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sqrt 3 3 1 displaystyle 1 cosecant undefined 2 displaystyle 2 2 displaystyle sqrt 2 2 3 3 displaystyle 2 sqrt 3 3 1 displaystyle 1 2 3 3 displaystyle 2 sqrt 3 3 2 displaystyle sqrt 2 2 displaystyle 2 undefined cotangent undefined 3 displaystyle sqrt 3 1 displaystyle 1 3 3 displaystyle sqrt 3 3 0 displaystyle 0 3 3 displaystyle sqrt 3 3 1 displaystyle 1 3 displaystyle sqrt 3 undefined trigonometric functions of real or complex variables main article trigonometric function using the unit circle one can extend the definitions of trigonometric ratios to all positive and negative arguments 37 see trigonometric function graphs of trigonometric functions the following table summarizes the properties of the graphs of the six main trigonometric functions 38 39 function period domain range graph sine 2 π displaystyle 2 pi displaystyle infty infty 1 1 displaystyle 1 1 cosine 2 π displaystyle 2 pi displaystyle infty infty 1 1 displaystyle 1 1 tangent π displaystyle pi x π 2 n π displaystyle x neq pi 2 n pi displaystyle infty infty secant 2 π displaystyle 2 pi x π 2 n π displaystyle x neq pi 2 n pi 1 1 displaystyle infty 1 cup 1 infty cosecant 2 π displaystyle 2 pi x n π displaystyle x neq n pi 1 1 displaystyle infty 1 cup 1 infty cotangent π displaystyle pi x n π displaystyle x neq n pi displaystyle infty infty inverse trigonometric functions main article inverse trigonometric functions because the six main trigonometric functions are periodic they are not injective or 1 to 1 and thus are not invertible by restricting the domain of a trigonometric function however they can be made invertible 40 48ff the names of the inverse trigonometric functions together with their domains and range can be found in the following table 40 48ff 41 521ff name usual notation definition domain of x for real result range of usual principal value radians range of usual principal value degrees arcsine y arcsin x x sin y 1 x 1 π 2 y π 2 90 y 90 arccosine y arccos x x cos y 1 x 1 0 y π 0 y 180 arctangent y arctan x x tan y all real numbers π 2 y π 2 90 y 90 arccotangent y arccot x x cot y all real numbers 0 y π 0 y 180 arcsecant y arcsec x x sec y x 1 or 1 x 0 y π 2 or π 2 y π 0 y 90 or 90 y 180 arccosecant y arccsc x x csc y x 1 or 1 x π 2 y 0 or 0 y π 2 90 y 0 or 0 y 90 power series representations when considered as functions of a real variable the trigonometric ratios can be represented by an infinite series for instance sine and cosine have the following representations 42 sin x x x 3 3 x 5 5 x 7 7 n 0 1 n x 2 n 1 2 n 1 displaystyle begin aligned sin x x frac x 3 3 frac x 5 5 frac x 7 7 cdots sum _ n 0 infty frac 1 n x 2n 1 2n 1 end aligned cos x 1 x 2 2 x 4 4 x 6 6 n 0 1 n x 2 n 2 n displaystyle begin aligned cos x 1 frac x 2 2 frac x 4 4 frac x 6 6 cdots sum _ n 0 infty frac 1 n x 2n 2n end aligned with these definitions the trigonometric functions can be defined for complex numbers 43 when extended as functions of real or complex variables the following formula holds for the complex exponential e x i y e x cos y i sin y displaystyle e x iy e x cos y i sin y this complex exponential function written in terms of trigonometric functions is particularly useful 44 45 calculating trigonometric functions main article trigonometric tables trigonometric functions were among the earliest uses for mathematical tables 46 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 47 slide rules had special scales for trigonometric functions 48 scientific calculators have buttons for calculating the main trigonometric functions sin cos tan and sometimes cis and their inverses 49 most allow a choice of angle measurement methods degrees radians and sometimes gradians most computer programming languages provide function libraries that include the trigonometric functions 50 the floating point unit hardware incorporated into the microprocessor chips used in most personal computers has built in instructions for calculating trigonometric functions 51 other trigonometric functions main article trigonometric functions history in addition to the six ratios listed earlier there are additional trigonometric functions that were historically important though seldom used today these include the chord crd θ 2 sin θ 2 the versine versin θ 1 cos θ 2 sin 2 θ 2 which appeared in the earliest tables 52 the coversine coversin θ 1 sin θ versin π 2 θ the haversine haversin θ 1 2 versin θ sin 2 θ 2 53 the exsecant exsec θ sec θ 1 and the excosecant excsc θ exsec π 2 θ csc θ 1 see list of