Trigonometry

{{main|History of trigonometry}}

File:Head of Hipparchus (cropped).jpg, credited with compiling the first 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]}}]]

Sumerian astronomers studied angle measure, using a division of circles into 360 degrees.{{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] 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.{{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}}

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.{{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.{{Citation|title=Ptolemy's Almagest|last1=Toomer |first1=G.|author-link=Gerald J. Toomer|publisher=Princeton University Press|year=1998|bibcode=1998ptal.book.....T |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.{{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, 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 mathematician and astronomer Aryabhata.{{sfnp|Boyer |1991|p=215}} These Greek and Indian works were translated and expanded by medieval Islamic mathematicians. In 830 AD, Persian mathematician Habash al-Hasib al-Marwazi produced the first table of cotangents.{{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}} 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{{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 }}Gingerich, Owen. "Islamic astronomy." Scientific American 254.4 (1986): 74–83{{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}} The Persian polymath Nasir al-Din al-Tusi has been described as the creator of trigonometry as a mathematical discipline in its own right.{{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.}}{{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. }}{{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.}} He was the first to treat trigonometry as a mathematical discipline independent from astronomy, and he developed spherical trigonometry into its present form.{{cite web |title=trigonometry |url=http://www.britannica.com/EBchecked/topic/605281/trigonometry |publisher=Encyclopædia Britannica |access-date=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.{{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}} 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.{{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 Greek scholar cardinal Basilios Bessarion with whom he lived for several years.{{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}} 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.{{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}} Bartholomaeus Pitiscus was the first to use the word, publishing his Trigonometria in 1595.{{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}} 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.{{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}} Also in the 18th century, Brook Taylor defined the general Taylor series.{{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}}

{{main|Trigonometric function}}

File:Trigonometry triangle.svg

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 similar.{{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}}

So, these ratios define functions of this angle that are called trigonometric functions. 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.

• Sine (denoted sin), defined as the ratio of the side opposite the angle to the hypotenuse.

:: $\backslash sin\; A=\backslash frac\{\backslash textrm\{opposite\}\}\{\backslash textrm\{hypotenuse\}\}=\backslash frac\{a\}\{h\}.$

• 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.

:: $\backslash cos\; A=\backslash frac\{\backslash textrm\{adjacent\}\}\{\backslash textrm\{hypotenuse\}\}=\backslash frac\{b\}\{h\}.$

• Tangent (denoted tan), defined as the ratio of the opposite leg to the adjacent leg.

::$\backslash tan\; A=\backslash frac\{\backslash textrm\{opposite\}\}\{\backslash textrm\{adjacent\}\}=\backslash frac\{a\}\{b\}=\backslash frac\{a/h\}\{b/h\}=\backslash frac\{\backslash sin\; A\}\{\backslash cos\; A\}.$

The reciprocals of these ratios are named the cosecant (csc), secant (sec), and cotangent (cot), respectively:

:$\backslash csc\; A=\backslash frac\{1\}\{\backslash sin\; A\}=\backslash frac\{\backslash textrm\{hypotenuse\}\}\{\backslash textrm\{opposite\}\}=\backslash frac\{h\}\{a\}\; ,$

:$\backslash sec\; A=\backslash frac\{1\}\{\backslash cos\; A\}=\backslash frac\{\backslash textrm\{hypotenuse\}\}\{\backslash textrm\{adjacent\}\}=\backslash frac\{h\}\{b\}\; ,$

:$\backslash cot\; A=\backslash frac\{1\}\{\backslash tan\; A\}=\backslash frac\{\backslash textrm\{adjacent\}\}\{\backslash textrm\{opposite\}\}=\backslash frac\{\backslash cos\; A\}\{\backslash sin\; A\}=\backslash frac\{b\}\{a\}\; .$

The cosine, cotangent, and cosecant are so named because they are respectively the sine, tangent, and secant of the complementary angle abbreviated to "co-".{{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}}

With these functions, one can answer virtually all questions about arbitrary triangles by using the law of sines and the law of cosines.{{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}} 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.

{{main|Mnemonics in trigonometry}}

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|ref=none|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. {{IPAc-en|ˌ|s|oʊ|k|ə|ˈ|t|oʊ|ə}} {{respell|SOH|kə|TOH|ə}}, similar to Krakatoa).{{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}} 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". {{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}}

{{main|Unit circle}}

File:Sin-cos-defn-I.png

File:Math Trigonometry Unit Circle Rotation Sign Indication.svg

Trigonometric ratios can also be represented using the unit circle, which is the circle of radius 1 centered at the origin in the plane.{{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}} 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 $x\; =\; \backslash cos\; A$ and $y\; =\; \backslash sin\; A$. This representation allows for the calculation of commonly found trigonometric values, such as those in the following table:{{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}}

{{main|Trigonometric function}}

Using the unit circle, one can extend the definitions of trigonometric ratios to all positive and negative arguments{{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}} (see trigonometric function).

