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{{Trigonometry}}

'''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]] concerned with relationships between [[angle]]s and side lengths of triangles. In particular, the [[trigonometric functions]] 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}}}}

Throughout history, trigonometry has been applied in areas such as [[geodesy]], [[surveying]], [[celestial mechanics]], and [[navigation]].<ref name="Hackley1853">{{cite book|author=Charles William Hackley|title=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|url=https://books.google.com/books?id=Q4FTAAAAYAAJ|year=1853|publisher=G. P. Putnam}}</ref>

Trigonometry is known for its many [[identity (mathematics)|identities]]. These 
[[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> 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>

== History ==
{{main|History of trigonometry}}
[[File:Head of Hipparchus (cropped).jpg|thumb|upright=0.8|right|[[Hipparchus]], credited with compiling the first [[Trigonometric tables|trigonometric table]], has been described as "the [[List of people considered father or mother of a field|father]] of trigonometry".{{sfnp|Boyer|1991 |loc="Greek Trigonometry and Mensuration" |p=[https://archive.org/details/historyofmathema00boye/page/162 162]}}]]

[[Sumer]]ian astronomers studied angle measure, using a division of circles into 360 degrees.<ref>{{cite book |title=Cambridge IGCSE Core Mathematics |edition=4th |first1=Ric |last1=Pimentel |first2=Terry |last2=Wall |publisher=Hachette UK |year=2018 |isbn=978-1-5104-2058-8 |page=275 |url=https://books.google.com/books?id=WcJWDwAAQBAJ}} [https://books.google.com/books?id=WcJWDwAAQBAJ&pg=PA275 Extract of page 275]</ref> They, and later the [[Babylonians]], studied the ratios of the sides of [[Similarity (geometry)|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 [[Nubia|ancient Nubians]] used a similar method.<ref>{{cite book|author=Otto Neugebauer |title=A history of ancient mathematical astronomy. 1 |url=https://books.google.com/books?id=vO5FCVIxz2YC&pg=PA744 |year=1975 |publisher=Springer-Verlag |isbn=978-3-540-06995-9 |page=744}}</ref>

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 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 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 definition of the sine is first attested in the ''[[Surya Siddhanta]]'', and its properties were further documented in 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, in the work of Persian mathematician [[Abū al-Wafā' al-Būzjānī]], all six [[trigonometric functions]] were used.{{sfn|Boyer|1991|p=238}} Abu al-Wafa had sine tables in 0.25° increments, to 8 decimal places of accuracy, and accurate tables of tangent values.{{sfn|Boyer|1991|p=238}} He also made important innovations in [[spherical trigonometry]]<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): 74–83</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. 208–214) 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> 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>{{Cite book|last=Ewald|first=William Bragg|url=https://books.google.com/books?id=AcuF0w-Qg08C&pg=PA93|title=From Kant to Hilbert Volume 1: A Source Book in the Foundations of Mathematics|date=2005-04-21|publisher=OUP Oxford|isbn=978-0-19-152309-0|language=en |page=93}}</ref> Also in the 18th century, [[Brook Taylor]] defined the general [[Taylor series]].<ref>{{Cite book|last=Dempski|first=Kelly|url=https://books.google.com/books?id=zxdigX-KSZYC&pg=PA29|title=Focus on Curves and Surfaces|date=November 2002|publisher=Premier Press|isbn=978-1-59200-007-4|language=en |page=29}}</ref>

== Trigonometric ratios ==
{{main|Trigonometric function}}
[[File:Trigonometry triangle.svg|thumb|In this right triangle: {{math|1= sin ''A'' = ''a''/''h'';}} {{math|1= cos ''A'' = ''b''/''h'';}} {{math|1= tan ''A'' = ''a''/''b''.}}]]
Trigonometric ratios are the ratios between edges of a right triangle. These ratios depend only on one acute angle of the right triangle, since any two right triangles with the same acute angle are [[Similarity (geometry)|similar]].<ref name="StewartRedlin2015">{{cite book|author1=James Stewart|author2=Lothar Redlin|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> 

So, these ratios define [[function (mathematics)|function]]s of this angle that are called [[trigonometric function]]s. Explicitly, they are defined below as functions of the known angle ''A'', where ''a'', '' b'' and ''h'' refer to the lengths of the sides in the accompanying figure.

