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“Trig” redirects here. For other uses, see Trig (disam- tive curvature, in elliptic (a fundamental part biguation). of astronomy and ). Trigonometry on surfaces Trigonometry (from Greek trigōnon, “” and of negative curvature is part of . Trigonometry basics are often taught in schools, either as F cot a separate course or as a part of a precalculus course. excsc cvs H A G crd tan csc 1 sin 1 History arc θ C Main article: O cos versin DEexsec Sumerian astronomers studied measure, using a di-

sec

B

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

metron, “measure”[1]) is a branch of that studies relationships involving lengths and of . The field emerged in the Hellenistic world dur- ing 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 be- tween 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 de- termined algorithmically. These calculations soon came to be defined as the trigonometric functions and today are 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 across , credited with compiling the first trigonometric table, many applications in fields as diverse as physics, mechan- is known as “the father of trigonometry”.[3] ical and electrical engineering, music and acoustics, as- tronomy, ecology, and biology. Trigonometry is also the vision of into 360 degrees.[4] They, and later the foundation of . Babylonians, studied the ratios of the sides of similar tri- Trigonometry is most simply associated with planar right- angles and discovered some properties of these ratios but angle triangles (each of which is a two-dimensional tri- did not turn that into a systematic method for finding sides and angles of triangles. The ancient Nubians used a sim- angle with one angle equal to 90 degrees). The appli- [5] cability to non-right-angle triangles exists, but, since any ilar method. non-right-angle triangle (on a flat ) can be bisected In the 3rd century BC, Hellenistic Greek mathematicians to create two right-angle triangles, most problems can be such as (from Alexandria, Egypt) and reduced to calculations on right-angle triangles. Thus (from Syracuse, Sicily) studied the properties of chords the majority of applications relate to right-angle trian- and inscribed angles in circles, and they proved theo- gles. One exception to this is , the rems that are equivalent to modern trigonometric formu- study of triangles on , surfaces of constant posi- lae, although they presented them geometrically rather

1 2 2 OVERVIEW than algebraically. In 140 BC Hipparchus (from Iznik, velopment of trigonometric .[15] Also in the 18th Turkey) gave the first tables of chords, analogous to mod- century, defined the general Taylor se- ern tables of values, and used them to solve prob- ries.[16] lems in trigonometry and spherical trigonometry.[6] In the 2nd century AD the Greco-Egyptian astronomer (from Alexandria, Egypt) printed detailed trigonometric 2 Overview tables (Ptolemy’s table of chords) in Book 1, chapter 11 of his .[7] Ptolemy used length to define his trigonometric functions, a minor difference from the Main article: Trigonometric function sine convention we use today.[8] (The value we call sin(θ) If one angle of a triangle is 90 degrees and one of the can be found by looking up the chord length for twice the angle of interest (2θ) in Ptolemy’s table, and then di- viding that value by two.) Centuries passed before more detailed tables were produced, and Ptolemy’s treatise re- B mained in use for performing trigonometric calculations in astronomy throughout the next 1200 years in the me- Opposite dieval Byzantine, Islamic, and, later, Western European c worlds. a The modern sine convention is first attested in the , and its properties were further documented Adjacent by the 5th century (AD) Indian mathematician and as- A C b tronomer .[9] These Greek and Indian works were translated and expanded by medieval Islamic math- ematicians. By the 10th century, Islamic mathemati- cians were using all six trigonometric functions, had tab- In this : sin A = a/ c; cos A = b/ c; tan A = a/ b. ulated their values, and were applying them to prob- lems in . At about the same time, other angles is known, the third is thereby fixed, because Chinese mathematicians developed trigonometry inde- the three angles of any triangle add up to 180 degrees. pendently, although it was not a major field of study The two acute angles therefore add up to 90 degrees: they for them. Knowledge of trigonometric functions and are complementary angles. The shape of a triangle is methods reached Western Europe via Latin translations completely determined, except for similarity, by the an- of Ptolemy’s Greek Almagest as well as the works of gles. Once the angles are known, the ratios of the sides Persian and Arabic astronomers such as Al Battani and [10] are determined, regardless of the overall size of the trian- Nasir al-Din al-Tusi. One of the earliest works on gle. If the length of one of the sides is known, the other trigonometry by a northern European mathematician is two are determined. These ratios are given by the follow- De Triangulis by the 15th century German mathemati- ing trigonometric functions of the known angle A, where cian , who was encouraged to write, and a, b and c refer to the lengths of the sides in the accom- provided with a copy of the Almagest, by the Byzantine panying figure: Greek scholar cardinal Basilios Bessarion with whom he lived for several years.[11] At the same time another trans- • lation of the Almagest from Greek into Latin was com- Sine function (sin), defined as the ratio of the side pleted by the Cretan George of Trebizond.[12] Trigonom- opposite the angle to the hypotenuse. etry was still so little known in 16th-century northern Eu- rope that devoted two chapters of De revolutionibus orbium coelestium to explain its basic opposite a concepts. sin A = = . hypotenuse c Driven by the demands of navigation and the grow- ing need for accurate maps of large geographic • Cosine function (cos), defined as the ratio of the , trigonometry grew into a major branch of adjacent leg to the hypotenuse. 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 still used today in surveying. It was adjacent b who fully incorporated complex num- cos A = = . hypotenuse c bers into trigonometry. The works of the Scottish math- ematicians James Gregory in the 17th century and Colin • Maclaurin in the 18th century were influential in the de- Tangent function (tan), defined as the ratio of the opposite leg to the adjacent leg. 2.2 Mnemonics 3

