Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Trigonometry “Trig” redirects here. For other uses, see Trig (disam- tive curvature, in elliptic geometry (a fundamental part biguation). of astronomy and navigation). Trigonometry on surfaces Trigonometry (from Greek trigōnon, “triangle” and of negative curvature is part of hyperbolic geometry. Trigonometry basics are often taught in schools, either as a separate course or as a part of a precalculus course. d cr Fc excsc H ot A cvs G csc 1 sin tan 1 History arc θ C O cos versin D exsec Main article: History of trigonometry Sumerian astronomers studied angle measure, using a di- E sec B All of the trigonometric functions of an angle θ can be constructed geometrically in terms of a unit circle centered at O. metron, “measure”[1] ) is a branch of mathematics that studies relationships involving 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 3rd-century astronomers first noted that the lengths of the sides of a right-angle triangle and the angles between those sides have fixed relationships: that is, if at least the length of one side and the value of one angle is known, then all other angles and lengths can be determined algorithmically. These calculations soon came to be defined as the trigonometric functions and today are pervasive in both pure and applied mathematics: fundamental methods of analysis such as the Fourier transform, for example, or the wave equation, use trigonometric functions to understand cyclical phenomena across many applications in fields as diverse as physics, mechanical and electrical engineering, music and acoustics, astronomy, ecology, and biology. Trigonometry is also the foundation of surveying. 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 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.[5] Trigonometry is most simply associated with planar rightangle triangles (each of which is a two-dimensional triangle with one angle equal to 90 degrees). The applicability to non-right-angle triangles exists, but, since any non-right-angle triangle (on a flat plane) can be bisected to create two right-angle triangles, most problems can be reduced to calculations on right-angle triangles. Thus the majority of applications relate to right-angle triangles. One exception to this is spherical trigonometry, the study of triangles on spheres, surfaces of constant posi- In the 3rd century BC, Hellenistic Greek mathematicians such as Euclid (from Alexandria, Egypt) and Archimedes (from Syracuse, Sicily) 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 1 2 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.[9] 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. At about the same time, Chinese mathematicians developed trigonometry independently, although it was not a major field of study for them. 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 Arabic astronomers such as Al Battani and Nasir al-Din al-Tusi.[10] 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.[11] At the same time another translation of the Almagest from Greek into Latin was completed by the Cretan George of Trebizond.[12] 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.[13] Bartholomaeus Pitiscus was the first to use the word, publishing his Trigonometria in 1595.[14] 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 de- OVERVIEW velopment of trigonometric series.[15] Also in the 18th century, Brook Taylor defined the general Taylor series.[16] 2 Overview Main article: Trigonometric function If one angle of a triangle is 90 degrees and one of the B c Hy A e us en t po Adjacent b Opposite than algebraically. In 140 BC Hipparchus (from Iznik, Turkey) gave the first tables of chords, analogous to modern tables of sine values, and used them to solve problems in trigonometry and spherical trigonometry.[6] In the 2nd century AD the Greco-Egyptian astronomer Ptolemy (from Alexandria, Egypt) printed detailed trigonometric tables (Ptolemy’s table of chords) in Book 1, chapter 11 of his Almagest.[7] Ptolemy used chord length to define his trigonometric functions, a minor difference from the sine convention we use today.[8] (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. 2 a C 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, because the three angles of any triangle add up to 180 degrees. The two acute angles therefore add up to 90 degrees: they are complementary angles. The shape of a triangle is completely determined, except for similarity, by the angles. Once the angles are known, the ratios of the sides are determined, regardless of the overall size of the triangle. If the length of one of the sides is known, the other two are determined. 