In the following definitions, 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]].

* '''[[Sine]]''' (denoted sin), defined as the ratio of the side opposite the angle to the hypotenuse.
:: <math>\sin A=\frac{\textrm{opposite}}{\textrm{hypotenuse}}=\frac{a}{h}.</math>
* '''[[Cosine]]''' (denoted cos), defined as the ratio of the adjacent leg (the side of the triangle joining the angle to the right angle) to the hypotenuse.
:: <math>\cos A=\frac{\textrm{adjacent}}{\textrm{hypotenuse}}=\frac{b}{h}.</math>
* '''[[Tangent (trigonometric function)|Tangent]]''' (denoted tan), defined as the ratio of the opposite leg to the adjacent leg.

::<math>\tan A=\frac{\textrm{opposite}}{\textrm{adjacent}}=\frac{a}{b}=\frac{a/h}{b/h}=\frac{\sin A}{\cos A}.</math>

The [[Multiplicative inverse|reciprocals]] of these ratios are named the '''cosecant''' (csc), '''secant''' (sec), and '''cotangent''' (cot), respectively:
:<math>\csc A=\frac{1}{\sin A}=\frac{\textrm{hypotenuse}}{\textrm{opposite}}=\frac{h}{a} ,</math>

:<math>\sec A=\frac{1}{\cos A}=\frac{\textrm{hypotenuse}}{\textrm{adjacent}}=\frac{h}{b} ,</math>

:<math>\cot A=\frac{1}{\tan A}=\frac{\textrm{adjacent}}{\textrm{opposite}}=\frac{\cos A}{\sin A}=\frac{b}{a} .</math>

The cosine, cotangent, and cosecant are so named because they are respectively the sine, tangent, and secant of the complementary angle abbreviated to "co-".<ref name="JardineShell-Gellasch2011">{{cite book|author1=Dick Jardine|author2=Amy Shell-Gellasch|title=Mathematical Time Capsules: Historical Modules for the Mathematics Classroom|url=https://books.google.com/books?id=Aa_VmrWEdvEC&pg=PA182|year=2011|publisher=MAA|isbn=978-0-88385-984-1|page=182}}</ref>

With these functions, one can answer virtually all questions about arbitrary triangles by using the [[law of sines]] and the [[law of cosines]].<ref name="ForsethBurger2008">{{cite book|author1=Krystle Rose Forseth |author2=Christopher Burger|author3=Michelle Rose Gilman|author4=Deborah J. Rumsey|title=Pre-Calculus For Dummies|url=https://books.google.com/books?id=nfwGEJaLlgsC&pg=PA218|year=2008|publisher=John Wiley & Sons|isbn=978-0-470-16984-1|page=218}}</ref> 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.

=== {{anchor|SOHCAHTOA}}Mnemonics ===
<!--------
  Please note: The one mnemonic already provided is enough. DO NOT ADD YOUR OWN MNEMONICS, especially ridiculous ones such as "Sex on holidays Comes after having Tons of alcohol", etcetera. Such edits DO NOT improve the article but add obfuscation and, in some cases, constitute subversive vandalism. They WILL be reverted.
 --------->
{{main|Mnemonics in trigonometry}}
A common use of [[mnemonic]]s 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:<ref>{{MathWorld|ref=none|title=SOHCAHTOA|urlname=SOHCAHTOA}}</ref>

:'''S'''ine = '''O'''pposite ÷ '''H'''ypotenuse
:'''C'''osine = '''A'''djacent ÷ '''H'''ypotenuse
:'''T'''angent = '''O'''pposite ÷ '''A'''djacent

