Trigonometry Information
Trigonometry 

Reference 
Laws and theorems 
Calculus 
Trigonometry (from Greek trigōnon, "triangle" and metron, "measure"^{ [1]}) is a branch of mathematics that studies relationships between side lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies.^{ [2]} The Greeks focused on the calculation of chords, while mathematicians in India created the earliestknown tables of values for trigonometric ratios (also called trigonometric functions) such as sine.^{ [3]}
Throughout history, trigonometry has been applied in areas such as geodesy, surveying, celestial mechanics, and navigation.^{ [4]}
Trigonometry is known for its many identities. These trigonometric identities^{ [5]}^{ [6]} are commonly used for rewriting trigonometrical expressions with the aim to simplify an expression, to find a more useful form of an expression, or to solve an equation.^{ [7]}
History
Sumerian astronomers studied angle measure, using a division of circles into 360 degrees.^{ [9]} They, and later the Babylonians, studied the ratios of the sides of similar triangles and discovered some properties of these ratios but did not turn that into a systematic method for finding sides and angles of triangles. The ancient Nubians used a similar method.^{ [10]}
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.^{ [11]} In the 2nd century AD, the GrecoEgyptian astronomer Ptolemy (from Alexandria, Egypt) constructed detailed trigonometric tables ( Ptolemy's table of chords) in Book 1, chapter 11 of his Almagest.^{ [12]} Ptolemy used chord length to define his trigonometric functions, a minor difference from the sine convention we use today.^{ [13]} (The value we call sin(θ) can be found by looking up the chord length for twice the angle of interest (2θ) in Ptolemy's table, and then dividing that value by two.) Centuries passed before more detailed tables were produced, and Ptolemy's treatise remained in use for performing trigonometric calculations in astronomy throughout the next 1200 years in the medieval Byzantine, Islamic, and, later, Western European worlds.
The modern sine convention is first attested in the Surya Siddhanta, and its properties were further documented by the 5th century (AD) Indian mathematician and astronomer Aryabhata.^{ [14]} These Greek and Indian works were translated and expanded by medieval Islamic mathematicians. By the 10th century, Islamic mathematicians were using all six trigonometric functions, had tabulated their values, and were applying them to problems in spherical geometry.^{ [15]}^{ [16]} The Persian polymath Nasir alDin alTusi has been described as the creator of trigonometry as a mathematical discipline in its own right.^{ [17]}^{ [18]}^{ [19]} Nasīr alDīn alTūsī was the first to treat trigonometry as a mathematical discipline independent from astronomy, and he developed spherical trigonometry into its present form.^{ [20]} He listed the six distinct cases of a rightangled 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.^{ [21]} 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 alDin alTusi.^{ [22]} 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.^{ [23]} At the same time, another translation of the Almagest from Greek into Latin was completed by the Cretan George of Trebizond.^{ [24]} Trigonometry was still so little known in 16thcentury 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.^{ [25]} Bartholomaeus Pitiscus was the first to use the word, publishing his Trigonometria in 1595.^{ [26]} 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.^{ [27]} Also in the 18th century, Brook Taylor defined the general Taylor series.^{ [28]}
Trigonometric ratios
Trigonometric ratios are the ratios between edges of a right triangle. These ratios are given by the following trigonometric functions of the known angle A, where a, b and c refer to the lengths of the sides in the accompanying figure:
 Sine function (sin), defined as the ratio of the side opposite the angle to the hypotenuse.
 Cosine function (cos), defined as the ratio of the adjacent leg (the side of the triangle joining the angle to the right angle) to the hypotenuse.
 Tangent function (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.
Since any two right triangles with the same acute angle A are similar,^{ [29]} the value of a trigonometric ratio depends only on the angle A.
The reciprocals of these functions are named the cosecant (csc), secant (sec), and cotangent (cot), respectively:
The cosine, cotangent, and cosecant are so named because they are respectively the sine, tangent, and secant of the complementary angle abbreviated to "co".^{ [30]}
With these functions, one can answer virtually all questions about arbitrary triangles by using the law of sines and the law of cosines.^{ [31]} 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.