trigonometric identities for more relations between these functions applications main article uses of trigonometry astronomy main article astronomy for centuries spherical trigonometry has been used for locating solar lunar and stellar positions 54 predicting eclipses and describing the orbits of the planets 55 in modern times the technique of triangulation is used in astronomy to measure the distance to nearby stars 56 as well as in satellite navigation systems 16 navigation main article navigation 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 historically trigonometry has been used for locating latitudes and longitudes of sailing vessels plotting courses and calculating distances during navigation 57 trigonometry is still used in navigation through such means as the global positioning system and artificial intelligence for autonomous vehicles 58 surveying main article surveying in land surveying trigonometry is used in the calculation of lengths areas and relative angles between objects 59 on a larger scale trigonometry is used in geography to measure distances between landmarks 60 periodic functions main articles fourier series and fourier transform function s x displaystyle s x 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 s f displaystyle s f in blue which depicts amplitude vs frequency reveals the 6 frequencies at odd harmonics and their amplitudes 1 odd number the sine and cosine functions are fundamental to the theory of periodic functions 61 such as those that describe sound and light waves fourier discovered that every continuous periodic function could be described as an infinite sum of trigonometric functions even non periodic functions can be represented as an integral of sines and cosines through the fourier transform this has applications to quantum mechanics 62 and communications 63 among other fields optics and acoustics main articles optics and acoustics trigonometry is useful in many physical sciences 64 including acoustics 65 and optics 65 in these areas they are used to describe sound and light waves and to solve boundary and transmission related problems 66 other applications other fields that use trigonometry or trigonometric functions include music theory 67 geodesy audio synthesis 68 architecture 69 electronics 67 biology 70 medical imaging ct scans and ultrasound 71 chemistry 72 number theory and hence cryptology 73 seismology 65 meteorology 74 oceanography 75 image compression 76 phonetics 77 economics 78 electrical engineering mechanical engineering civil engineering 67 computer graphics 79 cartography 67 crystallography 80 and game development 79 identities main article list of trigonometric identities triangle with sides a b c and respectively opposite angles a b c trigonometry has been noted for its many identities that is equations that are true for all possible inputs 81 identities involving only angles are known as trigonometric identities other equations known as triangle identities 82 relate both the sides and angles of a given triangle triangle identities in the following identities a b and c are the angles of a triangle and a b and c are the lengths of sides of the triangle opposite the respective angles as shown in the diagram 83 law of sines the law of sines also known as the sine rule for an arbitrary triangle states 84 a sin a b sin b c sin c 2 r a b c 2 δ displaystyle frac a sin a frac b sin b frac c sin c 2r frac abc 2 delta where δ displaystyle delta is the area of the triangle and r is the radius of the circumscribed circle of the triangle r a b c a b c a b c a b c b c a displaystyle r frac abc sqrt a b c a b c a b c b c a 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 84 c 2 a 2 b 2 2 a b cos c displaystyle c 2 a 2 b 2 2ab cos c or equivalently cos c a 2 b 2 c 2 2 a b displaystyle cos c frac a 2 b 2 c 2 2ab law of tangents the law of tangents developed by françois viète is an alternative to the law of cosines when solving for the unknown edges of a triangle providing simpler computations when using trigonometric tables 85 it is given by a b a b tan 1 2 a b tan 1 2 a b displaystyle frac a b a b frac tan left tfrac 1 2 a b right tan left tfrac 1 2 a b right area given two sides a and b and the angle between the sides c the area of the triangle is given by half the product of the lengths of two sides and the sine of the angle between the two sides 84 heron s formula is another method that may be used to calculate the area of a triangle this formula states that if a triangle has sides of lengths a b and c