Trigonometry

“Trig” redirects here. For other uses, see Trig (disam-biguation).Trigonometry (from Greek trigōnon, “triangle” and

Oθ

A

B

CD E

F

GH

1sin

cvsexcsc

csc

cos versin exsec

sec

crd

arc

cot

tan

All of the trigonometric functions of an angle θ can be con-structed geometrically in terms of a unit circle centered at O.

metron, “measure”[1]) is a branch of mathematics thatstudies relationships involving lengths and angles oftriangles. The field emerged in the Hellenistic world dur-ing the 3rd century BC from applications of geometry toastronomical studies.[2]

The 3rd-century astronomers first noted that the lengthsof the sides of a right-angle triangle and the angles be-tween those sides have fixed relationships: that is, if atleast the length of one side and the value of one angleis known, then all other angles and lengths can be de-termined algorithmically. These calculations soon cameto be defined as the trigonometric functions and todayare pervasive in both pure and applied mathematics: fun-damental methods of analysis such as the Fourier trans-form, for example, or the wave equation, use trigono-metric functions to understand cyclical phenomena acrossmany applications in fields as diverse as physics, mechan-ical and electrical engineering, music and acoustics, as-tronomy, ecology, and biology. Trigonometry is also thefoundation of surveying.Trigonometry is most simply associated with planar right-angle triangles (each of which is a two-dimensional tri-angle with one angle equal to 90 degrees). The appli-cability to non-right-angle triangles exists, but, since anynon-right-angle triangle (on a flat plane) can be bisectedto create two right-angle triangles, most problems can bereduced to calculations on right-angle triangles. Thusthe majority of applications relate to right-angle trian-gles. One exception to this is spherical trigonometry, thestudy of triangles on spheres, surfaces of constant posi-

tive curvature, in elliptic geometry (a fundamental partof astronomy and navigation). Trigonometry on surfacesof negative curvature is part of hyperbolic geometry.Trigonometry basics are often taught in schools, either asa separate course or as a part of a precalculus course.

1 History

Main article: History of trigonometrySumerian astronomers studied angle measure, using a di-

Hipparchus, credited with compiling the first trigonometric table,is known as “the father of trigonometry”.[3]

vision of circles into 360 degrees.[4] They, and later theBabylonians, studied the ratios of the sides of similar tri-angles and discovered some properties of these ratios butdid not turn that into a systematic method for finding sidesand angles of triangles. The ancient Nubians used a sim-ilar method.[5]

In the 3rd century BC, Hellenistic Greek mathematicianssuch as Euclid (from Alexandria, Egypt) and Archimedes(from Syracuse, Sicily) studied the properties of chordsand inscribed angles in circles, and they proved theo-rems that are equivalent to modern trigonometric formu-lae, although they presented them geometrically rather

1

2 2 OVERVIEW

than algebraically. In 140 BC Hipparchus (from Iznik,Turkey) gave the first tables of chords, analogous to mod-ern tables of sine values, and used them to solve prob-lems in trigonometry and spherical trigonometry.[6] In the2nd century AD the Greco-Egyptian astronomer Ptolemy(from Alexandria, Egypt) printed detailed trigonometrictables (Ptolemy’s table of chords) in Book 1, chapter 11of his Almagest.[7] Ptolemy used chord length to definehis trigonometric functions, a minor difference from thesine convention we use today.[8] (The value we call sin(θ)can be found by looking up the chord length for twicethe angle of interest (2θ) in Ptolemy’s table, and then di-viding that value by two.) Centuries passed before moredetailed tables were produced, and Ptolemy’s treatise re-mained in use for performing trigonometric calculationsin astronomy throughout the next 1200 years in the me-dieval Byzantine, Islamic, and, later, Western Europeanworlds.The modern sine convention is first attested in the SuryaSiddhanta, and its properties were further documentedby the 5th century (AD) Indian mathematician and as-tronomer Aryabhata.[9] These Greek and Indian workswere translated and expanded by medieval Islamic math-ematicians. By the 10th century, Islamic mathemati-cians were using all six trigonometric functions, had tab-ulated their values, and were applying them to prob-lems in spherical geometry. At about the same time,Chinese mathematicians developed trigonometry inde-pendently, although it was not a major field of studyfor them. Knowledge of trigonometric functions andmethods reached Western Europe via Latin translationsof Ptolemy’s Greek Almagest as well as the works ofPersian and Arabic astronomers such as Al Battani andNasir al-Din al-Tusi.[10] One of the earliest works ontrigonometry by a northern European mathematician isDe Triangulis by the 15th century German mathemati-cian Regiomontanus, who was encouraged to write, andprovided with a copy of the Almagest, by the ByzantineGreek scholar cardinal Basilios Bessarion with whom helived for several years.[11] At the same time another trans-lation of the Almagest from Greek into Latin was com-pleted by the Cretan George of Trebizond.[12] Trigonom-etry was still so little known in 16th-century northern Eu-rope that Nicolaus Copernicus devoted two chapters ofDe revolutionibus orbium coelestium to explain its basicconcepts.Driven by the demands of navigation and the grow-ing need for accurate maps of large geographicareas, trigonometry grew into a major branch ofmathematics.[13] Bartholomaeus Pitiscus was the first touse the word, publishing his Trigonometria in 1595.[14]Gemma Frisius described for the first time the methodof triangulation still used today in surveying. It wasLeonhard Euler who fully incorporated complex num-bers into trigonometry. The works of the Scottish math-ematicians James Gregory in the 17th century and ColinMaclaurin in the 18th century were influential in the de-