Mnemonics
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 SOHCAHTOA:^{ [32]}
 Sine = Opposite ÷ Hypotenuse
 Cosine = Adjacent ÷ Hypotenuse
 Tangent = Opposite ÷ Adjacent
One way to remember the letters is to sound them out phonetically (i.e., SOHCAHTOA, which is pronounced 'sokatoeuh' /soʊkæˈtoʊə/). Another method is to expand the letters into a sentence, such as "Some Old Hippie Caught Another Hippie Trippin' On Acid".^{ [33]}
The unit circle and common trigonometric values
Trigonometric ratios can also be represented using the unit circle, which is the circle of radius 1 centered at the origin in the plane.^{ [34]} 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 and .^{ [34]} This representation allows for the calculation of commonly found trigonometric values, such as those in the following table:^{ [35]}
Function  0  

sine  0  1  0  
cosine  1  0  1  
tangent  0  undefined  0  
secant  1  undefined  1  
cosecant  undefined  1  undefined  
cotangent  undefined  0  undefined 
Trigonometric functions of real or complex variables
Using the unit circle, one can extend the definitions of trigonometric ratios to all positive and negative arguments^{ [36]} (see trigonometric function).
Graphs of trigonometric functions
The following table summarizes the properties of the graphs of the six main trigonometric functions:^{ [37]}^{ [38]}
Function  Period  Domain  Range  Graph 

sine  
cosine  
tangent  
secant  
cosecant  
cotangent 
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.^{ [39]}^{:48ff}
The names of the inverse trigonometric functions, together with their domains and range, can be found in the following table:^{ [39]}^{:48ff}^{ [40]}^{:521ff}
Name  Usual notation  Definition  Domain of x for real result  Range of usual principal value ( radians) 
Range of usual principal value ( degrees) 

arcsine  y = arcsin(x)  x = sin(y)  −1 ≤ x ≤ 1  −π/2 ≤ y ≤ π/2  −90° ≤ y ≤ 90° 
arccosine  y = arccos(x)  x = cos(y)  −1 ≤ x ≤ 1  0 ≤ y ≤ π  0° ≤ y ≤ 180° 
arctangent  y = arctan(x)  x = tan(y)  all real numbers  −π/2 < y < π/2  −90° < y < 90° 
arccotangent  y = arccot(x)  x = cot(y)  all real numbers  0 < y < π  0° < y < 180° 
arcsecant  y = arcsec(x)  x = sec(y)  x ≤ −1 or 1 ≤ x  0 ≤ y < π/2 or π/2 < y ≤ π  0° ≤ y < 90° or 90° < y ≤ 180° 
arccosecant  y = arccsc(x)  x = csc(y)  x ≤ −1 or 1 ≤ x  −π/2 ≤ y < 0 or 0 < y ≤ π/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:^{ [41]}
With these definitions the trigonometric functions can be defined for complex numbers.^{ [42]} When extended as functions of real or complex variables, the following formula holds for the complex exponential:
This complex exponential function, written in terms of trigonometric functions, is particularly useful.^{ [43]}^{ [44]}
Calculating trigonometric functions
Trigonometric functions were among the earliest uses for mathematical tables.^{ [45]} 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.^{ [46]} Slide rules had special scales for trigonometric functions.^{ [47]}
Scientific calculators have buttons for calculating the main trigonometric functions (sin, cos, tan, and sometimes cis and their inverses).^{ [48]} 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.^{ [49]} The floating point unit hardware incorporated into the microprocessor chips used in most personal computers has builtin instructions for calculating trigonometric functions.^{ [50]}
Other trigonometric functions
In addition to the six ratios listed earlier, there are additional trigonometric functions that were historically important, though seldom used today. These include the chord (crd(θ) = 2 sin(θ/2)), the versine (versin(θ) = 1 − cos(θ) = 2 sin^{2}(θ/2)) (which appeared in the earliest tables^{ [51]}), the coversine (coversin(θ) = 1 − sin(θ) = versin(π/2 − θ)), the haversine (haversin(θ) = 1/2versin(θ) = sin^{2}(θ/2)),^{ [52]} the exsecant (exsec(θ) = sec(θ) − 1), and the excosecant (excsc(θ) = exsec(π/2 − θ) = csc(θ) − 1). See List of trigonometric identities for more relations between these functions.
