Sumerian astronomers studied angle measure, using a division of circles into 360 degrees.[8] They, and later theBabylonians, studied the ratios of the sides ofsimilar triangles and discovered some properties of these ratios but did not turn that into a systematic method for finding sides and angles of triangles. Theancient Nubians used a similar method.[9]
In the 3rd century BC,Hellenistic mathematicians such asEuclid andArchimedes studied the properties ofchords andinscribed 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 (fromNicaea, Asia Minor) gave the first tables of chords, analogous to moderntables of sine values, and used them to solve problems in trigonometry andspherical trigonometry.[10] In the 2nd century AD, the Greco-Egyptian astronomerPtolemy (from Alexandria, Egypt) constructed detailed trigonometric tables (Ptolemy's table of chords) in Book 1, chapter 11 of hisAlmagest.[11] Ptolemy usedchord length to define his trigonometric functions, a minor difference from thesine convention we use today.[12] (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 medievalByzantine,Islamic, and, later, Western European worlds.
The modern definition of the sine is first attested in theSurya Siddhanta, and its properties were further documented in the 5th century (AD) byIndian mathematician and astronomerAryabhata.[13] These Greek and Indian works were translated and expanded bymedieval Islamic mathematicians. In 830 AD, Persian mathematicianHabash al-Hasib al-Marwazi produced the first table of cotangents.[14][15] By the 10th century AD, in the work of Persian mathematicianAbū al-Wafā' al-Būzjānī, all sixtrigonometric functions were used.[16] Abu al-Wafa had sine tables in 0.25° increments, to 8 decimal places of accuracy, and accurate tables of tangent values.[16] He also made important innovations inspherical trigonometry[17][18][19] ThePersianpolymathNasir al-Din al-Tusi has been described as the creator of trigonometry as a mathematical discipline in its own right.[20][21][22] He was the first to treat trigonometry as a mathematical discipline independent from astronomy, and he developed spherical trigonometry into its present form.[15] He listed the six distinct cases of a right-angled triangle in spherical trigonometry, and in hisOn the Sector Figure, he stated the law of sines for plane and spherical triangles, discovered thelaw of tangents for spherical triangles, and provided proofs for both these laws.[23] Knowledge of trigonometric functions and methods reachedWestern Europe viaLatin translations of Ptolemy's GreekAlmagest as well as the works ofPersian and Arab astronomers such asAl Battani andNasir al-Din al-Tusi.[24] One of the earliest works on trigonometry by a northern European mathematician isDe Triangulis by the 15th century German mathematicianRegiomontanus, who was encouraged to write, and provided with a copy of theAlmagest, by theByzantine Greek scholar cardinalBasilios Bessarion with whom he lived for several years.[25] At the same time, another translation of theAlmagest from Greek into Latin was completed by the CretanGeorge of Trebizond.[26] Trigonometry was still so little known in 16th-century northern Europe thatNicolaus Copernicus devoted two chapters ofDe revolutionibus orbium coelestium to explain its basic concepts.
In this right triangle:sinA =a/h;cosA =b/h;tanA =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 aresimilar.[31]
So, these ratios definefunctions of this angle that are calledtrigonometric functions. Explicitly, they are defined below as functions of the known angleA, wherea, b andh refer to the lengths of the sides in the accompanying figure.
In the following definitions, thehypotenuse 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 angleA. Theadjacent leg is the other side that is adjacent to angleA. Theopposite side is the side that is opposite to angleA. The termsperpendicular andbase are sometimes used for the opposite and adjacent sides respectively. See below underMnemonics.
Sine (denoted sin), defined as the ratio of the side opposite the angle to the hypotenuse.
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.
Tangent (denoted tan), defined as the ratio of the opposite leg to the adjacent leg.
Thereciprocals of these ratios are named thecosecant (csc),secant (sec), andcotangent (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-".[32]
With these functions, one can answer virtually all questions about arbitrary triangles by using thelaw of sines and thelaw of cosines.[33] These laws can be used to compute the remaining angles and sides of any triangle as soon as two sides and their included angle or two angles and a side or three sides are known.