One way to remember the letters is to sound them out phonetically (i.e. {{IPAc-en|ˌ|s|oʊ|k|ə|ˈ|t|oʊ|ə}} {{respell|SOH|kə|TOH|ə}}, similar to [[Krakatoa]]).<ref>{{Cite book |last=Humble |first=Chris |url=https://www.worldcat.org/oclc/47985033 |title=Key Maths : GCSE, Higher. |date=2001 |publisher=Stanley Thornes Publishers |others=Fiona McGill |isbn=0-7487-3396-5 |location=Cheltenham |oclc=47985033|page=51}}</ref> Another method is to expand the letters into a sentence, such as "'''S'''ome '''O'''ld '''H'''ippie '''C'''aught '''A'''nother '''H'''ippie '''T'''rippin' '''O'''n '''A'''cid".<ref>A sentence more appropriate for high schools is "'''S'''ome '''O'''ld '''H'''orse '''C'''ame '''A'''''''H'''opping '''T'''hrough '''O'''ur '''A'''lley". {{cite book |title=Memory: A Very Short Introduction|first=Jonathan K.|last=Foster|publisher=Oxford|year=2008|isbn=978-0-19-280675-8|page=128}}</ref>

===The unit circle and common trigonometric values===
{{main|Unit circle}}
[[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 Sign Indication.svg|thumb|Indication of the sign and amount of key angles according to rotation 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>

{| class="wikitable"
|-
! Function
! 0
! <math>\pi/6</math>
! <math>\pi/4</math>
! <math>\pi/3</math>
! <math>\pi/2</math>
! <math>2\pi/3</math>
! <math>3\pi/4</math>
! <math>5\pi/6</math>
! <math>\pi</math>
|-
! sine
| <math>0</math>
| <math>1/2</math>
| <math>\sqrt{2}/2</math>
| <math>\sqrt{3}/2</math>
| <math>1</math>
| <math>\sqrt{3}/2</math>
| <math>\sqrt{2}/2</math>
| <math>1/2</math>
| <math>0</math>
|-
! cosine
|<math>1</math>
| <math>\sqrt{3}/2</math>
| <math>\sqrt{2}/2</math>
| <math>1/2</math>
| <math>0</math>
| <math>-1/2</math>
| <math>-\sqrt{2}/2</math>
| <math>-\sqrt{3}/2</math>
| <math>-1</math>
|-
! tangent
| <math>0</math>
| <math>\sqrt{3}/3</math>
| <math>1</math>
| <math>\sqrt{3}</math>
| undefined
| <math>-\sqrt{3}</math>
| <math>-1</math>
| <math>-\sqrt{3}/3</math>
| <math>0</math>
|-
! secant
| <math>1</math>
| <math>2\sqrt{3}/3</math>
| <math>\sqrt{2}</math>
| <math>2</math>
| undefined
| <math>-2</math>
| <math>-\sqrt{2}</math>
| <math>-2\sqrt{3}/3</math>
| <math>-1</math>
|-
! cosecant
| undefined
| <math>2</math>
| <math>\sqrt{2}</math>
| <math>2\sqrt{3}/3</math>
| <math>1</math>
| <math>2\sqrt{3}/3</math>
| <math>\sqrt{2}</math>
| <math>2</math>
| undefined
|-
! cotangent
| undefined
| <math>\sqrt{3}</math>
| <math>1</math>
| <math>\sqrt{3}/3</math>
| <math>0</math>
| <math>-\sqrt{3}/3</math>
| <math>-1</math>
| <math>-\sqrt{3}</math>
| undefined
|}

== Trigonometric functions of real or complex variables ==
{{main|Trigonometric function}}

Using the [[unit circle]], one can extend the definitions of trigonometric ratios to all positive and negative arguments<ref name="Olive2003">{{cite book|author=Jenny Olive|title=Maths: A Student's Survival Guide: A Self-Help Workbook for Science and Engineering Students|url=https://books.google.com/books?id=_Ir7euRke_oC&pg=PA175|date=18 September 2003|publisher=Cambridge University Press|isbn=978-0-521-01707-7|page=175}}</ref> (see [[trigonometric function]]).