Applications
Astronomy
For centuries, spherical trigonometry has been used for locating solar, lunar, and stellar positions,^{ [53]} predicting eclipses, and describing the orbits of the planets.^{ [54]}
In modern times, the technique of triangulation is used in astronomy to measure the distance to nearby stars,^{ [55]} as well as in satellite navigation systems.^{ [16]}
Historically, trigonometry has been used for locating latitudes and longitudes of sailing vessels, plotting courses, and calculating distances during navigation.^{ [56]}
Trigonometry is still used in navigation through such means as the Global Positioning System and artificial intelligence for autonomous vehicles.^{ [57]}
Surveying
In land surveying, trigonometry is used in the calculation of lengths, areas, and relative angles between objects.^{ [58]}
On a larger scale, trigonometry is used in geography to measure distances between landmarks.^{ [59]}
Periodic functions
The sine and cosine functions are fundamental to the theory of periodic functions,^{ [60]} 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 nonperiodic functions can be represented as an integral of sines and cosines through the Fourier transform. This has applications to quantum mechanics^{ [61]} and communications,^{ [62]} among other fields.
Optics and acoustics
Trigonometry is useful in many physical sciences,^{ [63]} including acoustics,^{ [64]} and optics.^{ [64]} In these areas, they are used to describe sound and light waves, and to solve boundary and transmissionrelated problems.^{ [65]}
Other applications
Other fields that use trigonometry or trigonometric functions include music theory,^{ [66]} geodesy, audio synthesis,^{ [67]} architecture,^{ [68]} electronics,^{ [66]} biology,^{ [69]} medical imaging ( CT scans and ultrasound),^{ [70]} chemistry,^{ [71]} number theory (and hence cryptology),^{ [72]} seismology,^{ [64]} meteorology,^{ [73]} oceanography,^{ [74]} image compression,^{ [75]} phonetics,^{ [76]} economics,^{ [77]} electrical engineering, mechanical engineering, civil engineering,^{ [66]} computer graphics,^{ [78]} cartography,^{ [66]} crystallography^{ [79]} and game development.^{ [78]}
Identities
Trigonometry has been noted for its many identities, that is, equations that are true for all possible inputs.^{ [80]}
Identities involving only angles are known as trigonometric identities. Other equations, known as triangle identities,^{ [81]} relate both the sides and angles of a given triangle.
Triangle identities
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).^{ [82]}
Law of sines
The law of sines (also known as the "sine rule") for an arbitrary triangle states:^{ [83]}
where is the area of the triangle and R is the radius of the circumscribed circle of the triangle:
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:^{ [83]}
or equivalently:
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.^{ [84]} It is given by:
Area
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:^{ [83]}
Heron's formula is another method that may be used to calculate the area of a triangle. This formula states that if a triangle has sides of lengths a, b, and c, and if the semiperimeter is
then the area of the triangle is:^{ [85]}
 ,
where R is the radius of the circumcircle of the triangle.
Trigonometric identities
Pythagorean identities
The following trigonometric identities are related to the Pythagorean theorem and hold for any value:^{ [86]}
Euler's formula
Euler's formula, which states that , produces the following analytical identities for sine, cosine, and tangent in terms of e and the imaginary unit i:
Other trigonometric identities
Other commonly used trigonometric identities include the halfangle identities, the angle sum and difference identities, and the producttosum identities.^{ [29]}
See also
References
 ^ "trigonometry". Online Etymology Dictionary.
 ^ R. Nagel (ed.), Encyclopedia of Science, 2nd Ed., The Gale Group (2002)
 ^ Boyer (1991), p. ^{[ page needed]}.
 ^ Charles William Hackley (1853). 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. G. P. Putnam.
 ^ Mary Jane Sterling (24 February 2014). Trigonometry For Dummies. John Wiley & Sons. p. 185. ISBN 9781118827413.
 ^ P.R. Halmos (1 December 2013). I Want to be a Mathematician: An Automathography. Springer Science & Business Media. ISBN 9781461210849.
 ^ Ron Larson; Robert P. Hostetler (10 March 2006). Trigonometry. Cengage Learning. p. 230. ISBN 061864332X.
 ^ Boyer (1991), p. 162, "Greek Trigonometry and Mensuration".
 ^ Aaboe, Asger (2001). Episodes from the Early History of Astronomy. New York: Springer. ISBN 0387951369
 ^ Otto Neugebauer (1975). A history of ancient mathematical astronomy. 1. SpringerVerlag. p. 744. ISBN 9783540069959.
 ^ Thurston (1996), pp. 235–236, "Appendix 1: Hipparchus's Table of Chords".
 ^ Toomer, G. (1998), Ptolemy's Almagest, Princeton University Press, ISBN 9780691002606
 ^ Thurston (1996), pp. 239–243, "Appendix 3: Ptolemy's Table of Chords".
 ^ Boyer (1991), p. 215.