velopment of trigonometric series.[15] Also in the 18thcentury, Brook Taylor defined the general Taylor se-ries.[16]

2 Overview

Main article: Trigonometric functionIf one angle of a triangle is 90 degrees and one of the

A

B

C

ca

b

Adjacent

Opposite

Hypoten

use

In this right triangle: sin A = a/ c; cos A = b/ c; tan A = a/ b.

other angles is known, the third is thereby fixed, becausethe three angles of any triangle add up to 180 degrees.The two acute angles therefore add up to 90 degrees: theyare complementary angles. The shape of a triangle iscompletely determined, except for similarity, by the an-gles. Once the angles are known, the ratios of the sidesare determined, regardless of the overall size of the trian-gle. If the length of one of the sides is known, the othertwo are determined. These ratios are given by the follow-ing trigonometric functions of the known angle A, wherea, b and c refer to the lengths of the sides in the accom-panying figure:

• Sine function (sin), defined as the ratio of the sideopposite the angle to the hypotenuse.

sinA =oppositehypotenuse =

a

c.

• Cosine function (cos), defined as the ratio of theadjacent leg to the hypotenuse.

cosA =adjacenthypotenuse =

b

c.

• Tangent function (tan), defined as the ratio of theopposite leg to the adjacent leg.

2.2 Mnemonics 3

tanA =oppositeadjacent =

a

b=

a

c∗ c

b=

a

c/b

c=

sinAcosA .

The hypotenuse is the side opposite to the 90 degree an-gle in a right triangle; it is the longest side of the trian-gle and one of the two sides adjacent to angle A. Theadjacent leg is the other side that is adjacent to an-gle A. The opposite side is the side that is opposite toangle A. The terms perpendicular and base are some-times used for the opposite and adjacent sides respec-tively. Many people find it easy to remember what sidesof the right triangle are equal to sine, cosine, or tangent,by memorizing the word SOH-CAH-TOA (see below un-der Mnemonics).The reciprocals of these functions are named the cose-cant (csc or cosec), secant (sec), and cotangent (cot),respectively:

cscA =1

sinA =hypotenuseopposite =

c

a,

secA =1

cosA =hypotenuseadjacent =

c

b,

cotA =1

tanA =adjacentopposite =

cosAsinA =

b

a.

The inverse functions are called the arcsine, arccosine,and arctangent, respectively. There are arithmetic re-lations between these functions, which are known astrigonometric identities. The cosine, cotangent, and cose-cant are so named because they are respectively the sine,tangent, and secant of the complementary angle abbrevi-ated to “co-".With these functions one can answer virtually all ques-tions about arbitrary triangles by using the law of sinesand the law of cosines. These laws can be used to com-pute the remaining angles and sides of any triangle as soonas two sides and their included angle or two angles and aside or three sides are known. These laws are useful in allbranches of geometry, since every polygon may be de-scribed as a finite combination of triangles.

2.1 Extending the definitions

The above definitions only apply to angles between 0 and90 degrees (0 and π/2 radians). Using the unit circle,one can extend them to all positive and negative argu-ments (see trigonometric function). The trigonometricfunctions are periodic, with a period of 360 degrees or2π radians. That means their values repeat at those in-tervals. The tangent and cotangent functions also have ashorter period, of 180 degrees or π radians.The trigonometric functions can be defined in other waysbesides the geometrical definitions above, using tools

Fig. 1a – Sine and cosine of an angle θ defined using the unitcircle.

from calculus and infinite series. With these definitionsthe trigonometric functions can be defined for complexnumbers. The complex exponential function is particu-larly useful.

ex+iy = ex(cos y + i sin y).