=== Graphs of trigonometric functions ===

The following table summarizes the properties of the graphs of the six main trigonometric functions:<ref name="Attenborough2003">{{cite book|author=Mary P Attenborough|title=Mathematics for Electrical Engineering and Computing|url=https://books.google.com/books?id=CcwBLa1G8BUC&pg=PA418|date=30 June 2003|publisher=Elsevier|isbn=978-0-08-047340-6|page=418}}</ref><ref name="LarsonEdwards2008">{{cite book|author1=Ron Larson|author2=Bruce H. Edwards|title=Calculus of a Single Variable|url=https://books.google.com/books?id=gR7nGg5_9xcC&pg=PA21|date=10 November 2008|publisher=Cengage Learning|isbn=978-0-547-20998-2|page=21}}</ref>

{| class="wikitable"
|-
! Function
! Period
! Domain
! Range
! Graph
|-
! sine
| <math>2\pi</math>
| <math>(-\infty,\infty)</math>
| <math>[-1,1]</math>
| [[Image:Sine one period.svg|200 px]]
|-
! cosine
| <math>2\pi</math>
| <math>(-\infty,\infty)</math>
| <math>[-1,1]</math>
| [[Image:Cosine one period.svg|200 px]]
|-
! tangent
| <math>\pi</math>
| <math>x \neq \pi/2+n\pi</math>
| <math>(-\infty,\infty)</math>
| [[Image:Tangent-plot.svg|200 px]]
|-
! secant
| <math>2\pi</math>
| <math>x \neq \pi/2 + n\pi</math>
| <math>(-\infty,-1] \cup [1,\infty)</math>
| [[Image:Secant.svg|200 px]]
|-
! cosecant
| <math>2\pi</math>
| <math>x \neq n\pi</math>
| <math>(-\infty,-1] \cup [1,\infty)</math>
| [[Image:Cosecant.svg|200 px]]
|-
! cotangent
| <math>\pi</math>
| <math>x \neq n\pi</math>
| <math>(-\infty,\infty)</math>
| [[Image:Cotangent.svg|200 px]]
|}

=== Inverse trigonometric functions ===
{{main|Inverse trigonometric functions}}

Because the six main trigonometric functions are periodic, they are not [[injective function|injective]] (or, 1 to 1), and thus are not invertible. By [[Restriction (mathematics)|restricting]] the domain of a trigonometric function, however, they can be made invertible.<ref name="BremiganBremigan2011">{{cite book|author1=Elizabeth G. Bremigan|author2=Ralph J. Bremigan|author3=John D. Lorch|title=Mathematics for Secondary School Teachers|url=https://books.google.com/books?id=OfFEC5drTVMC&pg=PR48|year=2011|publisher=MAA|isbn=978-0-88385-773-1}}</ref>{{rp|48ff}}

The names of the inverse trigonometric functions, together with their domains and range, can be found in the following table:<ref name="BremiganBremigan2011"/>{{rp|48ff}}<ref name="BrokateManchanda2019">{{cite book|author1=Martin Brokate|author2=Pammy Manchanda|author3=Abul Hasan Siddiqi|title=Calculus for Scientists and Engineers|url=https://books.google.com/books?id=7DenDwAAQBAJ&pg=PA521|date=3 August 2019|publisher=Springer|isbn=9789811384646}}</ref>{{rp|521ff}}
{| class="wikitable" style="text-align:center"
|- 
!Name
!Usual notation
!Definition
!Domain of ''x'' for real result
!Range of usual principal value <br /> ([[radian]]s)
!Range of usual principal value <br /> ([[Degree (angle)|degrees]])
|-
| '''arcsine''' || ''y'' = {{math|arcsin(''x'')}} || ''x'' = {{math|[[sine|sin]](''y'')}} || −1 ≤ ''x'' ≤ 1 || −{{sfrac|{{pi}}|2}} ≤ ''y'' ≤ {{sfrac|{{pi}}|2}} || −90° ≤ ''y'' ≤ 90°
|-
| '''arccosine''' || ''y'' = {{math|arccos(''x'')}} || ''x'' = {{math|[[cosine|cos]](''y'')}} || −1 ≤ ''x'' ≤ 1 || 0 ≤ ''y'' ≤ {{pi}} || 0° ≤ ''y'' ≤ 180°
|-
| '''arctangent''' || ''y'' = {{math|arctan(''x'')}} || ''x'' = {{math|[[Tangent (trigonometry)|tan]](''y'')}} || all real numbers || −{{sfrac|{{pi}}|2}} < ''y'' < {{sfrac|{{pi}}|2}} || −90° < ''y'' < 90°
|-
| '''arccotangent''' || ''y'' = {{math|arccot(''x'')}} ||''x'' = {{math|[[cotangent|cot]](''y'')}} || all real numbers
| 0 < ''y'' < {{pi}} || 0° < ''y'' < 180°
|-
| '''arcsecant''' || ''y'' = {{math|arcsec(''x'')}} || ''x'' = {{math|[[Secant (trigonometry)|sec]](''y'')}} || ''x'' ≤ −1 or 1 ≤ ''x'' || 0 ≤ ''y'' < {{sfrac|{{pi}}|2}} or {{sfrac|{{pi}}|2}} < ''y'' ≤ {{pi}} || 0° ≤ ''y'' < 90° or 90° < ''y'' ≤ 180°
|-
| '''arccosecant''' || ''y'' = {{math|arccsc(''x'')}} || ''x'' = {{math|[[cosecant|csc]](''y'')}} || ''x'' ≤ −1 or 1 ≤ ''x'' || −{{sfrac|{{pi}}|2}} ≤ ''y'' < 0 or 0 < ''y'' ≤ {{sfrac|{{pi}}|2}} || −90° ≤ ''y'' < 0° or 0° < ''y'' ≤ 90°
|-
|}