 ^ Gingerich, Owen. "Islamic astronomy." Scientific American 254.4 (1986): 7483
 ^ ^{a} ^{b} Michael Willers (13 February 2018). Armchair Algebra: Everything You Need to Know From Integers To Equations. Book Sales. p. 37. ISBN 9780785835950.

^
"Nasir alDin alTusi".
MacTutor History of Mathematics archive. Retrieved 20210108.
One of alTusi'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 alTusi 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 rightangled spherical triangle are set forth.
 ^ "the cambridge history of science". October 2013.

^
"ṬUSI, NAṢIRALDIN i. Biography". Encyclopaedia Iranica. Retrieved 20180805.
His major contribution in mathematics (Nasr, 1996, pp. 208214) 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 rightangled triangles.
 ^ "trigonometry". Encyclopædia Britannica. Retrieved 20080721.
 ^ Berggren, J. Lennart (2007). "Mathematics in Medieval Islam". The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook. Princeton University Press. p. 518. ISBN 9780691114859.
 ^ Boyer (1991), pp. 237, 274.
 ^ "Johann Müller Regiomontanus". MacTutor History of Mathematics archive. Retrieved 20210108.
 ^ N.G. Wilson (1992). From Byzantium to Italy. Greek Studies in the Italian Renaissance, London. ISBN 0715624180
 ^ GrattanGuinness, Ivor (1997). The Rainbow of Mathematics: A History of the Mathematical Sciences. W.W. Norton. ISBN 9780393320305.
 ^ Robert E. Krebs (2004). Groundbreaking Scientific Experiments, Inventions, and Discoveries of the Middle Ages and the Renaissance. Greenwood Publishing Group. p. 153. ISBN 9780313324338.
 ^ William Bragg Ewald (2007). From Kant to Hilbert: a source book in the foundations of mathematics. Oxford University Press US. p. 93. ISBN 0198505353
 ^ Kelly Dempski (2002). Focus on Curves and Surfaces. p. 29. ISBN 159200007X
 ^ ^{a} ^{b} James Stewart; Lothar Redlin; Saleem Watson (16 January 2015). Algebra and Trigonometry. Cengage Learning. p. 448. ISBN 9781305537033.
 ^ Dick Jardine; Amy ShellGellasch (2011). Mathematical Time Capsules: Historical Modules for the Mathematics Classroom. MAA. p. 182. ISBN 9780883859841.
 ^ Krystle Rose Forseth; Christopher Burger; Michelle Rose Gilman; Deborah J. Rumsey (2008). PreCalculus For Dummies. John Wiley & Sons. p. 218. ISBN 9780470169841.
 ^ Weisstein, Eric W. "SOHCAHTOA". MathWorld.
 ^ A sentence more appropriate for high schools is "'Some Old Horse Came A''Hopping Through Our Alley". Foster, Jonathan K. (2008). Memory: A Very Short Introduction. Oxford. p. 128. ISBN 9780192806758.
 ^ ^{a} ^{b} David Cohen; Lee B. Theodore; David Sklar (17 July 2009). Precalculus: A ProblemsOriented Approach, Enhanced Edition. Cengage Learning. ISBN 9781439044605.
 ^ W. Michael Kelley (2002). The Complete Idiot's Guide to Calculus. Alpha Books. p. 45. ISBN 9780028643656.
 ^ Jenny Olive (18 September 2003). Maths: A Student's Survival Guide: A SelfHelp Workbook for Science and Engineering Students. Cambridge University Press. p. 175. ISBN 9780521017077.
 ^ Mary P Attenborough (30 June 2003). Mathematics for Electrical Engineering and Computing. Elsevier. p. 418. ISBN 9780080473406.
 ^ Ron Larson; Bruce H. Edwards (10 November 2008). Calculus of a Single Variable. Cengage Learning. p. 21. ISBN 9780547209982.
 ^ ^{a} ^{b} Elizabeth G. Bremigan; Ralph J. Bremigan; John D. Lorch (2011). Mathematics for Secondary School Teachers. MAA. ISBN 9780883857731.
 ^ Martin Brokate; Pammy Manchanda; Abul Hasan Siddiqi (3 August 2019). Calculus for Scientists and Engineers. Springer. ISBN 9789811384646.
 ^ Serge Lang (14 March 2013). Complex Analysis. Springer. p. 63. ISBN 9783642592737.
