57-396: Trigonometry (from Ancient Greek τρίγωνον ( trígōnon ) 'triangle' and μέτρον ( métron ) 'measure') is a branch of mathematics concerned with relationships between angles and side lengths of triangles. In particular, the trigonometric functions relate the angles of a right triangle with ratios of its side lengths. The field emerged in
114-543: A pitch accent . In Modern Greek, all vowels and consonants are short. Many vowels and diphthongs once pronounced distinctly are pronounced as /i/ ( iotacism ). Some of the stops and glides in diphthongs have become fricatives , and the pitch accent has changed to a stress accent . Many of the changes took place in the Koine Greek period. The writing system of Modern Greek, however, does not reflect all pronunciation changes. The examples below represent Attic Greek in
171-477: A lack of contemporaneous evidence. Several theories exist about what Hellenic dialect groups may have existed between the divergence of early Greek-like speech from the common Proto-Indo-European language and the Classical period. They have the same general outline but differ in some of the detail. The only attested dialect from this period is Mycenaean Greek , but its relationship to the historical dialects and
228-419: A lesser degree. Pamphylian Greek , spoken in a small area on the southwestern coast of Anatolia and little preserved in inscriptions, may be either a fifth major dialect group, or it is Mycenaean Greek overlaid by Doric, with a non-Greek native influence. Regarding the speech of the ancient Macedonians diverse theories have been put forward, but the epigraphic activity and the archaeological discoveries in
285-467: A minor difference from the sine convention we use today. (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
342-550: A prefix /e-/, called the augment . This was probably originally a separate word, meaning something like "then", added because tenses in PIE had primarily aspectual meaning. The augment is added to the indicative of the aorist, imperfect, and pluperfect, but not to any of the other forms of the aorist (no other forms of the imperfect and pluperfect exist). The two kinds of augment in Greek are syllabic and quantitative. The syllabic augment
399-608: A separate historical stage, though its earliest form closely resembles Attic Greek , and its latest form approaches Medieval Greek . There were several regional dialects of Ancient Greek; Attic Greek developed into Koine. Ancient Greek was a pluricentric language , divided into many dialects. The main dialect groups are Attic and Ionic , Aeolic , Arcadocypriot , and Doric , many of them with several subdivisions. Some dialects are found in standardized literary forms in literature , while others are attested only in inscriptions. There are also several historical forms. Homeric Greek
456-630: A standard subject of study in educational institutions of the Western world since the Renaissance . This article primarily contains information about the Epic and Classical periods of the language, which are the best-attested periods and considered most typical of Ancient Greek. From the Hellenistic period ( c. 300 BC ), Ancient Greek was followed by Koine Greek , which is regarded as
513-471: A systematic method for finding sides and angles of triangles. The ancient Nubians used a similar method. 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
570-510: A vowel or /n s r/ ; final stops were lost, as in γάλα "milk", compared with γάλακτος "of milk" (genitive). Ancient Greek of the classical period also differed in both the inventory and distribution of original PIE phonemes due to numerous sound changes, notably the following: The pronunciation of Ancient Greek was very different from that of Modern Greek . Ancient Greek had long and short vowels ; many diphthongs ; double and single consonants; voiced, voiceless, and aspirated stops ; and
627-766: 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. At the same time, another translation of the Almagest from Greek into Latin was completed by the Cretan George of Trebizond . 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
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#1732779916206684-570: Is a literary form of Archaic Greek (derived primarily from Ionic and Aeolic) used in the epic poems , the Iliad and the Odyssey , and in later poems by other authors. Homeric Greek had significant differences in grammar and pronunciation from Classical Attic and other Classical-era dialects. The origins, early form and development of the Hellenic language family are not well understood because of