See Euler’s and De Moivre’s formulas.

• Graphing process of y = sin(x) using a unit circle.

• Graphing process of y = csc(x), the reciprocal ofsine, using a unit circle.

• Graphing process of y = tan(x) using a unit circle.

2.2 Mnemonics

Main article: Mnemonics in trigonometry

A common use of mnemonics is to remember facts andrelationships in trigonometry. For example, the sine, co-sine, and tangent ratios in a right triangle can be remem-bered by representing them and their corresponding sidesas strings of letters. For instance, a mnemonic is SOH-CAH-TOA:[17]

Sine = Opposite ÷ HypotenuseCosine = Adjacent ÷ HypotenuseTangent = Opposite ÷ Adjacent

One way to remember the letters is to sound them outphonetically (i.e., SOH-CAH-TOA, which is pronounced'so-kə-toe-uh' /soʊkəˈtoʊə/). Another method is to ex-pand the letters into a sentence, such as "Some OldHippie Caught Another Hippie Trippin' On Acid”.[18]

4 6 COMMON FORMULAE

2.3 Calculating trigonometric functions

Main article: Trigonometric tables

Trigonometric functions were among the earliest uses formathematical tables. Such tables were incorporated intomathematics textbooks and students were taught to lookup values and how to interpolate between the values listedto get higher accuracy. Slide rules had special scales fortrigonometric functions.Today scientific calculators have buttons for calculatingthe main trigonometric functions (sin, cos, tan, and some-times cis and their inverses). Most allow a choice of an-gle measurement methods: degrees, radians, and some-times gradians. Most computer programming languagesprovide function libraries that include the trigonometricfunctions. The floating point unit hardware incorporatedinto the microprocessor chips used in most personal com-puters has built-in instructions for calculating trigonomet-ric functions.[19]

3 Applications of trigonometry

Sextants are used to measure the angle of the sun or stars with re-spect to the horizon. Using trigonometry and a marine chronome-ter, the position of the ship can be determined from suchmeasure-ments.

Main article: Uses of trigonometry

There is an enormous number of uses of trigonometry andtrigonometric functions. For instance, the technique oftriangulation is used in astronomy to measure the distanceto nearby stars, in geography to measure distances be-tween landmarks, and in satellite navigation systems. Thesine and cosine functions are fundamental to the theory ofperiodic functions such as those that describe sound andlight waves.Fields that use trigonometry or trigonometric functionsinclude astronomy (especially for locating apparent po-sitions of celestial objects, in which spherical trigonom-

etry is essential) and hence navigation (on the oceans,in aircraft, and in space), music theory, audio synthe-sis, acoustics, optics, electronics, biology, medical imag-ing (CAT scans and ultrasound), pharmacy, chemistry,number theory (and hence cryptology), seismology,meteorology, oceanography, many physical sciences,land surveying and geodesy, architecture, image com-pression, phonetics, economics, electrical engineering,mechanical engineering, civil engineering, computergraphics, cartography, crystallography and game devel-opment.

4 Pythagorean identities

Identities are those equations that hold true for any value.

sin2 A+ cos2 A = 1

(The following two can be derived from the first.)

sec2 A− tan2 A = 1

csc2 A− cot2 A = 1

5 Angle transformation formulae

sin(A±B) = sinA cosB ± cosA sinB

cos(A±B) = cosA cosB ∓ sinA sinB

tan(A±B) =tanA± tanB1∓ tanA tanB

cot(A±B) =cotA cotB ∓ 1

cotB ± cotA

6 Common formulae

Certain equations involving trigonometric functions aretrue for all angles and are known as trigonometric identi-ties. Some identities equate an expression to a differentexpression involving the same angles. These are listed inList of trigonometric identities. Triangle identities thatrelate the sides and angles of a given triangle are listedbelow.In the following identities, A, B and C are the angles ofa triangle and a, b and c are the lengths of sides of thetriangle opposite the respective angles (as shown in thediagram).