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. sin A = a opposite = . hypotenuse c • Cosine function (cos), defined as the ratio of the adjacent leg to the hypotenuse. cos A = b adjacent = . hypotenuse c • Tangent function (tan), defined as the ratio of the opposite leg to the adjacent leg. 2.2 Mnemonics tan A = 3 opposite a a c a b sin A = = ∗ = / = . adjacent b c b c c cos A 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. Many people find it easy to remember what sides of the right triangle are equal to sine, cosine, or tangent, by memorizing the word SOH-CAH-TOA (see below under Mnemonics). The reciprocals of these functions are named the cosecant (csc or cosec), secant (sec), and cotangent (cot), Fig. 1a – Sine and cosine of an angle θ defined using the unit respectively: circle. csc A = 1 hypotenuse c = = , sin A opposite a sec A = 1 hypotenuse c = = , cos A adjacent b cot A = 1 adjacent cos A b = = = . tan A opposite sin A a from calculus and infinite series. With these definitions the trigonometric functions can be defined for complex numbers. The complex exponential function is particularly useful. ex+iy = ex (cos y + i sin y). The inverse functions are called the arcsine, arccosine, See Euler’s and De Moivre’s formulas. and arctangent, respectively. There are arithmetic re• Graphing process of y = sin(x) using a unit circle. lations between these functions, which are known as trigonometric identities. The cosine, cotangent, and cose• Graphing process of y = csc(x), the reciprocal of cant are so named because they are respectively the sine, sine, using a unit circle. tangent, and secant of the complementary angle abbrevi• Graphing process of y = tan(x) using a unit circle. ated to “co-". With these functions one can answer virtually all questions about arbitrary triangles by using the law of sines and the law of cosines. 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. These laws are useful in all branches of geometry, since every polygon may be described as a finite combination of triangles. 2.1 Extending the definitions 2.2 Mnemonics Main article: 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 SOHCAH-TOA:[17] The above definitions only apply to angles between 0 and Sine = Opposite ÷ Hypotenuse 90 degrees (0 and π/2 radians). Using the unit circle, one can extend them to all positive and negative arguCosine = Adjacent ÷ Hypotenuse ments (see trigonometric function). The trigonometric Tangent = Opposite ÷ Adjacent functions are periodic, with a period of 360 degrees or 2π radians. That means their values repeat at those in- One way to remember the letters is to sound them out tervals. The tangent and cotangent functions also have a phonetically (i.e., SOH-CAH-TOA, which is pronounced shorter period, of 180 degrees or π radians. 'so-kə-toe-uh' /soʊkəˈtoʊə/). Another method is to exThe trigonometric functions can be defined in other ways pand the letters into a sentence, such as "Some Old besides the geometrical definitions above, using tools Hippie Caught Another Hippie Trippin' On Acid”.[18] 4 2.3 6 Calculating trigonometric functions Main article: Trigonometric tables Trigonometric functions were among the earliest uses for mathematical tables. 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. Slide rules had special scales for trigonometric functions. COMMON FORMULAE etry is essential) and hence navigation (on the oceans, in aircraft, and in space), music theory, audio synthesis, acoustics, optics, electronics, biology, medical imaging (CAT scans and ultrasound), pharmacy, chemistry, number theory (and hence cryptology), seismology, meteorology, oceanography, many physical sciences, land surveying and geodesy, architecture, image compression, phonetics, economics, electrical engineering, mechanical engineering, civil engineering, computer graphics, cartography, crystallography and game development. Today scientific calculators have buttons for calculating the main trigonometric functions (sin, cos, tan, and sometimes cis and their inverses). Most allow a choice of angle measurement methods: degrees, radians, and some- 4 Pythagorean identities times gradians. Most computer programming languages provide function libraries that include the trigonometric Identities are those equations that hold true for any value. functions. The floating point unit hardware incorporated into the microprocessor chips used in most personal computers has built-in instructions for calculating trigonometsin2 A + cos2 A = 1 ric functions.