opposite a a c a b sin A tan A = = = ∗ = / = . adjacent b c b c c cos A

The hypotenuse is the side opposite to the 90 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. The adjacent leg is the other side that is adjacent to an- gle A. The opposite side is the side that is opposite to angle A. The terms and base are some- times used for the opposite and adjacent sides respec- tively. 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 un- der Mnemonics). The reciprocals of these functions are named the cose- cant (csc or cosec), secant (sec), and cotangent (cot), respectively: Fig. 1a – Sine and cosine of an angle θ defined using the .

1 hypotenuse c from and infinite series. With these definitions csc A = = = , sin A opposite a the trigonometric functions can be defined for complex numbers. The complex exponential function is particu- 1 hypotenuse c sec A = = = , larly useful. cos A adjacent b 1 adjacent cos A b cot A = = = = . x+iy x tan A opposite sin A a e = e (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- lations between these functions, which are known as • Graphing process of y = sin(x) using a unit circle. 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- ated to “co-". • Graphing process of y = tan(x) using a unit circle. With these functions one can answer virtually all ques- tions about arbitrary triangles by using the law of 2.2 Mnemonics and the . These laws can be used to com- pute the remaining angles and sides of any triangle as soon Main article: Mnemonics in trigonometry 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 may be de- A common use of mnemonics is to remember facts and scribed as a finite combination of triangles. relationships 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 sides 2.1 Extending the definitions as strings of letters. For instance, a mnemonic is SOH- CAH-TOA:[17] The above definitions only apply to angles between 0 and 90 degrees (0 and π/2 ). Using the unit circle, Sine = Opposite ÷ Hypotenuse one can extend them to all positive and negative argu- Cosine = 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 ex- The 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 6 COMMON FORMULAE

2.3 Calculating trigonometric functions etry is essential) and hence navigation (on the oceans, in aircraft, and in space), music theory, audio synthe- Main article: sis, acoustics, optics, electronics, biology, medical imag- ing (CAT scans and ultrasound), pharmacy, chemistry, Trigonometric functions were among the earliest uses for number theory (and hence cryptology), seismology, mathematical tables. Such tables were incorporated into meteorology, oceanography, many physical , mathematics textbooks and students were taught to look land surveying and , architecture, image com- up values and how to interpolate between the values listed pression, phonetics, economics, electrical engineering, to get higher accuracy. Slide rules had special scales for mechanical engineering, civil engineering, computer trigonometric functions. graphics, cartography, crystallography and game devel- opment. Today scientific have buttons for calculating the 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- 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 com- puters has built-in instructions for calculating trigonomet- 2 2 ric functions.[19] sin A + cos A = 1

(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

tan A  tan B tan(A  B) = 1 ∓ tan A tan B Sextants are used to measure the angle of the or with re- ∓  cot A cot B 1 spect to the horizon. Using trigonometry and a marine chronome- cot(A B) =  ter, the position of the ship can be determined from such measure- cot B cot A ments.