6.3 Law of tangents 5

Triangle with sides a,b,c and respectively opposite angles A,B,C

6.1 Law of sines

The law of sines (also known as the “sine rule”) for anarbitrary triangle states:

a

sinA =b

sinB =c

sinC = 2R =abc

2∆,

where ∆ is the area of the triangle and R is the radius ofthe circumscribed circle of the triangle:

R =abc√

(a+ b+ c)(a− b+ c)(a+ b− c)(b+ c− a).

Another law involving sines can be used to calculate thearea of a triangle. Given two sides a and b and the anglebetween the sides C, the area of the triangle is given byhalf the product of the lengths of two sides and the sineof the angle between the two sides:

Area = ∆ =1

2ab sinC.

6.2 Law of cosines

The law of cosines (known as the cosine formula, or the“cos rule”) is an extension of the Pythagorean theorem toarbitrary triangles:

c2 = a2 + b2 − 2ab cosC,

or equivalently:

cosC =a2 + b2 − c2

2ab.

The law of cosines may be used to prove Heron’s formula,which is another method that may be used to calculate thearea of a triangle. This formula states that if a triangle hassides of lengths a, b, and c, and if the semiperimeter is

s =1

2(a+ b+ c),

then the area of the triangle is:

Area = ∆ =√s(s− a)(s− b)(s− c) =

abc

4R

where R is the radius of the circumcircle of the triangle.

6.3 Law of tangents

The law of tangents:

a− b

a+ b=tan

[12 (A−B)

]tan

[12 (A+B)

]6.4 Euler’s formula

Euler’s formula, which states that eix = cosx + i sinx, produces the following analytical identities for sine, co-sine, and tangent in terms of e and the imaginary unit i:

sinx =eix − e−ix

2i, cosx =

eix + e−ix

2, tanx =

i(e−ix − eix)

eix + e−ix.

7 See also

• 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

• Trigonometry in Galois fields

• Unit circle

• Uses of trigonometry

6 10 EXTERNAL LINKS

8 References

[1] “trigonometry”. Online Etymology Dictionary.

[2] R. Nagel (ed.), Encyclopedia of Science, 2nd Ed., TheGale Group (2002)

[3] Boyer (1991). “Greek Trigonometry and Mensuration”.A History of Mathematics. p. 162.

[4] Aaboe, Asger. Episodes from the Early History of As-tronomy. New York: Springer, 2001. ISBN 0-387-95136-9

[5] Otto Neugebauer (1975). A history of ancient mathemat-ical astronomy. 1. Springer-Verlag. pp. 744–. ISBN978-3-540-06995-9.

[6] Thurston, pp. 235–236.

[7] Toomer, G. J. (1998), Ptolemy’s Almagest, Princeton Uni-versity Press, ISBN 0-691-00260-6

[8] Thurston, pp. 239–243.

[9] Boyer p. 215

[10] Boyer pp. 237, 274

[11] http://www-history.mcs.st-and.ac.uk/Biographies/Regiomontanus.html

[12] N.G.Wilson, FromByzantium to Italy. Greek Studies in theItalian Renaissance, London, 1992. ISBN 0-7156-2418-0

[13] Grattan-Guinness, Ivor (1997). The Rainbow of Mathe-matics: A History of the Mathematical Sciences. W.W.Norton. ISBN 0-393-32030-8.

[14] Robert E. Krebs (2004). Groundbreaking Scientific Ex-periments, Inventions, and Discoveries of the Middle Agesand the Renaissance. Greenwood Publishing Group. pp.153–. ISBN 978-0-313-32433-8.

[15] William Bragg Ewald (2008). From Kant to Hilbert: asource book in the foundations of mathematics. OxfordUniversity Press US. p. 93. ISBN 0-19-850535-3

[16] Kelly Dempski (2002). Focus on Curves and Surfaces. p.29. ISBN 1-59200-007-X

[17] Weisstein, Eric W., “SOHCAHTOA”, MathWorld.

[18] A sentence more appropriate for high schools is "'SomeOld Horse Came A''Hopping Through Our Alley”. Fos-ter, Jonathan K. (2008). Memory: A Very Short Introduc-tion. Oxford. p. 128. ISBN 0-19-280675-0.

[19] Intel® 64 and IA-32 Architectures Software Developer’sManual Combined Volumes: 1, 2A, 2B, 2C, 3A, 3B and3C (PDF). Intel. 2013.