[19] (The following two can be derived from the first.) 3 Applications of trigonometry sec2 A − tan2 A = 1 csc2 A − cot2 A = 1 5 Angle transformation formulae sin(A ± B) = sin A cos B ± cos A sin B cos(A ± B) = cos A cos B ∓ sin A sin B Sextants 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. tan(A ± B) = tan A ± tan B 1 ∓ tan A tan B cot(A ± B) = cot A cot B ∓ 1 cot B ± cot A Main article: Uses of trigonometry 6 Common formulae There is an enormous number of uses of trigonometry and trigonometric functions. For instance, the technique of triangulation is used in astronomy to measure the distance to nearby stars, in geography to measure distances between landmarks, and in satellite navigation systems. The sine and cosine functions are fundamental to the theory of periodic functions such as those that describe sound and light waves. Certain equations involving trigonometric functions are true for all angles and are known as trigonometric identities. Some identities equate an expression to a different expression involving the same angles. These are listed in List of trigonometric identities. Triangle identities that relate the sides and angles of a given triangle are listed below. In the following identities, A, B and C are the angles of Fields that use trigonometry or trigonometric functions a triangle and a, b and c are the lengths of sides of the include astronomy (especially for locating apparent po- triangle opposite the respective angles (as shown in the sitions of celestial objects, in which spherical trigonom- diagram). 6.3 Law of tangents 5 The law of cosines may be used to prove Heron’s formula, which 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 s= 1 (a + b + c), 2 then the area of the triangle is: Area = ∆ = √ abc s(s − a)(s − b)(s − c) = 4R where R is the radius of the circumcircle of the triangle. Triangle with sides a,b,c and respectively opposite angles A,B,C 6.3 Law of tangents 6.1 Law of sines The law of sines (also known as the “sine rule”) for an arbitrary triangle states: a b c abc = = = 2R = , sin A sin B sin C 2∆ The law of tangents: [ ] tan 12 (A − B) a−b [1 ] = a+b tan 2 (A + B) 6.4 Euler’s formula where ∆ is the area of the triangle and R is the radius of Euler’s formula, which states that eix = cos x + i sin x the circumscribed circle of the triangle: , produces the following analytical identities for sine, cosine, and tangent in terms of e and the imaginary unit i: abc . R= √ (a + b + c)(a − b + c)(a + b − c)(b + c − a) eix − e−ix eix + e−ix i(e−ix − eix ) sin x = , cos x = , tan x = ix Another law involving sines can be used to calculate the 2i 2 e + e−ix area of a triangle. 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 7 See also of the angle between the two sides: • Aryabhata’s sine table Area = ∆ = 6.2 1 ab sin C. 2 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: • Generalized trigonometry • Lénárt sphere • List of triangle topics • List of trigonometric identities • Rational trigonometry • Skinny triangle • Small-angle approximation c2 = a2 + b2 − 2ab cos C, or equivalently: a2 + b2 − c2 cos C = . 2ab • Trigonometric functions • Trigonometry in Galois fields • Unit circle • Uses of trigonometry 6 8 10 References [1] “trigonometry”. Online Etymology Dictionary. [2] R. Nagel (ed.), Encyclopedia of Science, 2nd Ed., The Gale Group (2002) EXTERNAL LINKS 9 Bibliography • Boyer, Carl B. (1991). A History of Mathematics (Second ed.). John Wiley & Sons, Inc. ISBN 0471-54397-7. [3] Boyer (1991). “Greek Trigonometry and Mensuration”. A History of Mathematics. p. 162. • Hazewinkel, Michiel, ed. (2001), “Trigonometric functions”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4 [4] Aaboe, Asger. Episodes from the Early History of Astronomy. New York: Springer, 2001. ISBN 0-38795136-9 • Christopher M. Linton (2004). From Eudoxus to Einstein: A History of Mathematical Astronomy . Cambridge University Press. [5] Otto Neugebauer (1975). A history of ancient mathematical astronomy. 1. Springer-Verlag. pp. 744–. ISBN 978-3-540-06995-9. [6] Thurston, pp. 235–236. • Weisstein, Eric W., “Trigonometric Addition Formulas”, MathWorld. 