 ^ Silvia Maria Alessio (9 December 2015). Digital Signal Processing and Spectral Analysis for Scientists: Concepts and Applications. Springer. p. 339. ISBN 9783319254685.
 ^ K. RAJA RAJESWARI; B. VISVESVARA RAO (24 March 2014). SIGNALS AND SYSTEMS. PHI Learning. p. 263. ISBN 9788120349414.
 ^ John Stillwell (23 July 2010). Mathematics and Its History. Springer Science & Business Media. p. 313. ISBN 9781441960535.
 ^ Martin CampbellKelly; Professor Emeritus of Computer Science Martin CampbellKelly; Visiting Fellow Department of Computer Science Mary Croarken; Raymond Flood; Eleanor Robson (2 October 2003). The History of Mathematical Tables: From Sumer to Spreadsheets. OUP Oxford. ISBN 9780198508410.
 ^ George S. Donovan; Beverly Beyreuther Gimmestad (1980). Trigonometry with calculators. Prindle, Weber & Schmidt. ISBN 9780871502841.
 ^ Ross Raymond Middlemiss (1945). Instructions for Posttrig and Mannheimtrig Slide Rules. Frederick Post Company.
 ^ "Calculator keys—what they do". Popular Science. Bonnier Corporation. April 1974. p. 125.
 ^ Steven S. Skiena; Miguel A. Revilla (18 April 2006). Programming Challenges: The Programming Contest Training Manual. Springer Science & Business Media. p. 302. ISBN 9780387220819.
 ^ Intel® 64 and IA32 Architectures Software Developer's Manual Combined Volumes: 1, 2A, 2B, 2C, 3A, 3B and 3C (PDF). Intel. 2013.
 ^ Boyer (1991), pp. xxiii–xxiv.
 ^ Nielsen (1966), pp. xxiii–xxiv.
 ^ Olinthus Gregory (1816). 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. Baldwin, Cradock, and Joy.
 ^ Neugebauer, Otto (1948). "Mathematical methods in ancient astronomy". Bulletin of the American Mathematical Society. 54 (11): 1013–1041.
 ^ Michael Seeds; Dana Backman (5 January 2009). Astronomy: The Solar System and Beyond. Cengage Learning. p. 254. ISBN 9780495562030.
 ^ John Sabine (1800). The Practical Mathematician, Containing Logarithms, Geometry, Trigonometry, Mensuration, Algebra, Navigation, Spherics and Natural Philosophy, Etc. p. 1.
 ^ Mordechai BenAri; Francesco Mondada (2018). Elements of Robotics. Springer. p. 16. ISBN 9783319625331.
 ^ George Roberts Perkins (1853). Plane Trigonometry and Its Application to Mensuration and Land Surveying: Accompanied with All the Necessary Logarithmic and Trigonometric Tables. D. Appleton & Company.
 ^ Charles W. J. Withers; Hayden Lorimer (14 December 2015). Geographers: Biobibliographical Studies. A&C Black. p. 6. ISBN 9781441107855.
 ^ H. G. ter Morsche; J. C. van den Berg; E. M. van de Vrie (7 August 2003). Fourier and Laplace Transforms. Cambridge University Press. p. 61. ISBN 9780521534413.
 ^ Bernd Thaller (8 May 2007). Visual Quantum Mechanics: Selected Topics with ComputerGenerated Animations of QuantumMechanical Phenomena. Springer Science & Business Media. p. 15. ISBN 9780387227702.
 ^ M. Rahman (2011). Applications of Fourier Transforms to Generalized Functions. WIT Press. ISBN 9781845645649.
 ^ Lawrence Bornstein; Basic Systems, Inc (1966). Trigonometry for the Physical Sciences. AppletonCenturyCrofts.
 ^ ^{a} ^{b} ^{c} John J. Schiller; Marie A. Wurster (1988). College Algebra and Trigonometry: Basics Through Precalculus. Scott, Foresman. ISBN 9780673183934.
 ^ Dudley H. Towne (5 May 2014). Wave Phenomena. Dover Publications. ISBN 9780486145150.
 ^ ^{a} ^{b} ^{c} ^{d} E. Richard Heineman; J. Dalton Tarwater (1 November 1992). Plane Trigonometry. McGrawHill. ISBN 9780070281875.
 ^ Mark Kahrs; Karlheinz Brandenburg (18 April 2006). Applications of Digital Signal Processing to Audio and Acoustics. Springer Science & Business Media. p. 404. ISBN 9780306470424.