Main article: Uses of trigonometry 6 Common formulae

There is an enormous number of uses of trigonometry and Certain equations involving trigonometric functions are trigonometric functions. For instance, the technique of true for all angles and are known as trigonometric identi- triangulation is used in astronomy to measure the distance ties. Some identities equate an expression to a different to nearby stars, in geography to measure distances be- expression involving the same angles. These are listed in tween landmarks, and in satellite navigation systems. The List of trigonometric identities. Triangle identities that sine and cosine functions are fundamental to the theory of relate the sides and angles of a given triangle are listed periodic functions such as those that describe sound and below. light waves. 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 5

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

1 s = (a + b + c), 2 then the area of the triangle is:

√ abc Area = ∆ = s(s − a)(s − b)(s − c) = 4R where R is the 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 The law of tangents: The law of sines (also known as the “sine rule”) for an arbitrary triangle states: [ ] − tan 1 (A − B) a b [ 2 ] = 1 a + b tan 2 (A + B) a b c abc = = = 2R = , sin A sin B sin C 2∆ 6.4 Euler’s formula where ∆ is the area of the triangle and R is the radius of the of the triangle: Euler’s formula, which states that eix = cos x + i sin x , produces the following analytical identities for sine, co- sine, 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 = . Another law involving sines can be used to calculate the 2i 2 eix + 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 • 1 Area = ∆ = ab sin C. 2 • Lénárt

• List of triangle topics 6.2 Law of cosines • List of trigonometric identities The law of cosines (known as the cosine formula, or the “cos rule”) is an extension of the Pythagorean to • Rational trigonometry arbitrary triangles: • Skinny triangle

• Small-angle approximation c2 = a2 + b2 − 2ab cos C, • Trigonometric functions or equivalently: • Trigonometry in Galois fields

• Unit circle a2 + b2 − c2 cos C = . • 2ab Uses of trigonometry 6 10 EXTERNAL LINKS

8 References 9 Bibliography

[1] “trigonometry”. Online Etymology Dictionary. • Boyer, Carl B. (1991). A (Second ed.). John Wiley & Sons, Inc. ISBN 0- [2] R. Nagel (ed.), Encyclopedia of , 2nd Ed., The 471-54397-7. Gale Group (2002) • Hazewinkel, Michiel, ed. (2001), “Trigonometric [3] Boyer (1991). “Greek Trigonometry and Mensuration”. functions”, Encyclopedia of Mathematics, Springer, A History of Mathematics. p. 162. ISBN 978-1-55608-010-4 • Christopher M. Linton (2004). From Eudoxus to [4] Aaboe, Asger. Episodes from the Early History of As- Einstein: A History of Mathematical Astronomy . tronomy. New York: Springer, 2001. ISBN 0-387- 95136-9 Cambridge University Press. • Weisstein, Eric W., “Trigonometric Addition For- [5] Otto Neugebauer (1975). A history of ancient mathemat- mulas”, MathWorld. ical astronomy. 1. Springer-Verlag. pp. 744–. ISBN 978-3-540-06995-9.

[6] Thurston, pp. 235–236. 10 External links

[7] Toomer, G. J. (1998), Ptolemy’s Almagest, Princeton Uni- • Khan Academy: Trigonometry, free online micro versity Press, ISBN 0-691-00260-6 lectures

[8] Thurston, pp. 239–243. • Trigonometry by Alfred Monroe Kenyon and Louis Ingold, The Macmillan Company, 1914. In images, [9] Boyer p. 215 full text presented. • [10] Boyer pp. 237, 274 Benjamin Banneker’s Trigonometry Puzzle at Convergence

[11] http://www-history.mcs.st-and.ac.uk/Biographies/ • Dave’s Short Course in Trigonometry by David Regiomontanus.html Joyce of Clark University

[12] N.G. Wilson, From Byzantium to Italy. Greek Studies in the • Trigonometry, by Michael Corral, Covers elemen- Italian , London, 1992. ISBN 0-7156-2418-0 tary trigonometry, Distributed under GNU Free Documentation License [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 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”. 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’s Manual Combined Volumes: 1, 2A, 2B, 2C, 3A, 3B and 3C (PDF). Intel. 2013. 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, Trevor MacInnis, 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, 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, Peti- atil, 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, 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 [email protected], 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, Doc Tropics, Petrsw, Jj137, Dylan Lake, LibLord, Spencer, BrittonLaRoche, JAnDbot, Dan D. 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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 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-by- sa-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: Open- Clipart 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