10 External links [7] Toomer, G. J. (1998), Ptolemy’s Almagest, Princeton University Press, ISBN 0-691-00260-6 • Khan Academy: Trigonometry, free online micro lectures [8] Thurston, pp. 239–243. • Trigonometry by Alfred Monroe Kenyon and Louis Ingold, The Macmillan Company, 1914. In images, full text presented. [9] Boyer p. 215 [10] Boyer pp. 237, 274 • Benjamin Banneker’s Trigonometry Puzzle at Convergence [11] http://www-history.mcs.st-and.ac.uk/Biographies/ Regiomontanus.html • Dave’s Short Course in Trigonometry by David Joyce of Clark University [12] N.G. Wilson, From Byzantium to Italy. Greek Studies in the Italian Renaissance, London, 1992. ISBN 0-7156-2418-0 • Trigonometry, by Michael Corral, Covers elementary trigonometry, Distributed under GNU Free Documentation License [13] Grattan-Guinness, Ivor (1997). The Rainbow of Mathematics: A History of the Mathematical Sciences. W.W. Norton. ISBN 0-393-32030-8. [14] Robert E. Krebs (2004). Groundbreaking Scientific Experiments, Inventions, and Discoveries of the Middle Ages and the Renaissance. Greenwood Publishing Group. pp. 153–. ISBN 978-0-313-32433-8. [15] William Bragg Ewald (2008). From Kant to Hilbert: a source book in the foundations of mathematics. Oxford University 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 "'Some Old Horse Came A''Hopping Through Our Alley”. Foster, Jonathan K. (2008). Memory: A Very Short Introduction. Oxford. p. 128. ISBN 0-19-280675-0. [19] Intel® 64 and IA-32 Architectures Software Developer’s Manual Combined Volumes: 1, 2A, 2B, 2C, 3A, 3B and 3C (PDF). Intel. 2013. 7 11 11.1 Text and image sources, contributors, and licenses 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, Tobias 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, LucasVB, Quadell, Noe, Antandrus, OverlordQ, APH, Secfan, Maximaximax, Halo, Icairns, Joyous!, Imjustmatthew, Jh51681, Trevor MacInnis, Canterbury Tail, Grstain, Brianjd, Mindspillage, Discospinster, ArnoldReinhold, Gianluigi, Mani1, Paul August, Dmr2, Bender235, Zaslav, Brian0918, Lycurgus, Art LaPella, Renfield, Bobo192, Longhair, TheSolomon, C S, SpeedyGonsales, Jeffgoin, Jojit fb, 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, Petiatil, Hornandsoccer, Pigman, GLaDOS, Sinecostan, Stephenb, Polluxian, Giro720, Rsrikanth05, Gustavb, NawlinWiki, Shreshth91, Wiki alf, 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, Loadmaster, 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 [email protected], RobertLovesPi, JettaMann, ThatOneGuy, Cksilver, Funnyfarmofdoom, Doctormatt, Fnlayson, Benzi455, Ausmitra, Goataraju, Gogo Dodo, Anonymi, Pascal.Tesson, Tawkerbot4, Doug Weller, Christian75, 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, Doc Tropics, Petrsw, Jj137, Dylan Lake, LibLord, Spencer, BrittonLaRoche, JAnDbot, Dan D. Ric, Barek, MER-C, The Transhumanist, Ericoides, BeeArkKey, Hut 8.5, PhilKnight, Kdarche~enwiki, Kerotan, Acroterion, Bencherlite, Penubag, Pedro, Murgh, Bennybp, Bongwarrior, VoABot II, AuburnPilot, JNW, RMN, Hmo, JamesBWatson, Kajasudhakarababu, Swpb, Twsx, Indon, Animum, Styrofoam1994, DerHexer, 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, InternationalEducation, 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., JohnBlackburne, Nburden, TheOtherJesse, Amishbhadeshia, Barneca, Sześćsetsześćdziesiątsześć, NYDirk, Ampersandfrabjous(mark), Philip Trueman, TXiKiBoT, Malinaccier, Editor plus, Technopat, Xerxesnine, Miranda, Anonymous Dissident, UYHAT238, Clark Kimberling, Anna Lincoln, DennyColt, Corvus cornix, Abdullais4u, LeaveSleaves, Rustysrfbrds99, Chriscooperlondon, Cremepuff222, Thiefer, R0ssar00, Isis4563, CO, Larklight, Enigmaman, ImmortalKnight, SmileToday, Enviroboy, Sallz0r, Sylent, Spinningspark, Insanity Incarnate, 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, Superbeecat, Nic bor, Wahrmund, Denisarona, Besimmons, 3rdAlcove, Explicit, Faithlessthewonderboy, Owenshahim, Church, Martarius, 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 albert square, 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, Citation bot, Akilaa, E2eamon, Frankenpuppy, ArthurBot, Xqbot, Sathimantha, TinucherianBot II, Techsmoke15, JimVC3, Nasnema, Ksagittariusr, 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, Calmer Waters, A8UDI, POOEATZA, Jschnur, Impala2009, Nosoup4uNOOB, SpaceFlight89, Σ, Rohitphy, Prin09, Maheshnathwani, Dude1818, Dysfunktionall, Ykhatana1, FoxBot, TobeBot, PorkHeart, Vrenator, Leebian, Diannaa, Ammodramus, Pedro999999999, Stupid geometric shapes, Uber smart man, Sideways713, Minerva27, TjBot, Beyond