741-418: Is added to stems beginning with consonants, and simply prefixes e (stems beginning with r , however, add er ). The quantitative augment is added to stems beginning with vowels, and involves lengthening the vowel: Some verbs augment irregularly; the most common variation is e → ei . The irregularity can be explained diachronically by the loss of s between vowels, or that of the letter w , which affected
798-666: Is called 'East Greek'. Arcadocypriot apparently descended more closely from the Mycenaean Greek of the Bronze Age. Boeotian Greek had come under a strong Northwest Greek influence, and can in some respects be considered a transitional dialect, as exemplified in the poems of the Boeotian poet Pindar who wrote in Doric with a small Aeolic admixture. Thessalian likewise had come under Northwest Greek influence, though to
855-448: Is considered by some linguists to have been closely related to Greek . Among Indo-European branches with living descendants, Greek is often argued to have the closest genetic ties with Armenian (see also Graeco-Armenian ) and Indo-Iranian languages (see Graeco-Aryan ). Ancient Greek differs from Proto-Indo-European (PIE) and other Indo-European languages in certain ways. In phonotactics , ancient Greek words could end only in
912-789: Is given by: 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: The following trigonometric identities are related to the Pythagorean theorem and hold for any value: 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. Ancient Greek language Ancient Greek ( Ἑλληνῐκή , Hellēnikḗ ; [hellɛːnikɛ́ː] ) includes
969-560: Is known for its many identities . These trigonometric identities are commonly used for rewriting trigonometrical expressions with the aim to simplify an expression, to find a more useful form of an expression, or to solve an equation . Sumerian astronomers studied angle measure, using a division of circles into 360 degrees. 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
1026-653: 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". Trigonometric ratios can also be represented using the unit circle , which is the circle of radius 1 centered at the origin in the plane. 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} . This representation allows for
1083-438: 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: 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 ). Another method
1140-1140: The Fourier transform . This has applications to quantum mechanics and communications , among other fields. Trigonometry is useful in many physical sciences , including acoustics , and optics . In these areas, they are used to describe sound and light waves , and to solve boundary- and transmission-related problems. Other fields that use trigonometry or trigonometric functions include music theory , geodesy , audio synthesis , architecture , electronics , biology , medical imaging ( CT scans and ultrasound ), chemistry , number theory (and hence cryptology ), seismology , meteorology , oceanography , image compression , phonetics , economics , electrical engineering , mechanical engineering , civil engineering , computer graphics , cartography , crystallography and game development . Trigonometry has been noted for its many identities, that is, equations that are true for all possible inputs. Identities involving only angles are known as trigonometric identities . Other equations, known as triangle identities , relate both
1197-753: The Global Positioning System and artificial intelligence for autonomous vehicles . In land surveying , trigonometry is used in the calculation of lengths, areas, and relative angles between objects. On a larger scale, trigonometry is used in geography to measure distances between landmarks. The sine and cosine functions are fundamental to the theory of periodic functions , 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
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#17327799162061254-759: The Greek region of Macedonia during the last decades has brought to light documents, among which the first texts written in Macedonian , such as the Pella curse tablet , as Hatzopoulos and other scholars note. Based on the conclusions drawn by several studies and findings such as Pella curse tablet , Emilio Crespo and other scholars suggest that ancient Macedonian was a Northwest Doric dialect , which shares isoglosses with its neighboring Thessalian dialects spoken in northeastern Thessaly . Some have also suggested an Aeolic Greek classification. The Lesbian dialect