9 Bibliography• Boyer, Carl B. (1991). A History of Mathematics(Second ed.). John Wiley & Sons, Inc. ISBN 0-471-54397-7.

• Hazewinkel, Michiel, ed. (2001), “Trigonometricfunctions”, Encyclopedia of Mathematics, Springer,ISBN 978-1-55608-010-4

• Christopher M. Linton (2004). From Eudoxus toEinstein: A History of Mathematical Astronomy .Cambridge University Press.

• Weisstein, Eric W., “Trigonometric Addition For-mulas”, MathWorld.

10 External links• Khan Academy: Trigonometry, free online microlectures

• Trigonometry by Alfred Monroe Kenyon and LouisIngold, The Macmillan Company, 1914. In images,full text presented.

• Benjamin Banneker’s Trigonometry Puzzle atConvergence

• Dave’s Short Course in Trigonometry by DavidJoyce of Clark University

• Trigonometry, by Michael Corral, Covers elemen-tary trigonometry, Distributed under GNU FreeDocumentation License

7

11 Text and image sources, contributors, and licenses

11.1 Text• Trigonometry Source: https://en.wikipedia.org/wiki/Trigonometry?oldid=698136325 Contributors: AxelBoldt, Zundark, Andre Engels,Danny, XJaM, Christian List, Aldie, Boleslav Bobcik, DrBob, Michael Hardy, Wshun, Ixfd64, TakuyaMurata, Eric119, ArnoLagrange,Ahoerstemeier, Ronz, Stevenj, Snoyes, Александър, Kwekubo, Evercat, Ideyal, Hydnjo, Dtgm, Tpbradbury, Furrykef, Lewisdg2000,Bevo, Pakaran, Donarreiskoffer, Robbot, Fredrik, Arkuat, Gandalf61, Sparquin, Academic Challenger, Catbar, UtherSRG, Fuelbottle, To-bias Bergemann, Marc Venot, Connelly, Giftlite, Smjg, BenFrantzDale, Herbee, Ayman, Everyking, Dratman, Guanaco, Al-khowarizmi,Eequor, Bgoldenberg, Matt Crypto, Mckaysalisbury, Lakefall~enwiki, Edcolins, Utcursch, SoWhy, SarekOfVulcan, Slowking Man, Lu-casVB, Quadell, Noe, Antandrus, OverlordQ, APH, Secfan, Maximaximax, Halo, Icairns, Joyous!, Imjustmatthew, Jh51681, TrevorMacInnis, Canterbury Tail, Grstain, Brianjd, Mindspillage, Discospinster, ArnoldReinhold, Gianluigi, Mani1, Paul August, Dmr2, Ben-der235, Zaslav, Brian0918, Lycurgus, Art LaPella, Renfield, Bobo192, Longhair, TheSolomon, C S, SpeedyGonsales, Jeffgoin, Jojitfb, 3mta3, Cherlin, Polylerus, M vitaly, Nsaa, Oolong, Alansohn, Happenstantially~enwiki, AzaToth, Krischik, Wtmitchell, Velella,Almafeta, Cburnett, Yuckfoo, Dirac1933, Zereshk, HenryLi, Dan100, Oleg Alexandrov, Joriki, Uncle G, Kokoriko, Davidkazuhiro,Jeff3000, Mpatel, Striver, Prashanthns, Palica, MassGalactusUniversum, Graham87, Chun-hian, David Levy, Banana!, OneWeirdDude,Vary, MarSch, KamasamaK, Salix alba, Bhadani, MarnetteD, Matt Deres, MapsMan, Maurog, Yamamoto Ichiro, FlaBot, TiagoTiago,Old Moonraker, Mathbot, Nihiltres, SouthernNights, Nivix, RexNL, Alphachimp, Salvatore Ingala, Chobot, Jersey Devil, DVdm, Korg,Cactus.man, Philten, Abu Amaal, WriterHound, Debivort, YurikBot, Wavelength, TexasAndroid, Joerow, Sceptre, Phantomsteve, Peti-atil, Hornandsoccer, Pigman, GLaDOS, Sinecostan, Stephenb, Polluxian, Giro720, Rsrikanth05, Gustavb, NawlinWiki, Shreshth91, Wikialf, Msikma, Grafen, MathMan64, RazorICE, Joelr31, Tearlach, Rufua, Misza13, Crasshopper, DeadEyeArrow, Xiankai, Wknight94,Igiffin, Tetracube, Alecmconroy, Lt-wiki-bot, Ninly, Spondoolicks, Arthur Rubin, Dspradau, Haddock420, Skittle, Katieh5584, Cmglee,Mejor