from the [[sine]] convention we use today.{{sfnp|Thurston|1996|loc="Appendix 3: Ptolemy's Table of Chords"|pp=[https://books.google.com/books?id=rNpHjqxQQ9oC&pg=PA239#v=onepage&q&f=false 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 Golden Age|Islamic]], and, later, Western European worlds.

The modern sinedefinition conventionof the sine is first attested in the ''[[Surya Siddhanta]]'', and its properties were further documented byin the 5th century (AD) by [[Indian mathematics|Indian mathematician]] and astronomer [[Aryabhata]].{{sfnp|Boyer |1991|p=215}} These Greek and Indian works were translated and expanded by [[Mathematics in medieval Islam|medieval Islamic mathematicians]]. In 830 AD, Persian mathematician [[Habash al-Hasib al-Marwazi]] produced the first table of cotangents.<ref name="Sesiano">{{Cite book |author=Jacques Sesiano |chapter=Islamic mathematics |page=157 |title=Mathematics Across Cultures: The History of Non-western Mathematics |editor1-first=Helaine |editor1-last=Selin |editor1-link=Helaine Selin |editor2-first=Ubiratan |editor2-last=D'Ambrosio |editor2-link=Ubiratan D'Ambrosio |year=2000 |publisher=[[Springer Science+Business Media]] |isbn=978-1-4020-0260-1}}</ref><ref name="Britannica"/> By the 10th century AD, Islamicin mathematiciansthe werework usingof Persian mathematician [[Abū al-Wafā' al-Būzjānī]], all six [[trigonometric functions,]] were used.{{sfn|Boyer|1991|p=238}} Abu al-Wafa had tabulatedsine theirtables valuesin 0.25° increments, to 8 decimal places of accuracy, and wereaccurate applyingtables themof totangent problemsvalues.{{sfn|Boyer|1991|p=238}} He also made important innovations in [[spherical geometrytrigonometry]]<ref name="musa">{{cite journal |last=Moussa |first=Ali |title=Mathematical Methods in Abū al-Wafāʾ's Almagest and the Qibla Determinations |journal=Arabic Sciences and Philosophy |year=2011 |volume=21 |issue=1 |pages=1–56 |publisher=[[Cambridge University Press]] |doi=10.1017/S095742391000007X|s2cid=171015175 }}</ref><ref>Gingerich, Owen. "Islamic astronomy." Scientific American 254.4 (1986): 74-8374–83</ref><ref name="Willers2018">{{cite book|author=Michael Willers|title=Armchair Algebra: Everything You Need to Know From Integers To Equations|url=https://books.google.com/books?id=45R2DwAAQBAJ&pg=PA37|date=13 February 2018|publisher=Book Sales|isbn=978-0-7858-3595-0|page=37}}</ref> The [[Persian people|Persian]] [[polymath]] [[Nasir al-Din al-Tusi]] has been described as the creator of trigonometry as a mathematical discipline in its own right.<ref>{{Cite web|title=Nasir al-Din al-Tusi |website=[[MacTutor History of Mathematics archive]] |url=https://mathshistory.st-andrews.ac.uk/Biographies/Al-Tusi_Nasir/ |access-date=2021-01-08|quote=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.}}</ref><ref>{{Cite book|title=the cambridge history of science|chapter=Islamic Mathematics |date=October 2013|volume=2 |pages=62–83 |publisher=Cambridge University Press |doi=10.1017/CHO9780511974007.004 |isbn=9780521594486 |url=https://www.cambridge.org/core/books/the-cambridge-history-of-science/islamic-mathematics/4BF4D143150C0013552902EE270AF9C2|last1=Berggren |first1=J. L. }}</ref><ref>{{Cite encyclopedia|title=ṬUSI, NAṢIR-AL-DIN i. Biography |encyclopedia=Encyclopaedia Iranica |url=http://www.iranicaonline.org/articles/tusi-nasir-al-din-bio |access-date=2018-08-05|quote=His major contribution in mathematics (Nasr, 1996, pp. 208-214208–214) 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.