1311-515: The Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies . The Greeks focused on the calculation of chords , while mathematicians in India created the earliest-known tables of values for trigonometric ratios (also called trigonometric functions ) such as sine . Throughout history, trigonometry has been applied in areas such as geodesy , surveying , celestial mechanics , and navigation . Trigonometry
1368-399: The law of tangents for spherical triangles, and provided proofs for both these laws. 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 . One of the earliest works on trigonometry by a northern European mathematician
1425-501: The present , future , and imperfect are imperfective in aspect; the aorist , present perfect , pluperfect and future perfect are perfective in aspect. Most tenses display all four moods and three voices, although there is no future subjunctive or imperative. Also, there is no imperfect subjunctive, optative or imperative. The infinitives and participles correspond to the finite combinations of tense, aspect, and voice. The indicative of past tenses adds (conceptually, at least)
1482-702: The versine ( versin( θ ) = 1 − cos( θ ) = 2 sin( θ / 2 ) ) (which appeared in the earliest tables), the coversine ( coversin( θ ) = 1 − sin( θ ) = versin( π / 2 − θ ) ), the haversine ( haversin( θ ) = 1 / 2 versin( θ ) = sin( θ / 2 ) ), 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. For centuries, spherical trigonometry has been used for locating solar, lunar, and stellar positions, predicting eclipses, and describing
1539-1031: The 5th century BC. Ancient pronunciation cannot be reconstructed with certainty, but Greek from the period is well documented, and there is little disagreement among linguists as to the general nature of the sounds that the letters represent. /oː/ raised to [uː] , probably by the 4th century BC. Greek, like all of the older Indo-European languages , is highly inflected. It is highly archaic in its preservation of Proto-Indo-European forms. In ancient Greek, nouns (including proper nouns) have five cases ( nominative , genitive , dative , accusative , and vocative ), three genders ( masculine , feminine , and neuter ), and three numbers (singular, dual , and plural ). Verbs have four moods ( indicative , imperative , subjunctive , and optative ) and three voices (active, middle, and passive ), as well as three persons (first, second, and third) and various other forms. Verbs are conjugated through seven combinations of tenses and aspect (generally simply called "tenses"):
1596-495: The Archaic period of ancient Greek (see Homeric Greek for more details): Μῆνιν ἄειδε, θεά, Πηληϊάδεω Ἀχιλῆος οὐλομένην, ἣ μυρί' Ἀχαιοῖς ἄλγε' ἔθηκε, πολλὰς δ' ἰφθίμους ψυχὰς Ἄϊδι προΐαψεν ἡρώων, αὐτοὺς δὲ ἑλώρια τεῦχε κύνεσσιν οἰωνοῖσί τε πᾶσι· Διὸς δ' ἐτελείετο βουλή· ἐξ οὗ δὴ τὰ πρῶτα διαστήτην ἐρίσαντε Ἀτρεΐδης τε ἄναξ ἀνδρῶν καὶ δῖος Ἀχιλλεύς. The beginning of Apology by Plato exemplifies Attic Greek from
1653-661: The Classical period of ancient Greek. (The second line is the IPA , the third is transliterated into the Latin alphabet using a modern version of the Erasmian scheme .) Ὅτι [hóti Hóti μὲν men mèn ὑμεῖς, hyːmêːs hūmeîs, Ab%C5%AB al-Waf%C4%81%27 al-B%C5%ABzj%C4%81n%C4%AB Too Many Requests If you report this error to the Wikimedia System Administrators, please include
1710-534: The Scottish mathematicians James Gregory in the 17th century and Colin Maclaurin in the 18th century were influential in the development of trigonometric series . Also in the 18th century, Brook Taylor defined the general Taylor series . Trigonometric ratios are the ratios between edges of a right triangle. These ratios depend only on one acute angle of the right triangle, since any two right triangles with
1767-550: The aorist. Following Homer 's practice, the augment is sometimes not made in poetry , especially epic poetry. The augment sometimes substitutes for reduplication; see below. Almost all forms of the perfect, pluperfect, and future perfect reduplicate the initial syllable of the verb stem. (A few irregular forms of perfect do not reduplicate, whereas a handful of irregular aorists reduplicate.) The three types of reduplication are: Irregular duplication can be understood diachronically. For example, lambanō (root lab ) has
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1824-577: The area of the triangle and R is the radius of the circumscribed circle of the triangle: The law of cosines (known as the cosine formula, or the "cos rule") is an extension of the Pythagorean theorem to arbitrary triangles: or equivalently: 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. It