=== 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:<ref name="Lang2013">{{cite book|author=Serge Lang|title=Complex Analysis|url=https://books.google.com/books?id=0qx3BQAAQBAJ&pg=PA63|date=14 March 2013|publisher=Springer|isbn=978-3-642-59273-7|page=63}}</ref>

:<math>
\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}
</math>

:<math>
\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}
</math>

With these definitions the trigonometric functions can be defined for [[complex number]]s.<ref name="Alessio2015">{{cite book|author=Silvia Maria Alessio|title=Digital Signal Processing and Spectral Analysis for Scientists: Concepts and Applications|url=https://books.google.com/books?id=ja8vCwAAQBAJ&pg=PA339|date=9 December 2015|publisher=Springer|isbn=978-3-319-25468-5|page=339}}</ref> When extended as functions of real or complex variables, the following [[Euler's formula|formula]] holds for the complex exponential:

: <math>e^{x+iy} = e^x(\cos y + i \sin y).</math>

This complex exponential function, written in terms of trigonometric functions, is particularly useful.<ref name="RAJESWARIRAO2014">{{cite book|author1=K. RAJA RAJESWARI|author2=B. VISVESVARA RAO|title=SIGNALS AND SYSTEMS|url=https://books.google.com/books?id=QZBeBAAAQBAJ&pg=PA263|date=24 March 2014|publisher=PHI Learning|isbn=978-81-203-4941-4|page=263}}</ref><ref name="Stillwell2010">{{cite book|author=John Stillwell|title=Mathematics and Its History|url=https://books.google.com/books?id=3bE_AAAAQBAJ&pg=PA313|date=23 July 2010|publisher=Springer Science & Business Media|isbn=978-1-4419-6053-5|page=313}}</ref>

=== Calculating trigonometric functions ===
{{main|Trigonometric tables}}
Trigonometric functions were among the earliest uses for [[mathematical table]]s.<ref name="Campbell-KellyCampbell-Kelly2003">{{cite book|author1=Martin Campbell-Kelly|author2=[[Mary Croarken]]|author3= Raymond Flood|author4= [[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>