The following table summarizes the properties of the graphs of the six main trigonometric functions:{{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}}{{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}}

{{main|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.{{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}}{{rp|48ff}}

The names of the inverse trigonometric functions, together with their domains and range, can be found in the following table:{{rp|48ff}}{{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}}{{rp|521ff}}

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:{{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}}

:$$

\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.{{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}} When extended as functions of real or complex variables, the following formula holds for the complex exponential:

: $e^\{x+iy\}\; =\; e^x(\backslash cos\; y\; +\; i\; \backslash sin\; y).$

This complex exponential function, written in terms of trigonometric functions, is particularly useful.{{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}}{{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}}

{{main|Trigonometric tables}}

Trigonometric functions were among the earliest uses for mathematical tables.{{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}} 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.{{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}} Slide rules had special scales for trigonometric functions.{{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}}

Scientific calculators have buttons for calculating the main trigonometric functions (sin, cos, tan, and sometimes cis and their inverses).{{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}} 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.{{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}} The floating point unit hardware incorporated into the microprocessor chips used in most personal computers has built-in instructions for calculating trigonometric functions.{{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}}

{{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 ({{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.

{{main|Uses of trigonometry}}

{{main|Astronomy}}

For centuries, spherical trigonometry has been used for locating solar, lunar, and stellar positions,{{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}} predicting eclipses, and describing the orbits of the planets.{{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 }}

In modern times, the technique of triangulation is used in astronomy to measure the distance to nearby stars,{{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}} as well as in satellite navigation systems.

{{main|Navigation}}

File:Frieberger drum marine sextant.jpgs 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.{{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}}

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

{{main|Surveying}}

In land surveying, trigonometry is used in the calculation of lengths, areas, and relative angles between objects.{{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}}

On a larger scale, trigonometry is used in geography to measure distances between landmarks.{{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}}

{{main|Fourier series|Fourier transform}}

File:Fourier series and transform.gif 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,{{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}} 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 mechanics{{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}} and communications,{{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}} among other fields.

{{main|optics|acoustics}}

Trigonometry is useful in many physical sciences,{{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}} including acoustics,{{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}} and optics. In these areas, they are used to describe sound and light waves, and to solve boundary- and transmission-related problems.{{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}}

Other fields that use trigonometry or trigonometric functions include music theory,{{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}} geodesy, audio synthesis,{{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}} architecture,{{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}} electronics, biology,{{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}} medical imaging (CT scans and ultrasound),{{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}} chemistry,{{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}} number theory (and hence cryptology),{{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}} seismology, meteorology,{{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}} oceanography,{{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}} image compression,{{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}} phonetics,{{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}} economics,{{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}} electrical engineering, mechanical engineering, civil engineering, computer graphics, cartography, crystallography{{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]}} and game development.{{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]}}

{{Main|List of trigonometric identities}}

File:Triangle ABC with Sides a b c 2.png

Trigonometry has been noted for its many identities, that is, equations that are true for all possible inputs.{{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}}

Identities involving only angles are known as trigonometric identities. Other equations, known as triangle identities,{{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}} relate both the sides and angles of a given triangle.

{{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).

The law of sines (also known as the "sine rule") for an arbitrary triangle states:

:$\backslash frac\{a\}\{\backslash sin\; A\}\; =\; \backslash frac\{b\}\{\backslash sin\; B\}\; =\; \backslash frac\{c\}\{\backslash sin\; C\}\; =\; 2R\; =\; \backslash frac\{abc\}\{2\backslash Delta\},$

where $\backslash Delta$ is the area of the triangle and R is the radius of the circumscribed circle of the triangle:

:$R\; =\; \backslash frac\{abc\}\{\backslash sqrt\{(a+b+c)(a-b+c)(a+b-c)(b+c-a)\}\}.$

The law of cosines (known as the cosine formula, or the "cos rule") is an extension of the Pythagorean theorem to arbitrary triangles:

:$c^2=a^2+b^2-2ab\backslash cos\; C\; ,$

or equivalently:

:$\backslash cos\; C=\backslash frac\{a^2+b^2-c^2\}\{2ab\}.$

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.{{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}} It is given by:

:$\backslash frac\{a-b\}\{a+b\}=\backslash frac\{\backslash tan\backslash left[\backslash tfrac\{1\}\{2\}(A-B)\backslash right]\}\{\backslash tan\backslash left[\backslash tfrac\{1\}\{2\}(A+B)\backslash right]\}$

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:{{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}}

:$\backslash mbox\{Area\}\; =\; \backslash Delta\; =\; \backslash frac\{1\}\{2\}a\; b\backslash sin\; C$

The following trigonometric identities are related to the Pythagorean theorem and hold for any value:{{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]

:$\backslash sin^2\; A\; +\; \backslash cos^2\; A\; =\; 1\; \backslash$

:$\backslash tan^2\; A\; +\; 1\; =\; \backslash sec^2\; A\; \backslash$

:$\backslash cot^2\; A\; +\; 1\; =\; \backslash csc^2\; A\; \backslash$

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

Euler's formula, which states that $e^\{ix\}\; =\; \backslash cos\; x\; +\; i\; \backslash sin\; x$, produces the following analytical identities for sine, cosine, and tangent in terms of e and the imaginary unit i:

:$\backslash sin\; x\; =\; \backslash frac\{e^\{ix\}\; -\; e^\{-ix\}\}\{2i\},\; \backslash qquad\; \backslash cos\; x\; =\; \backslash frac\{e^\{ix\}\; +\; e^\{-ix\}\}\{2\},\; \backslash qquad\; \backslash tan\; x\; =\; \backslash frac\{i(e^\{-ix\}\; -\; e^\{ix\})\}\{e^\{ix\}\; +\; e^\{-ix\}\}.$

{{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.

{{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}}

{{reflist}}

• {{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}}

• {{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}}

{{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 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]

{{Geometry-footer}}

{{Areas of mathematics | state=collapsed}}

{{Authority control}}

Category:3rd-century BC introductions