}}</ref> [[Nasīr al-Dīn al-Tūsī]]He was the first to treat trigonometry as a mathematical discipline independent from astronomy, and he developed spherical trigonometry into its present form.<ref name="Britannica">{{cite web |title=trigonometry |url=http://www.britannica.com/EBchecked/topic/605281/trigonometry |publisher=[[Encyclopædia Britannica]] |access-date=2008-07-21}}</ref> 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.<ref>{{cite book |first=J. Lennart |last=Berggren |title=The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook |chapter=Mathematics in Medieval Islam |publisher=Princeton University Press |year=2007 |isbn=978-0-691-11485-9 |page=518}}</ref> Knowledge of trigonometric functions and methods reached [[Western Europe]] via [[Latin translations of the 12th century|Latin translations]] of Ptolemy's Greek ''Almagest'' as well as the works of [[Astronomy in medieval Islam|Persian and Arab astronomers]] such as [[Muhammad ibn Jābir al-Harrānī al-Battānī|Al Battani]] and [[Nasir al-Din al-Tusi]].{{sfnp|Boyer|1991|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 scholars in Renaissance|Byzantine Greek scholar]] cardinal [[Basilios Bessarion]] with whom he lived for several years.<ref>{{cite web |title=Johann Müller Regiomontanus |website=[[MacTutor History of Mathematics archive]] |url=https://mathshistory.st-andrews.ac.uk/Biographies/Regiomontanus/ |access-date=2021-01-08}}</ref> At the same time, another translation of the ''Almagest'' from Greek into Latin was completed by the Cretan [[George of Trebizond]].<ref>N.G. Wilson (1992). ''From Byzantium to Italy. Greek Studies in the Italian Renaissance'', London. {{isbn|0-7156-2418-0}}</ref> 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.<ref>{{cite book | last = Grattan-Guinness | first = Ivor | year = 1997 | title = The Rainbow of Mathematics: A History of the Mathematical Sciences | publisher = W.W. Norton | isbn = 978-0-393-32030-5}}</ref> [[Bartholomaeus Pitiscus]] was the first to use the word, publishing his ''Trigonometria'' in 1595.<ref>{{cite book|author=Robert E. Krebs |title=Groundbreaking Scientific Experiments, Inventions, and Discoveries of the Middle Ages and the Renaissance |url=https://books.google.com/books?id=MTXdplfiz-cC&pg=PA153 |year=2004 |publisher=Greenwood Publishing Group |isbn=978-0-313-32433-8 |page=153}}</ref> [[Gemma Frisius]] described for the first time the method of [[triangulation]] still used today in surveying. It was [[Leonhard Euler]] who fully incorporated [[complex number]]s into trigonometry. The works of the Scottish mathematicians [[James Gregory (astronomer and mathematician)|James Gregory]] in the 17th century and [[Colin Maclaurin]] in the 18th century were influential in the development of [[trigonometric series]].<ref>William Bragg{{Cite book|last=Ewald|first=William (2007). ''[Bragg|url=https://books.google.com/books?id=AcuF0w-Qg08C&pg=PA93 |title=From Kant to Hilbert Volume 1: aA sourceSource bookBook in the foundationsFoundations of mathematics]''.Mathematics|date=2005-04-21|publisher=OUP [[Oxford University Press US]]. p. 93. {{isbn|isbn=978-0-19-850535152309-30|language=en |page=93}}</ref> Also in the 18th century, [[Brook Taylor]] defined the general [[Taylor series]].<ref>Kelly{{Cite book|last=Dempski (2002). ''[|first=Kelly|url=https://books.google.com/books?id=zxdigX-KSZYC&pg=PA29 |title=Focus on Curves and Surfaces]''.|date=November p.2002|publisher=Premier 29. {{isbnPress|isbn=978-1-59200-007-X4|language=en |page=29}}</ref>