[[Scientific calculator]]s have buttons for calculating the main trigonometric functions (sin, cos, tan, and sometimes [[Euler's formula|cis]] and their inverses).<ref>{{cite magazine |title=Calculator keys—what they do |magazine=Popular Science |url=https://books.google.com/books?id=1T4ORu6EICkC&pg=PA125 |date=April 1974|publisher=Bonnier Corporation|page=125}}</ref> Most allow a choice of angle measurement methods: [[degree (angle)|degrees]], radians, and sometimes [[gradians]]. Most computer [[programming language]]s provide function libraries that include the trigonometric functions.<ref>{{cite book|author1=Steven S. Skiena |author2=Miguel A. Revilla|title=Programming Challenges: The Programming Contest Training Manual |url=https://books.google.com/books?id=dNoLBwAAQBAJ&pg=PA302 |date=18 April 2006|publisher=Springer Science & Business Media|isbn=978-0-387-22081-9|page=302}}</ref> The [[floating point unit]] hardware incorporated into the microprocessor chips used in most personal computers has built-in instructions for calculating trigonometric functions.<ref>{{cite book |title=Intel® 64 and IA-32 Architectures Software Developer's Manual Combined Volumes: 1, 2A, 2B, 2C, 3A, 3B and 3C |year=2013 |publisher=Intel |url=http://download.intel.com/products/processor/manual/325462.pdf}}</ref>

=== Other trigonometric functions ===
{{main|Trigonometric functions#History}}

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 sin<sup>2</sup>({{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'') = sin<sup>2</sup>({{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.

== Applications ==
{{main|Uses of trigonometry}}

===Astronomy===
{{main|Astronomy}}

For centuries, spherical trigonometry has been used for locating solar, lunar, and stellar positions,<ref name="Gregory1816">{{cite book|author=Olinthus Gregory|title=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|url=https://books.google.com/books?id=j3sAAAAAMAAJ|year=1816 |publisher=Baldwin, Cradock, and Joy}}</ref> predicting eclipses, and describing the orbits of the planets.<ref>{{cite journal |last=Neugebauer |first=Otto |title=Mathematical methods in ancient astronomy |journal=Bulletin of the American Mathematical Society |volume=54 |issue=11 |year=1948 |pages=1013–1041|doi=10.1090/S0002-9904-1948-09089-9 |doi-access=free }}</ref>

In modern times, the technique of [[triangulation]] is used in [[astronomy]] to measure the distance to nearby stars,<ref name="SeedsBackman2009">{{cite book|author1=Michael Seeds|author2=Dana Backman|title=Astronomy: The Solar System and Beyond|url=https://books.google.com/books?id=DajpkyXS-NUC&pg=PT254|date=5 January 2009|publisher=Cengage Learning|isbn=978-0-495-56203-0|page=254}}</ref> as well as in [[satellite navigation system]]s.<ref name="Willers2018"/>

===Navigation===
{{main|Navigation}}
[[File:Frieberger drum marine sextant.jpg|thumb|[[Sextant]]s are used to measure the angle of the sun or stars with respect to the horizon. Using trigonometry and a [[marine chronometer]], the position of the ship can be determined from such measurements.]]

Historically, trigonometry has been used for locating latitudes and longitudes of sailing vessels, plotting courses, and calculating distances during navigation.<ref name="Sabine1800">{{cite book|author=John Sabine|title=The Practical Mathematician, Containing Logarithms, Geometry, Trigonometry, Mensuration, Algebra, Navigation, Spherics and Natural Philosophy, Etc|url=https://books.google.com/books?id=d_9eAAAAcAAJ&pg=PR1|date=1800|page=1}}</ref>

Trigonometry is still used in navigation through such means as the [[Global Positioning System]] and [[artificial intelligence]] for [[autonomous vehicle]]s.<ref>{{cite book|author1=Mordechai Ben-Ari |author2=Francesco Mondada |year=2018<!--Google is wrong again--> |title=Elements of Robotics |url=https://books.google.com/books?id=itpCDwAAQBAJ&pg=PA16 |publisher=Springer |isbn=978-3-319-62533-1 |page=16}}</ref>

===Surveying===
{{main|Surveying}}

In land [[surveying]], trigonometry is used in the calculation of lengths, areas, and relative angles between objects.<ref name="Perkins1853">{{cite book|author=George Roberts Perkins|title=Plane Trigonometry and Its Application to Mensuration and Land Surveying: Accompanied with All the Necessary Logarithmic and Trigonometric Tables|url=https://archive.org/details/planetrigonometr00perk|year=1853|publisher=D. Appleton & Company}}</ref>

On a larger scale, trigonometry is used in [[geography]] to measure distances between landmarks.<ref name="WithersLorimer2015">{{cite book|author1=Charles W. J. Withers|author2=Hayden Lorimer|title=Geographers: Biobibliographical Studies|url=https://books.google.com/books?id=eidTTrsyTr4C&pg=PA6|date=14 December 2015|publisher=A&C Black|isbn=978-1-4411-0785-5|page=6}}</ref>

===Periodic functions===
{{main|Fourier series|Fourier transform}}
[[File:Fourier series and transform.gif|frame|right|Function <math>s(x)</math> (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, <math>S(f)</math> (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 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.