Los Indios, DVD R W, Luk, SG, Sardanaphalus, SmackBot, RDBury, Honza Záruba, Prodego, KnowledgeOfSelf, Unyoyega,Jagged 85, Davewild, Monz, Bakie, Cessator, Frymaster, Edgar181, HalfShadow, Septegram, Gaff, PeterSymonds, Gilliam, Hmains,Saros136, Bluebot, Keegan, Full Shunyata, Persian Poet Gal, SMP, MalafayaBot, SchfiftyThree, (boxed), Kungming2, DHN-bot~enwiki,Konstable, Darth Panda, D-Rock, Can't sleep, clown will eat me, Egsan Bacon, Timothy Clemans, Nick Levine, Shalom Yechiel, Ioscius,Geoboe84, Brutha~enwiki, SundarBot, PrometheusX303, DavidStern, Nakon, Valenciano, MichaelBillington, JanCeuleers, Richard001,Mwtoews, WoodyWerm, Vina-iwbot~enwiki, Ck lostsword, SashatoBot, Cyberdrummer, Lambiam, Rklawton, Mouse Nightshirt, Dbtfz,Kuru, Cronholm144, Gobonobo, Butko, Starhood`, Brian Gunderson, Noegenesis, Triacylglyceride, Jim.belk, Alpha Omicron, Loadmas-ter, Stupid Corn, Slakr, Dr Smith, Optimale, Childzy, Mets501, CmaccompH89, Dhp1080, Caiaffa, Inquisitus, Asyndeton, Paul Koning,Quantum Burrito, Malter, Joseph Solis in Australia, Shoeofdeath, Rhinny, Geekygator, PN123, Happy-melon, Quodfui, Tawkerbot2,Helierh, Conrad.Irwin, Pippin25, Robinhw, Tanthalas39, Sir Vicious, Hanspi, Nunquam Dormio, SHAMUUU, NickW557, McVities,MarsRover, Leujohn, RollEXE, Johner, Speedy mattie@hotmail.com, RobertLovesPi, JettaMann, ThatOneGuy, Cksilver, Funnyfarmof-doom, Doctormatt, Fnlayson, Benzi455, Ausmitra, Goataraju, Gogo Dodo, Anonymi, Pascal.Tesson, Tawkerbot4, Doug Weller, Chris-tian75, DBaba, Voldemortuet, Thijs!bot, Epbr123, Pstanton, Marek69, Tellyaddict, 49, Flszen, CharlotteWebb, WizardFusion, AbcXyz,Urdutext, Dantheman531, Hmrox, Sidasta, Ela112, AntiVandalBot, The Obento Musubi, RoMo37, Luna Santin, Nguyenthephuc, DocTropics, Petrsw, Jj137, Dylan Lake, LibLord, Spencer, BrittonLaRoche, JAnDbot, Dan D. Ric, Barek, MER-C, The Transhumanist, Eri-coides, BeeArkKey, Hut 8.5, PhilKnight, Kdarche~enwiki, Kerotan, Acroterion, Bencherlite, Penubag, Pedro, Murgh, Bennybp, Bongwar-rior, VoABot II, AuburnPilot, JNW, RMN, Hmo, JamesBWatson, Kajasudhakarababu, Swpb, Twsx, Indon, Animum, Styrofoam1994, Der-Hexer, Gjd001, Uber-Nerd, MartinBot, Shivdas, Miraculousrandomness, Rettetast, Ashwin.kj, Mschel, Kostisl, R'n'B, Deathnroll, Pekaje,PrestonH, Smokizzy, PStrait, J.delanoy, Kimse, Trusilver, Rgoodermote, Ali, Tikiwont, Terrek, Lightcatcher, Ajmint, InternationalEd-ucation, Krishnachandranvn, The Transhumanist (AWB), NewEnglandYankee, SJP, Policron, LeighvsOptimvsMaximvs, Sargefam6usa,Kraftlos, Jrc98, Mufka, FJPB, Han Solar de Harmonics, Shshshsh, NinjaTerdle, JavierMC, Kiwiinoz77, Vinsfan368, CP TTD, Xiahou,Squids and Chips, Idioma-bot, Xnuala, Lights, X!, Lord of Conquest, Deor, Rmirester, VolkovBot, ABF, Pleasantville, Jeff G., John-Blackburne, Nburden, TheOtherJesse, Amishbhadeshia, Barneca, Sześćsetsześćdziesiątsześć, NYDirk, Ampersandfrabjous(mark), PhilipTrueman, TXiKiBoT, Malinaccier, Editor plus, Technopat, Xerxesnine, Miranda, Anonymous Dissident, UYHAT238, Clark Kimber-ling, Anna Lincoln, DennyColt, Corvus cornix, Abdullais4u, LeaveSleaves, Rustysrfbrds99, Chriscooperlondon, Cremepuff222, Thiefer,R0ssar00, Isis4563, CO, Larklight, Enigmaman, ImmortalKnight, SmileToday, Enviroboy, Sallz0r, Sylent, Spinningspark, Insanity