 ^ Kim Williams; Michael J. Ostwald (9 February 2015). Architecture and Mathematics from Antiquity to the Future: Volume I: Antiquity to the 1500s. Birkhäuser. p. 260. ISBN 9783319001371.
 ^ Dan Foulder (15 July 2019). Essential Skills for GCSE Biology. Hodder Education. p. 78. ISBN 9781510460034.
 ^ Luciano Beolchi; Michael H. Kuhn (1995). Medical Imaging: Analysis of Multimodality 2D/3D Images. IOS Press. p. 122. ISBN 9789051992106.
 ^ Marcus Frederick Charles Ladd (2014). Symmetry of Crystals and Molecules. Oxford University Press. p. 13. ISBN 9780199670888.
 ^ Gennady I. Arkhipov; Vladimir N. Chubarikov; Anatoly A. Karatsuba (22 August 2008). Trigonometric Sums in Number Theory and Analysis. Walter de Gruyter. ISBN 9783110197983.
 ^ Study Guide for the Course in Meteorological Mathematics: Latest Revision, Feb. 1, 1943. 1943.
 ^ Mary Sears; Daniel Merriman; Woods Hole Oceanographic Institution (1980). Oceanography, the past. SpringerVerlag. ISBN 9780387904979.
 ^ "JPEG Standard (JPEG ISO/IEC 109181 ITUT Recommendation T.81)" (PDF). International Telecommunications Union. 1993. Retrieved 6 April 2019.
 ^ Kirsten Malmkjaer (4 December 2009). The Routledge Linguistics Encyclopedia. Routledge. p. 1. ISBN 9781134103713.
 ^ Kamran Dadkhah (11 January 2011). Foundations of Mathematical and Computational Economics. Springer Science & Business Media. p. 46. ISBN 9783642137488.
 ^ ^{a} ^{b} Christopher Griffith (12 November 2012). RealWorld Flash Game Development: How to Follow Best Practices AND Keep Your Sanity. CRC Press. p. 153. ISBN 9781136137020.
 ^ John Joseph Griffin (1841). A System of Crystallography, with Its Application to Mineralogy. R. Griffin. p. 119.
 ^ Dugopolski (July 2002). Trigonometry I/E Sup. Addison Wesley. ISBN 9780201786668.
 ^ V&S EDITORIAL BOARD (6 January 2015). CONCISE DICTIONARY OF MATHEMATICS. V&S Publishers. p. 288. ISBN 9789350574140.
 ^ Lecture 3  Quantum Entanglements, Part 1 (Stanford), Leonard Susskind, trigonometry in five minutes, law of sin, cos, euler formula 20061009.
 ^ ^{a} ^{b} ^{c} Cynthia Y. Young (19 January 2010). Precalculus. John Wiley & Sons. p. 435. ISBN 9780471756842.
 ^ Ron Larson (29 January 2010). Trigonometry. Cengage Learning. p. 331. ISBN 9781439049075.
 ^ Richard N. Aufmann; Vernon C. Barker; Richard D. Nation (5 February 2007). College Trigonometry. Cengage Learning. p. 306. ISBN 9780618825073.
 ^ Peterson, John C. (2004). Technical Mathematics with Calculus (illustrated ed.). Cengage Learning. p. 856. ISBN 9780766861893. Extract of page 856
Bibliography
 Boyer, Carl B. (1991). A History of Mathematics (Second ed.). John Wiley & Sons, Inc. ISBN 9780471543978.
 Nielsen, Kaj L. (1966). Logarithmic and Trigonometric Tables to Five Places (2nd ed.). New York: Barnes & Noble. LCCN 619103.
 Thurston, Hugh (1996). Early Astronomy. Springer Science & Business Media. ISBN 9780387948225.
Further reading
 "Trigonometric functions", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
 Linton, Christopher M. (2004). From Eudoxus to Einstein: A History of Mathematical Astronomy. Cambridge University Press.
 Weisstein, Eric W. "Trigonometric Addition Formulas". MathWorld.
External links
Library resources about Trigonometry 
 Khan Academy: Trigonometry, free online micro lectures
 Trigonometry by Alfred Monroe Kenyon and Louis Ingold, The Macmillan Company, 1914. In images, full text presented.
 Benjamin Banneker's Trigonometry Puzzle at Convergence
 Dave's Short Course in Trigonometry by David Joyce of Clark University
 Trigonometry, by Michael Corral, Covers elementary trigonometry, Distributed under GNU Free Documentation License