Even non-periodic functions can be represented as an [[integral]] of sines and cosines through the [[Fourier transform]]. This has applications to [[quantum mechanics]]<ref name="Thaller2007">{{cite book|author=Bernd Thaller|title=Visual Quantum Mechanics: Selected Topics with Computer-Generated Animations of Quantum-Mechanical Phenomena|url=https://books.google.com/books?id=GOfjBwAAQBAJ&pg=PA15|date=8 May 2007|publisher=Springer Science & Business Media|isbn=978-0-387-22770-2|page=15}}</ref> and [[telecommunication|communication]]s,<ref name="Rahman2011">{{cite book|author=M. Rahman|title=Applications of Fourier Transforms to Generalized Functions|url=https://books.google.com/books?id=k_rdcKaUdr4C|year=2011|publisher=WIT Press|isbn=978-1-84564-564-9}}</ref> among other fields.

===Optics and acoustics===
{{main|optics|acoustics}}

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 applications ===

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>

== Identities ==
{{Main|List of trigonometric identities}}
[[File:Triangle ABC with Sides a b c 2.png|thumb|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.<ref name="Dugopolski2002">{{cite book|author=Dugopolski|title=Trigonometry I/E Sup|url=https://books.google.com/books?id=_dXVeYx_kgoC|date=July 2002|publisher=Addison Wesley|isbn=978-0-201-78666-8}}</ref>

Identities involving only angles are known as ''trigonometric identities''. Other equations, known as ''triangle identities'',<ref name="BOARD2015">{{cite book|author=V&S EDITORIAL BOARD|title=CONCISE DICTIONARY OF MATHEMATICS|url=https://books.google.com/books?id=TJQ3DwAAQBAJ&pg=PA288|date=6 January 2015|publisher=V&S Publishers|isbn=978-93-5057-414-0|page=288}}</ref> 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).

==== Law of sines ====
The '''[[law of sines]]''' (also known as the "sine rule") for an arbitrary triangle states:<ref name="Young2010"/>

:<math>\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R = \frac{abc}{2\Delta},</math>
where <math>\Delta</math> is the area of the triangle and ''R'' is the radius of the [[circumscribed circle]] of the triangle:

:<math>R = \frac{abc}{\sqrt{(a+b+c)(a-b+c)(a+b-c)(b+c-a)}}.</math>

==== 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:<ref name="Young2010" />

:<math>c^2=a^2+b^2-2ab\cos C ,</math>

or equivalently:

:<math>\cos C=\frac{a^2+b^2-c^2}{2ab}.</math>

==== 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.<ref name="Larson2010">{{cite book|author=Ron Larson|title=Trigonometry|url=https://books.google.com/books?id=KzALQF4QresC&pg=PA331|date=29 January 2010|publisher=Cengage Learning|isbn=978-1-4390-4907-5|page=331}}</ref> It is given by:

:<math>\frac{a-b}{a+b}=\frac{\tan\left[\tfrac{1}{2}(A-B)\right]}{\tan\left[\tfrac{1}{2}(A+B)\right]}</math>

==== Area ====
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>\mbox{Area} = \Delta = \frac{1}{2}a b\sin C</math>

===Trigonometric identities===

==== Pythagorean identities ====

The following trigonometric [[Identity (mathematics)|identities]] are related to the [[Pythagorean theorem]] and hold for any value:<ref>{{cite book |title=Technical Mathematics with Calculus |edition=illustrated |first1=John C. |last1=Peterson |publisher=Cengage Learning |year=2004 |isbn=978-0-7668-6189-3 |page=856 |url=https://books.google.com/books?id=PGuSDjHvircC}} [https://books.google.com/books?id=PGuSDjHvircC&pg=PA856 Extract of page 856]</ref>

:<math>\sin^2 A + \cos^2 A = 1 \ </math>

:<math>\tan^2 A + 1 = \sec^2 A \ </math>

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


The second and third equations are derived from dividing the first equation by <math>\cos^2{A}</math> and <math>\sin^2{A}</math>, respectively.