In-carnate, Dmcq, AlleborgoBot, Praefectorian, Symane, W4chris, Hughey, Ken Kuniyuki, Kbrose, Netopalis, Minestrone Soup, SieBot,Coffee, Dusti, Nubiatech, Tiddly Tom, Work permit, Scarian, Sawjansee, BotMultichill, Dtreed, Viskonsas, Caltas, Yintan, Bentogoa,Tiptoety, Arbor to SJ, CutOffTies, Nuttycoconut, KPH2293, Bagatelle, Hobartimus, OKBot, Infemous, Kaycooksey, Randomblue, Su-perbeecat, Nic bor, Wahrmund, Denisarona, Besimmons, 3rdAlcove, Explicit, Faithlessthewonderboy, Owenshahim, Church, Martar-ius, DocRushing, ClueBot, NickCT, Kl4m, GorillaWarfare, Alpha Beta Epsilon, Fyyer, The Thing That Should Not Be, Helenabella,TrigWorks, Abhinav, Aditibhatia29, Blocklayer, R000t, Utoddl, LarnerGAL, Crumbinator, Ramiz.Ibrahim, Excirial, Alexbot, Abrech,ParisianBlade, Prancibaldfpants, Echion2, NuclearWarfare, CylonSix, Cenarium, Aurora2698, Jotterbot, Eustress, Razorflame, Dekisugi,DatDoo, BOTarate, Thingg, Kruusamägi, Ubardak, DumZiBoT, Kiensvay, Crazy Boris with a red beard, Bgreise24, BendersGame,Rankiri, Jovianeye, Duncan, Avoided, SilvonenBot, NellieBly, Dinosaursrule42, Sw8511, Matt5215, HexaChord, Simon12345, Addbot,Cxz111, Wigert, Physprob, Tcncv, Landon1980, Non-dropframe, Captain-tucker, AkhtaBot, Georgebush3, Jncraton, Fieldday-sunday,Catherinemandot, CanadianLinuxUser, Mohammed farag, Cst17, Mohamed Magdy, MrOllie, Glane23, Favonian, Kyle1278, 5 albertsquare, Sirturtle1990, Squandermania, Manorupa108, Ehrenkater, Timyxp, Tide rolls, Zorrobot, TeH nOmInAtOr, Quantumobserver,HerculeBot, Crt, Cooookie3001, Ben Ben, Math Champion, Luckas-bot, Yobot, Tohd8BohaithuGh1, Senator Palpatine, Gobbleswoggler,Mmxx, Ajh16, Goodberry, Wikihelper100, AnomieBOT, IRP, Proevaholic, AdjustShift, Kingpin13, Halfs, Materialscientist, Citationbot, Akilaa, E2eamon, Frankenpuppy, ArthurBot, Xqbot, Sathimantha, TinucherianBot II, Techsmoke15, JimVC3, Nasnema, Ksagittar-iusr, Tyrol5, Gap9551, Sabio101, Omnipaedista, RibotBOT, Charvest, Doulos Christos, WhizzWizard, Raulshc, Sophus Bie, GhalyBot,Shadowjams, Skepticalmouse, Nfgsurf7, Griffinofwales, Prari, FrescoBot, Tobby72, Pepper, Dan1232112321, Wikipe-tan, Oldlaptop321,Routerone, Biscuit555, Rigaudon, Swaying tree, TheStudentRoom, Jamesooders, Wireless Keyboard, Cannolis, HamburgerRadio, 4pi,Cobness, Dirtyasianmen, Watsup101, Pinethicket, I dream of horses, Boulaur, Hendvi, Kiefer.Wolfowitz, 01000100 W 01000010, CalmerWaters, A8UDI, POOEATZA, Jschnur, Impala2009, Nosoup4uNOOB, SpaceFlight89, Σ, Rohitphy, Prin09, Maheshnathwani, Dude1818,Dysfunktionall, Ykhatana1, FoxBot, TobeBot, PorkHeart, Vrenator, Leebian, Diannaa, Ammodramus, Pedro999999999, Stupid geomet-ric shapes, Uber smart man, Sideways713, Minerva27, TjBot, Beyond My Ken, Hajatvrc, Saulth, Balph Eubank, Ctdpmet2, Jack Schled-erer, Onef9day, EmausBot, Stryn, Immunize, Gfoley4, Ibbn, Faolin42, NotAnonymous0, Wikipelli, Pjboonstra, MonoALT, Sardar0987,JSquish, Fæ, Nicajelejunsutuc, Rosswante, Monterey Bay, Openstrings, Mlpearc Public, Colin.campbell.27, L Kensington, Donner60,Chewings72, Herebo, ChuispastonBot, Nicepepper, PaulAg54, ClueBot NG, Smtchahal, Kingofpoptarts, Mblum116, Arnavmishra2050,