Trigonometric ratios are the ratios between edges of a right triangle. These ratios aredepend givenonly byon theone followingacute [[trigonometric function]]sangle of the knownright angle ''A''triangle, wheresince ''a'',any ''two b''right andtriangles ''h''with referthe tosame theacute lengthsangle ofare the[[Similarity sides(geometry)|similar]].<ref inname="StewartRedlin2015">{{cite thebook|author1=James accompanyingStewart|author2=Lothar figureRedlin|author3=Saleem Watson|title=Algebra and Trigonometry|url=https://books.google.com/books?id=uJqaBAAAQBAJ&pg=PA448|date=16 January 2015|publisher=Cengage Learning|isbn=978-1-305-53703-3|page=448}}</ref>

* '''[[Cosine]]''' function (denoted cos), defined as the ratio of the [[adjacent side (right triangle)|adjacent]] leg (the side of the triangle joining the angle to the right angle) to the hypotenuse.

* '''[[Tangent (trigonometric function)|Tangent]]''' function (denoted tan), defined as the ratio of the opposite leg to the adjacent leg.

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|Mnemonics]].

The [[Multiplicative inverse|reciprocals]] of these functionsratios are named the '''cosecant''' (csc), '''secant''' (sec), and '''cotangent''' (cot), respectively:

[[File:Sin-cos-defn-I.png|right|thumb|Fig. 1a – Sine and cosine of an angle θ defined using the unit circle.]]

[[File:Math Trigonometry Unit Circle Rotation WiseSign Indication.pngsvg|thumb|Indication of clockwisethe sign and counterclockwise amountsamount of key rotationsangles inaccording degrees,to inrotation the unit circle.direction]]

Trigonometric ratios can also be represented using the [[unit circle]], which is the circle of radius 1 centered at the origin in the plane.<ref name="CohenTheodore2009">{{cite book|author1=David Cohen|author2=Lee B. Theodore|author3=David Sklar|title=Precalculus: A Problems-Oriented Approach, Enhanced Edition|url=https://books.google.com/books?id=-ZXNfthUCOMC|date=17 July 2009|publisher=Cengage Learning|isbn=978-1-4390-4460-5}}</ref> In this setting, the [[Angle#Positive and negative angles|terminal side]] of an angle ''A'' placed in [[Angle#Positive and negative angles|standard position]] will intersect the unit circle in a point (x,y), where <math>x = \cos A </math> and <math>y = \sin A </math>.<ref name="CohenTheodore2009" /> This representation allows for the calculation of commonly found trigonometric values, such as those in the following table:<ref name="Kelley2002">{{cite book|author=W. Michael Kelley|title=The Complete Idiot's Guide to Calculus|url=https://books.google.com/books?id=H-0L9Dxor6sC&pg=PA45|year=2002|publisher=Alpha Books|isbn=978-0-02-864365-6|page=45}}</ref>

Trigonometric functions were among the earliest uses for [[mathematical table]]s.<ref name="Campbell-KellyCampbell-Kelly2003">{{cite book|author1=Martin Campbell-Kelly|author2=Professor Emeritus of Computer Science Martin Campbell-Kelly|author3=Visiting Fellow Department of Computer Science [[Mary Croarken]]|author4author3= Raymond Flood|author5author4= [[Eleanor Robson]]|title=The History of Mathematical Tables: From Sumer to Spreadsheets|title-link= The History of Mathematical Tables |date=2 October 2003|publisher=OUP Oxford|isbn=978-0-19-850841-0}}</ref> 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.<ref name="DonovanGimmestad1980">{{cite book|author1=George S. Donovan|author2=Beverly Beyreuther Gimmestad|title=Trigonometry with calculators|url=https://books.google.com/books?id=zUruGK7TOTYC|year=1980|publisher=Prindle, Weber & Schmidt|isbn=978-0-87150-284-1}}</ref> [[Slide rule]]s had special scales for trigonometric functions.<ref name="Middlemiss1945">{{cite book|author=Ross Raymond Middlemiss|title=Instructions for Post-trig and Mannheim-trig Slide Rules|url=https://books.google.com/books?id=OH0_AAAAYAAJ|year=1945|publisher=Frederick Post Company}}</ref>

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 (geometry)#In trigonometry|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{{sfnp|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}})}}),{{sfnp|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.