==== Euler's formula ====
[[Euler's formula]], which states that <math>e^{ix} = \cos x + i \sin x</math>, produces the following [[mathematical analysis|analytical]] identities for sine, cosine, and tangent in terms of ''[[e (mathematics)|e]]'' and the [[imaginary unit]] ''i'':
:<math>\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}}.</math>
<!-- Note: the expression of tan(x) has i in the numerator, not in the denominator, because the order of the terms (and thus the sign) of the numerator is changed w.r.t. the expression of sin(x). -->

==== Other trigonometric identities ====
{{main | List of trigonometric identities}}
Other commonly used trigonometric identities include the half-angle identities, the angle sum and difference identities, and the product-to-sum identities.<ref name="StewartRedlin2015" />

== See also ==
{{div col|colwidth=22em}}
* [[Aryabhata's sine table]]
* [[Generalized trigonometry]]
* [[Lénárt sphere]]
* [[List of triangle topics]]
* [[List of trigonometric identities]]
* [[Rational trigonometry]]
* [[Skinny triangle]]
* [[Small-angle approximation]]
* [[Trigonometric functions]]
* [[Unit circle]]
* [[Uses of trigonometry]]
{{div col end}}

== References ==
{{reflist}}

== Bibliography ==
* {{cite book |first=Carl B. |last=Boyer |author-link=Carl Benjamin Boyer |title=A History of Mathematics |edition=Second |publisher=John Wiley & Sons, Inc. |year=1991 |isbn=978-0-471-54397-8 |url-access=registration |url=https://archive.org/details/historyofmathema00boye }}
* {{cite book |last1=Nielsen |first1=Kaj L. |title=Logarithmic and Trigonometric Tables to Five Places |edition=2nd |location=New York |publisher=[[Barnes & Noble]] |date=1966 |lccn=61-9103}}
* {{cite book |last=Thurston |first=Hugh |year=1996 |title=Early Astronomy |publisher=Springer Science & Business Media |isbn=978-0-387-94822-5}}

==Further reading==
* {{Springer |title=Trigonometric functions |id=p/t094210}}
* Linton, Christopher M. (2004). ''From Eudoxus to Einstein: A History of Mathematical Astronomy''. Cambridge University Press.
* {{MathWorld|ref=none|title=Trigonometric Addition Formulas|urlname=TrigonometricAdditionFormulas}}

== External links ==
{{Sister project links|Trigonometry}}
{{Library resources box |by=no |onlinebooks=no |others=no |about=yes |label=Trigonometry}}
* [http://www.khanacademy.org/math/trigonometry Khan Academy: Trigonometry, free online micro lectures]
* [https://web.archive.org/web/20071104225720/http://baqaqi.chi.il.us/buecher/mathematics/trigonometry/index.html Trigonometry] by Alfred Monroe Kenyon and Louis Ingold, The Macmillan Company, 1914. In images, full text presented.
* [http://www.maa.org/publications/periodicals/convergence/benjamin-bannekers-trigonometry-puzzle-introduction Benjamin Banneker's Trigonometry Puzzle] at [http://www.maa.org/loci-category/convergence?page=1 Convergence]
* [http://www.clarku.edu/~djoyce/trig/ Dave's Short Course in Trigonometry] by [[David E. Joyce (mathematician)|David Joyce]] of [[Clark University]]
* [https://web.archive.org/web/20201216180745/http://mecmath.net/trig/trigbook.pdf Trigonometry, by Michael Corral, Covers elementary trigonometry, Distributed under GNU Free Documentation License]

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{{Areas of mathematics | state=collapsed}}

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[[Category:3rd-century BC introductions]]
[[Category:Trigonometry| ]]