and if the semiperimeter is s 1 2 a b c displaystyle s frac 1 2 a b c then the area of the triangle is 86 area δ s s a s b s c a b c 4 r displaystyle mbox area delta sqrt s s a s b s c frac abc 4r where r is the radius of the circumcircle of the triangle area δ 1 2 a b sin c displaystyle mbox area delta frac 1 2 ab sin c trigonometric identities pythagorean identities the following trigonometric identities are related to the pythagorean theorem and hold for any value 87 sin 2 a cos 2 a 1 displaystyle sin 2 a cos 2 a 1 tan 2 a 1 sec 2 a displaystyle tan 2 a 1 sec 2 a cot 2 a 1 csc 2 a displaystyle cot 2 a 1 csc 2 a the second and third equations are derived from dividing the first equation by cos 2 a displaystyle cos 2 a and sin 2 a displaystyle sin 2 a respectively euler s formula euler s formula which states that e i x cos x i sin x displaystyle e ix cos x i sin x produces the following analytical identities for sine cosine and tangent in terms of e and the imaginary unit i sin x e i x e i x 2 i cos x e i x e i x 2 tan x i e i x e i x e i x e i x displaystyle sin x frac e ix e ix 2i qquad cos x frac e ix e ix 2 qquad tan x frac i e ix e ix e ix e ix other trigonometric identities main article list of trigonometric identities other commonly used trigonometric identities include the half angle identities the angle sum and difference identities and the product to sum identities 29 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 unit circle uses of trigonometry references harper douglas trigonometry online etymology dictionary retrieved 2022 03 18 r nagel ed encyclopedia of science 2nd ed the gale group 2002 boyer 1991 p page needed charles william hackley 1853 a treatise on trigonometry plane and spherical with its application to navigation and surveying nautical and practical astronomy and geodesy with logarithmic trigonometrical and nautical tables g p putnam mary jane sterling 24 february 2014 trigonometry for dummies john wiley sons p 185 isbn 978 1 118 82741 3 p r halmos 1 december 2013 i want to be a mathematician an automathography springer science business media isbn 978 1 4612 1084 9 ron larson robert p hostetler 10 march 2006 trigonometry cengage learning p 230 isbn 0 618 64332 x boyer 1991 p 162 greek trigonometry and mensuration pimentel ric wall terry 2018 cambridge igcse core mathematics 4th ed hachette uk p 275 isbn 978 1 5104 2058 8 extract of page 275 otto neugebauer 1975 a history of ancient mathematical astronomy 1 springer verlag p 744 isbn 978 3 540 06995 9 thurston 1996 pp 235 236 appendix 1 hipparchus s table of chords toomer g 1998 ptolemy s almagest princeton university press isbn 978 0 691 00260 6 thurston 1996 pp 239 243 appendix 3 ptolemy s table of chords boyer 1991 p 215 gingerich owen islamic astronomy scientific american 254 4 1986 74 83 a b michael willers 13 february 2018 armchair algebra everything you need to know from integers to equations book sales p 37 isbn 978 0 7858 3595 0 nasir al din al tusi mactutor history of mathematics archive retrieved 2021 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 berggren j l october 2013 islamic mathematics the cambridge history of science vol 2 cambridge university press pp 62 83 doi 10 1017 cho9780511974007 004 isbn 9780521594486 ṭusi naṣir al din i biography encyclopaedia iranica retrieved 2018 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 trigonometry encyclopædia britannica retrieved 2008 07 21 berggren j lennart 2007 mathematics in medieval islam the mathematics of egypt mesopotamia china india and islam a sourcebook princeton university press p 518 isbn 978 0 691 11485 9 boyer 1991 pp 237 274 johann müller regiomontanus mactutor history of mathematics archive retrieved 2021 01 08 n g wilson 1992 from byzantium to italy greek studies in the italian renaissance london isbn 0 7156 2418 0 grattan guinness ivor 1997 the rainbow of mathematics a history of the mathematical sciences w w norton isbn 978 0 393 32030 5 robert e krebs 2004 groundbreaking scientific experiments inventions and discoveries of the middle ages and the renaissance greenwood publishing group p 153 isbn 978 0 313 32433 8 william bragg ewald 2007 from kant to hilbert a source book in the foundations of mathematics oxford university press us p 93 isbn 0 19 850535 3 kelly dempski 2002 focus on curves and surfaces p 29 isbn 1 59200 007 x a b james stewart lothar redlin saleem watson 16 january 2015 algebra and trigonometry cengage learning p 448 isbn 978 1 305 53703 3 dick jardine amy shell gellasch 2011 math...