The sine and cosine functions are fundamental to the theory of [[periodic function]]s,<ref name="MorscheBerg2003">{{cite book|author1=H. G. ter Morsche|author2=J. C. van den Berg|author3=E. M. van de Vrie|title=Fourier and Laplace Transforms|url=https://books.google.com/books?id=frT5_rfyO4IC&pg=PA61|date=7 August 2003|publisher=Cambridge University Press|isbn=978-0-521-53441-3|page=61}}</ref> such as those that describe [[sound]] and [[light]] waves. [[Jean-Baptiste Joseph Fourier|Fourier]] discovered that every [[continuous function|continuous]], [[periodic function]] could be described as an [[infinite series|infinite sum]] of trigonometric functions.

Trigonometry is useful in many [[physical science]]s,<ref name="BornsteinInc1966">{{cite book|author1=Lawrence Bornstein|author2=Basic Systems, Inc|title=Trigonometry for the Physical Sciences|url=https://books.google.com/books?id=6I1GAAAAYAAJ|year=1966|publisher=Appleton-Century-Crofts}}</ref> including [[acoustics]],<ref name="SchillerWurster1988">{{cite book|author1=John J. Schiller|author2=Marie A. Wurster|title=College Algebra and Trigonometry: Basics Through Precalculus|url=https://books.google.com/books?id=-CXYAAAAMAAJ|year=1988|publisher=Scott, Foresman|isbn=978-0-673-18393-4}}</ref> and [[optics]].<ref name="SchillerWurster1988" /> In these areas, they are used to describe [[sound waves|sound]] and [[light wave]]s, and to solve boundary- and transmission-related problems.<ref name="Towne2014">{{cite book|author=Dudley H. Towne|title=Wave Phenomena|url=https://books.google.com/books?id=uZgJCAAAQBAJ|date=5 May 2014|publisher=Dover Publications|isbn=978-0-486-14515-0}}</ref>