My Ken, Hajatvrc, Saulth, Balph Eubank, Ctdpmet2, Jack Schlederer, 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, Helpful Pixie Bot, Thisthat2011, පසිඳු කාවින්ද, Ajoygphilip, Nightenbelle, KLBot2, DBigXray, Adamdude16, TarekHammad, SRWikis, John Cummings, 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 aurantiaca, 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, Pdecalculus, Neuroxic, SantosMabel, CC Scratch, Ugog Nizdast, Batrachomyomachia303, PolishPaul, Johnbrandow, Samayak, Dennisamilcar, Chintu211997, Jnaffin, Mahusha, TreebeardTheEnt, MaximusAlphus, Nickid12, Boscar the BA, Zx5rc, Amansingh935, Prashram, Justinm211, 0079.cooladitya, HMSLavender, Joshua.bransden, Shashank142, Whikie, Marcusman05054, Piledhighandeep, Mndata, Loraof, 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 Contributors: 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 Contributors: This is a vector graphic version of Image:Circle-trig6.png by user:Tttrung which was licensed under the GFDL. ; Based on en:Image:Circle-trig6.png, which was donated to Wikipedia under GFDL by Steven G. Johnson. Original artist: This is a vector graphic version of Image:Circle-trig6.png by user:Tttrung which was licensed under the GFDL. Based on en:Image:Circle-trig6.png, which was donated to Wikipedia under GFDL by Steven G. Johnson. • File:Commons-logo.svg Source: https://upload.wikimedia.org/wikipedia/en/4/4a/Commons-logo.svg License: ? Contributors: ? Original artist: ? • File:Folder_Hexagonal_Icon.svg Source: https://upload.wikimedia.org/wikipedia/en/4/48/Folder_Hexagonal_Icon.svg License: Cc-bysa-3.0 Contributors: ? Original artist: ? • File:Frieberger_drum_marine_sextant.jpg Source: https://upload.wikimedia.org/wikipedia/commons/2/2d/Frieberger_drum_ marine_sextant.jpg License: CC BY-SA 2.5 Contributors: Own work Original artist: Ken Walker [email protected] • File:Hipparchos_1.jpeg Source: https://upload.wikimedia.org/wikipedia/commons/5/50/Hipparchos_1.jpeg License: Public domain Contributors: ? Original artist: ? • File:People_icon.svg Source: https://upload.wikimedia.org/wikipedia/commons/3/37/People_icon.svg License: CC0 Contributors: OpenClipart Original artist: OpenClipart • File:Portal-puzzle.svg Source: https://upload.wikimedia.org/wikipedia/en/f/fd/Portal-puzzle.svg License: Public domain Contributors: ? Original artist: ? • File:Sin-cos-defn-I.png Source: https://upload.wikimedia.org/wikipedia/commons/b/b5/Sin-cos-defn-I.png License: CC-BY-SA-3.0 Contributors: Transferred from en.wikipedia to Commons. Original artist: 345Kai at English Wikipedia • File:Triangle_ABC_with_Sides_a_b_c.png Source: https://upload.wikimedia.org/wikipedia/en/9/9f/Triangle_ABC_with_Sides_a_b_ c.png License: PD Contributors: ? Original artist: ? • File:TrigonometryTriangle.svg Source: https://upload.wikimedia.org/wikipedia/commons/4/4f/TrigonometryTriangle.svg License: Public domain Contributors: Own work Original artist: TheOtherJesse • File:Wikibooks-logo.svg Source: https://upload.wikimedia.org/wikipedia/commons/f/fa/Wikibooks-logo.svg License: CC BY-SA 3.0 Contributors: Own work Original artist: User:Bastique, User:Ramac et al. • File:Wikinews-logo.svg Source: https://upload.wikimedia.org/wikipedia/commons/2/24/Wikinews-logo.svg License: CC BY-SA 3.0 Contributors: This is a cropped version of Image:Wikinews-logo-en.png. Original artist: Vectorized by Simon 01:05, 2 August 2006 (UTC) Updated by Time3000 17 April 2007 to use official Wikinews colours and appear correctly on dark backgrounds. Originally uploaded by Simon. • File:Wikiquote-logo.svg Source: https://upload.wikimedia.org/wikipedia/commons/f/fa/Wikiquote-logo.svg License: Public domain Contributors: ? Original artist: ? • File:Wikisource-logo.svg Source: https://upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg License: CC BY-SA 3.0 Contributors: Rei-artur Original artist: Nicholas Moreau • File:Wikiversity-logo-Snorky.svg Source: https://upload.wikimedia.org/wikipedia/commons/1/1b/Wikiversity-logo-en.svg License: CC BY-SA 3.0 Contributors: Own work Original artist: Snorky • File:Wiktionary-logo-en.svg Source: https://upload.wikimedia.org/wikipedia/commons/f/f8/Wiktionary-logo-en.svg License: Public domain Contributors: Vector version of Image:Wiktionary-logo-en.png. Original artist: Vectorized by Fvasconcellos (talk · contribs), based on original logo tossed together by Brion Vibber 11.3 Content license • Creative Commons Attribution-Share Alike 3.0