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to a systematic method for finding sides and angles of triangles the ancient nubians used a similar method 10 in the 3rd century bc hellenistic mathematicians such as euclid and archimedes 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 than algebraically in 140 bc hipparchus from nicaea asia minor gave the first tables of chords analogous to modern tables of sine values and used them to solve problems in trigonometry and spherical trigonometry 11 in the 2nd century ad the greco egyptian astronomer ptolemy from alexandria egypt constructed detailed trigonometric tables ptolemy s table of chords in book 1 chapter 11 of his almagest 12 ptolemy used chord length to define his trigonometric functions a minor difference from the sine convention we use today 13 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 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 14 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 15 16 the persian polymath nasir al din al tusi has been described as the creator of trigonometry as a mathematical discipline in its own right 17 18 19 he was the first to treat trigonometry as a mathematical discipline independent from astronomy and he developed spherical trigonometry into its present form 20 he listed the six distinct cases of a right angled triangle in spherical trigonometry and in his on the sector figure he stated the law of sines for plane and spherical triangles discovered the law of tangents for spherical triangles and provided proofs for both these laws 21 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 arab astronomers such as al battani and nasir al din al tusi 22 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 23 at the same time another translation of the almagest from greek into latin was completed by the cretan george of trebizond 24 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 25 bartholomaeus pitiscus was the first to use the word publishing his trigonometria in 1595 26 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 development of trigonometric series 27 also in the 18th century brook taylor defined the general taylor series 28 trigonometric ratios main article trigonometric function in this right triangle sin a a h cos a b h tan a a b trigonometric ratios are the ratios between edges of a right triangle these ratios are given by the following trigonometric functions of the known angle a where a b and h 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 opposite hypotenuse a h displaystyle sin a frac textrm opposite textrm hypotenuse frac a h cosine function cos defined as the ratio of the adjacent leg the side of the triangle joining the angle to the right angle to the hypotenuse cos a adjacent hypotenuse b h displaystyle cos a frac textrm adjacent textrm hypotenuse frac b h tangent function tan defined as the ratio of the opposite leg to the adjacent leg tan a opposite adjacent a b a h b h sin a cos a displaystyle tan a frac textrm opposite textrm adjacent frac a b frac a h b h frac sin a 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 see below under mnemonics since any two right triangles with the same acute angle a are similar 29 the value of a trigonometric ratio depends only on the angle a the reciprocals of these functions are named the cosecant csc secant sec and cotangent cot respectively csc a 1 sin a hypotenuse opposite h a displaystyle csc a frac 1 sin a frac textrm hypotenuse textrm opposite frac h a sec a 1 cos a hypotenuse adjacent h b displaystyle sec a frac 1 cos a frac textrm hypotenuse textrm adjacent frac h b cot a 1 tan a adjacent opposite cos a sin a b a displaystyle cot a frac 1 tan a frac textrm adjacent textrm opposite frac cos a sin a frac b a the cosine cotangent and cosecant are so named because they are respectively the sine tangent and secant of the complementary angle abbreviated to co 30 with these functions one can answer virtually all questions about arbitrary triangles by using the law of sines and the law of cosines 31 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 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 soh cah toa 32 s ine o pposite h ypotenuse c osine a djacent h ypotenuse t angent o pposite a djacent one way to remember the letters is to sound them out phonetically i e ˌ s oʊ k ə ˈ t oʊ ə soh kə toh ə similar to krakatoa 33 another method is to expand the letters into a sentence such as s ome o ld h ippie c aught a nother h ippie t rippin o n a cid 34 the unit circle and common trigonometric values main article unit circle fig 1a sine and cosine of an angle θ defined using the unit circle indication of the sign and amount of key angles according to rotation direction trigonometric ratios can also be represented using the unit circle which is the circle of radius 1 centered at the origin in the plane 35 in this setting the terminal side of an angle a placed in standard position will intersect the unit circle in a point x y where x cos a displaystyle x cos a and y sin a displaystyle y sin a 35 this representation allows for the calculation of commonly found trigonometric values such as those in the following table 36 function 0 π 6 displaystyle pi 6 π 4 displaystyle pi 4 π 3 displaystyle pi 3 π 2 displaystyle pi 2 2 π 3 displaystyle 2 pi 3 3 π 4 displaystyle 3 pi 4 5 π 6 displaystyle 5 pi 6 π displaystyle pi sine 0 displaystyle 0 1 2 displaystyle 1 2 2 2 displaystyle sqrt 2 2 3 2 displaystyle sqrt 3 2 1 displaystyle 1 3 2 displaystyle sqrt 3 2 2 2 displaystyle sqrt 2 2 1 2 displaystyle 1 2 0 displaystyle 0 cosine 1 displaystyle 1 3 2 displaystyle sqrt 3 2 2 2 displaystyle sqrt 2 2 1 2 displaystyle 1 2 0 displaystyle 0 1 2 displaystyle 1 2 2 2 displaystyle sqrt 2 2 3 2 displaystyle sqrt 3 2 1 displaystyle 1 tangent 0 displaystyle 0 3 3 displaystyle sqrt 3 3 1 displaystyle 1 3 displaystyle sqrt 3 undefined 3 displaystyle sqrt 3 1 displaystyle 1 3 3 displaystyle sqrt 3 3 0 displaystyle 0 secant 1 displaystyle 1 2 3 3 displaystyle 2 sqrt 3 3 2 displaystyle sqrt 2 2 displaystyle 2 undefined 2 displaystyle 2 2 displaystyle sqrt 2 2 3 3 displaystyle 2 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 ...
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