Other fields that use trigonometry or trigonometric functions include [[music theory]],<ref name="HeinemanTarwater1992">{{cite book|author1=E. Richard Heineman|author2=J. Dalton Tarwater|title=Plane Trigonometry|url=https://books.google.com/books?id=Hi7YAAAAMAAJ|date=1 November 1992|publisher=McGraw-Hill|isbn=978-0-07-028187-5}}</ref> [[geodesy]], [[audio synthesis]],<ref name="KahrsBrandenburg2006">{{cite book|author1=Mark Kahrs|author2=Karlheinz Brandenburg|title=Applications of Digital Signal Processing to Audio and Acoustics|url=https://books.google.com/books?id=UFwKBwAAQBAJ&pg=PA404|date=18 April 2006|publisher=Springer Science & Business Media|isbn=978-0-306-47042-4|page=404}}</ref> [[architecture]],<ref name="WilliamsOstwald2015">{{cite book|author1=Kim Williams|author1-link=Kim Williams (architect)|author2=Michael J. Ostwald|title=Architecture and Mathematics from Antiquity to the Future: Volume I: Antiquity to the 1500s|url=https://books.google.com/books?id=fWKYBgAAQBAJ&pg=PA260|date=9 February 2015|publisher=Birkhäuser|isbn=978-3-319-00137-1|page=260}}</ref> [[electronics]],<ref name="HeinemanTarwater1992" /> [[biology]],<ref name="Foulder2019">{{cite book|author=Dan Foulder|title=Essential Skills for GCSE Biology|url=https://books.google.com/books?id=teF6DwAAQBAJ&pg=PT78|date=15 July 2019|publisher=Hodder Education|isbn=978-1-5104-6003-4|page=78}}</ref> [[medical imaging]] ([[CT scan]]s and [[ultrasound]]),<ref name="BeolchiKuhn1995">{{cite book|author1=Luciano Beolchi|author2=Michael H. Kuhn|title=Medical Imaging: Analysis of Multimodality 2D/3D Images|url=https://books.google.com/books?id=HnRD08tDmlsC&pg=PA122|year=1995|publisher=IOS Press|isbn=978-90-5199-210-6|page=122}}</ref> [[chemistry]],<ref name="Ladd2014">{{cite book|author=Marcus Frederick Charles Ladd|title=Symmetry of Crystals and Molecules|url=https://books.google.com/books?id=7L3DAgAAQBAJ&pg=PA13|year=2014|publisher=Oxford University Press|isbn=978-0-19-967088-8|page=13}}</ref> [[number theory]] (and hence [[cryptology]]),<ref name="ArkhipovChubarikov2008">{{cite book|author1=Gennady I. Arkhipov|author2=Vladimir N. Chubarikov|author3=Anatoly A. Karatsuba|title=Trigonometric Sums in Number Theory and Analysis|url=https://books.google.com/books?id=G8j4Kqw45jwC|date=22 August 2008|publisher=Walter de Gruyter|isbn=978-3-11-019798-3}}</ref> [[seismology]],<ref name="SchillerWurster1988" /> [[meteorology]],<ref>{{cite book|title=Study Guide for the Course in Meteorological Mathematics: Latest Revision, Feb. 1, 1943|url=https://books.google.com/books?id=j-ow4TBWAbcC|year=1943}}</ref> [[oceanography]],<ref name="SearsMerriman1980">{{cite book|author1=Mary Sears|author2=Daniel Merriman|author3=Woods Hole Oceanographic Institution|title=Oceanography, the past|url=https://books.google.com/books?id=Z7dPAQAAIAAJ|year=1980|publisher=Springer-Verlag|isbn=978-0-387-90497-9}}</ref> [[image compression]],<ref>{{Cite web|url=https://www.w3.org/Graphics/JPEG/itu-t81.pdf|title=JPEG Standard (JPEG ISO/IEC 10918-1 ITU-T Recommendation T.81)|date=1993|publisher=[[International Telecommunication Union]]|access-date=6 April 2019}}</ref> [[phonetics]],<ref name="Malmkjaer2009">{{cite book|author=Kirsten Malmkjaer|title=The Routledge Linguistics Encyclopedia|url=https://books.google.com/books?id=O459AgAAQBAJ&pg=PA1|date=4 December 2009|publisher=Routledge|isbn=978-1-134-10371-3|page=1}}</ref> [[economics]],<ref name="Dadkhah2011">{{cite book|author=Kamran Dadkhah|title=Foundations of Mathematical and Computational Economics|url=https://books.google.com/books?id=Z76b-TGhs9sC&pg=PA46|date=11 January 2011|publisher=Springer Science & Business Media|isbn=978-3-642-13748-8|page=46}}</ref> [[electrical engineering]], [[mechanical engineering]], [[civil engineering]],<ref name="HeinemanTarwater1992" /> [[computer graphics]],<ref name="Griffith2012" /> [[cartography]],<ref name="HeinemanTarwater1992" /> [[crystallography]]<ref name="Griffin1841">{{cite book|author=John Joseph Griffin|title=A System of Crystallography, with Its Application to Mineralogy|url=https://archive.org/details/asystemcrystall03grifgoog|year=1841|publisher=R. Griffin|page=[https://archive.org/details/asystemcrystall03grifgoog/page/n157 119]}}</ref> and [[game development]].<ref name="Griffith2012">{{cite book|author=Christopher Griffith|title=Real-World Flash Game Development: How to Follow Best Practices AND Keep Your Sanity|url=https://archive.org/details/realworldflashga0000grif|url-access=registration|date=12 November 2012|publisher=CRC Press|isbn=978-1-136-13702-0|page=[https://archive.org/details/realworldflashga0000grif/page/153 153]}}</ref>

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).<ref>[https://www.youtube.com/watch?v=CaTF4QZ94Fk&t=245 Lecture 3 | Quantum Entanglements, Part 1 (Stanford)], [[Leonard Susskind]], trigonometry in five minutes, law of sin, cos, euler formula 2006-10-09.</ref>

Given two sides ''a'' and ''b'' and the angle between the sides ''C'', the [[area of a triangle|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:<ref name="Young2010">{{cite book|author=Cynthia Y. Young|author-link=Cynthia Y. Young|title=Precalculus|url=https://books.google.com/books?id=9HRLAn326zEC&pg=PA435|date=19 January 2010|publisher=John Wiley & Sons|isbn=978-0-471-75684-2|page=435}}</ref>

:<math>\cot^2 A + 1 = \csc^2 A \ </math>