8 11 TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES

Gareth Griffith-Jones, Gvsip, Wcherowi, Zg001, Movses-bot, Churrrp, Krishnan.adarsh, Muon, Marechal Ney, Widr, Jorgenev, HelpfulPixie Bot, Thisthat2011, පසිඳු කාවින්ද, Ajoygphilip, Nightenbelle, KLBot2, DBigXray, Adamdude16, TarekHammad, SRWikis, JohnCummings, Leonxlin, Hemil962, Cwm9, Stelpa, Wiki douglasm, SHUBHANKAN DAS, Mark Arsten, Shivanker1412, Altaïr, Theboorb,Deb 617, Brad7777, Glacialfox, Morning Sunshine, Jane33w, Pranavbhola, Fighterf4u, Bodema, Kc kennylau, Infolamer, Stigmatella au-rantiaca, Khazar2, Generaltater, Saung Tadashi, Letsbefiends, Mogism, Reniel09, UnbornA, Awesoham, Lugia2453, Frosty, Qadir.aqals,Gibbsncis, Yau Chun Yin, Piaractus, Puppuff, Abhisek365, Bilal harris, Mark viking, SomeDude123, Vishrocks, Makssr, Danerrific, Pde-calculus, Neuroxic, SantosMabel, CC Scratch, Ugog Nizdast, Batrachomyomachia303, PolishPaul, Johnbrandow, Samayak, Dennisamil-car, Chintu211997, Jnaffin, Mahusha, TreebeardTheEnt, MaximusAlphus, Nickid12, Boscar the BA, Zx5rc, Amansingh935, Prashram,Justinm211, 0079.cooladitya, HMSLavender, Joshua.bransden, Shashank142, Whikie, Marcusman05054, Piledhighandeep, Mndata, Lo-raof, Alastair winch, Fafa'Ron, Uditangshu.A, YErrrr, TaqPol, TomMarton167, Rayan1362000, Capacitor12, Parassavnani, For12for11aa,Lazzzboy, Trenteans123, Havalove, Deepak pandey mj, Bigsexii69, KasparBot, Btraegere, Fazbear7891, Mark2904, Abukooismyyeti,Xtarification, 1764cameron, Phasedeeznuts, Bossabdalla01, Salsbury saturday and Anonymous: 1403

11.2 Images• File:Cercle_trigo.png Source: https://upload.wikimedia.org/wikipedia/commons/0/05/Cercle_trigo.png License: CC-BY-SA-3.0 Con-tributors: réalisé avec un programme de dessin vectoriel par Cdang Original artist: Christophe Dang Ngoc Chan Cdang at fr.wikipedia

• File:Circle-trig6.svg Source: https://upload.wikimedia.org/wikipedia/commons/9/9d/Circle-trig6.svg License: CC-BY-SA-3.0 Contrib-utors: This is a vector graphic version of Image:Circle-trig6.png by user:Tttrung which was licensed under the GFDL. ; Based onen:Image:Circle-trig6.png, which was donated to Wikipedia under GFDL by Steven G. Johnson. Original artist: This is a vector graphicversion of Image:Circle-trig6.png by user:Tttrung which was licensed under the GFDL. Based on en:Image:Circle-trig6.png, which wasdonated to Wikipedia under GFDL by Steven G. Johnson.

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• File:TrigonometryTriangle.svg Source: https://upload.wikimedia.org/wikipedia/commons/4/4f/TrigonometryTriangle.svg License:Public domain Contributors: Own work Original artist: TheOtherJesse

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