A common use ofmnemonics is to remember facts and relationships in trigonometry. For example, thesine,cosine, andtangent 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:[34]
Sine =Opposite ÷Hypotenuse
Cosine =Adjacent ÷Hypotenuse
Tangent =Opposite ÷Adjacent
One way to remember the letters is to sound them out phonetically (i.e./ˌsoʊkəˈtoʊə/SOH-kə-TOH-ə, similar toKrakatoa).[35] Another method is to expand the letters into a sentence, such as "SomeOldHippieCaughtAnotherHippieTrippin'OnAcid".[36]
Fig. 1a – Sine and cosine of an angle θ defined using the unit circleIndication of the sign and amount of key angles according to rotation direction
Trigonometric ratios can also be represented using theunit circle, which is the circle of radius 1 centered at the origin in the plane.[37] In this setting, theterminal side of an angleA placed instandard position will intersect the unit circle in a point (x,y), where and.[37] This representation allows for the calculation of commonly found trigonometric values, such as those in the following table:[38]
Function
0
sine
cosine
tangent
undefined
secant
undefined
cosecant
undefined
undefined
cotangent
undefined
undefined
Trigonometric functions of real or complex variables
Because the six main trigonometric functions are periodic, they are notinjective (or, 1 to 1), and thus are not invertible. Byrestricting the domain of a trigonometric function, however, they can be made invertible.[42]: 48ff
The names of the inverse trigonometric functions, together with their domains and range, can be found in the following table:[42]: 48ff [43]: 521ff
When considered as functions of a real variable, the trigonometric ratios can be represented byMaclaurin series. For instance, sine and cosine have the following representations[44]
With these definitions the trigonometric functions can be defined forcomplex numbers.[45] When extended as functions of real or complex variables, the followingformula holds for the complex exponential:
This complex exponential function, written in terms of trigonometric functions, is particularly useful.[46][47]
Trigonometric functions were among the earliest uses formathematical tables.[48] Such tables were incorporated into mathematics textbooks and students were taught to look up values and how tointerpolate between the values listed to get higher accuracy.[49]Slide rules had special scales for trigonometric functions.[50]
Scientific calculators have buttons for calculating the main trigonometric functions (sin, cos, tan, and sometimescis and their inverses).[51] Most allow a choice of angle measurement methods:degrees, radians, and sometimesgradians. Most computerprogramming languages provide function libraries that include the trigonometric functions.[52] Thefloating point unit hardware incorporated into the microprocessor chips used in most personal computers has built-in instructions for calculating trigonometric functions.[53]
In addition to the six ratios listed earlier, there are additional trigonometric functions that were historically important, though seldom used today. These include thechord (crd(θ) = 2 sin(θ/2)), theversine (versin(θ) = 1 − cos(θ) = 2 sin2(θ/2)) (which appeared in the earliest tables[54]), thecoversine (coversin(θ) = 1 − sin(θ) = versin(π/2 −θ)), thehaversine (haversin(θ) =1/2versin(θ) = sin2(θ/2)),[55] theexsecant (exsec(θ) = sec(θ) − 1), and theexcosecant (excsc(θ) = exsec(π/2 −θ) = csc(θ) − 1). SeeList of trigonometric identities for more relations between these functions.
For centuries, spherical trigonometry has been used for locating solar, lunar, and stellar positions,[56] predicting eclipses, and describing the orbits of the planets.[57]
Sextants are used to measure the angle of the sun or stars with respect to the horizon. Using trigonometry and amarine 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.[59]
Function (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, (in blue), which depictsamplitude vsfrequency, reveals the 6 frequencies (at odd harmonics) and their amplitudes (1/odd number).
Triangle with sidesa,b,c and respectively opposite anglesA,B,C
Trigonometry has been noted for its many identities, that is, equations that are true for all possible inputs.[83]
Identities involving only angles are known astrigonometric identities. Other equations, known astriangle identities,[84] relate both the sides and angles of a given triangle.
Triangle identities
In the following identities,A,B andC are the angles of a triangle anda,b andc are the lengths of sides of the triangle opposite the respective angles (as shown in the diagram).
Law of sines
Thelaw of sines (also known as the "sine rule") for an arbitrary triangle states:[85]
where is the area of the triangle andR is the radius of thecircumscribed circle of the triangle:
Law of cosines
Thelaw of cosines (known as the cosine formula, or the "cos rule") is an extension of the Pythagorean theorem to arbitrary triangles:[85]
or equivalently:
Law of tangents
Thelaw of tangents, developed byFrançois 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.[86] It is given by:
Area
Given two sidesa andb and the angle between the sidesC, thearea 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:[85]
Other commonly used trigonometric identities include the half-angle identities, the angle sum and difference identities, and the product-to-sum identities.[31]
^"Nasir al-Din al-Tusi".MacTutor History of Mathematics archive. Retrieved2021-01-08.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.
^"ṬUSI, NAṢIR-AL-DIN i. Biography".Encyclopaedia Iranica. Retrieved2018-08-05.His major contribution in mathematics (Nasr, 1996, pp. 208–214) is said to be in trigonometry, which for the first time was compiled by him as a new discipline in its own right. Spherical trigonometry also owes its development to his efforts, and this includes the concept of the six fundamental formulas for the solution of spherical right-angled triangles.
^Berggren, J. Lennart (2007). "Mathematics in Medieval Islam".The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook. Princeton University Press. p. 518.ISBN978-0-691-11485-9.
^Humble, Chris (2001).Key Maths : GCSE, Higher. Fiona McGill. Cheltenham: Stanley Thornes Publishers. p. 51.ISBN0-7487-3396-5.OCLC47985033.
^A sentence more appropriate for high schools is "'SomeOldHorseCameA''HoppingThroughOurAlley".Foster, Jonathan K. (2008).Memory: A Very Short Introduction. Oxford. p. 128.ISBN978-0-19-280675-8.
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