1881-419: The augment when it was word-initial. In verbs with a preposition as a prefix, the augment is placed not at the start of the word, but between the preposition and the original verb. For example, προσ(-)βάλλω (I attack) goes to προσ έ βαλoν in the aorist. However compound verbs consisting of a prefix that is not a preposition retain the augment at the start of the word: αὐτο(-)μολῶ goes to ηὐ τομόλησα in
1938-494: The calculation of commonly found trigonometric values, such as those in the following table: Using the unit circle , one can extend the definitions of trigonometric ratios to all positive and negative arguments (see trigonometric function ). The following table summarizes the properties of the graphs of the six main 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
1995-438: The center of Greek scholarship, this division of people and language is quite similar to the results of modern archaeological-linguistic investigation. One standard formulation for the dialects is: West vs. non-West Greek is the strongest-marked and earliest division, with non-West in subsets of Ionic-Attic (or Attic-Ionic) and Aeolic vs. Arcadocypriot, or Aeolic and Arcado-Cypriot vs. Ionic-Attic. Often non-West
2052-491: The creator of trigonometry as a mathematical discipline in its own right. He was the first to treat trigonometry as a mathematical discipline independent from astronomy, and he developed spherical trigonometry into its present form. 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
2109-446: The demands of navigation and the growing need for accurate maps of large geographic areas, trigonometry grew into a major branch of mathematics. Bartholomaeus Pitiscus was the first to use the word, publishing his Trigonometria in 1595. 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
2166-615: The dialect of Sparta ), and Northern Peloponnesus Doric (including Corinthian ). All the groups were represented by colonies beyond Greece proper as well, and these colonies generally developed local characteristics, often under the influence of settlers or neighbors speaking different Greek dialects. After the conquests of Alexander the Great in the late 4th century BC, a new international dialect known as Koine or Common Greek developed, largely based on Attic Greek , but with influence from other dialects. This dialect slowly replaced most of
2223-416: The domain of a trigonometric function, however, they can be made invertible. The names of the inverse trigonometric functions, together with their domains and range, can be found in the following table: 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: With these definitions
2280-417: The first table of cotangents. By the 10th century AD, in the work of Persian mathematician Abū al-Wafā' al-Būzjānī , all six trigonometric functions were used. Abu al-Wafa had sine tables in 0.25° increments, to 8 decimal places of accuracy, and accurate tables of tangent values. He also made important innovations in spherical trigonometry The Persian polymath Nasir al-Din al-Tusi has been described as
2337-535: The first tables of chords, analogous to modern tables of sine values , and used them to solve problems in trigonometry and spherical trigonometry . 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 . Ptolemy used chord length to define his trigonometric functions,
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2394-673: The forms of the Greek language used in ancient Greece and the ancient world from around 1500 BC to 300 BC. It is often roughly divided into the following periods: Mycenaean Greek ( c. 1400–1200 BC ), Dark Ages ( c. 1200–800 BC ), the Archaic or Epic period ( c. 800–500 BC ), and the Classical period ( c. 500–300 BC ). Ancient Greek was the language of Homer and of fifth-century Athenian historians, playwrights, and philosophers . It has contributed many words to English vocabulary and has been
2451-561: The historical Dorians . The invasion is known to have displaced population to the later Attic-Ionic regions, who regarded themselves as descendants of the population displaced by or contending with the Dorians. The Greeks of this period believed there were three major divisions of all Greek people – Dorians, Aeolians, and Ionians (including Athenians), each with their own defining and distinctive dialects. Allowing for their oversight of Arcadian, an obscure mountain dialect, and Cypriot, far from
2508-476: The historical circumstances of the times imply that the overall groups already existed in some form. Scholars assume that major Ancient Greek period dialect groups developed not later than 1120 BC, at the time of the Dorian invasions —and that their first appearances as precise alphabetic writing began in the 8th century BC. The invasion would not be "Dorian" unless the invaders had some cultural relationship to
2565-506: The medieval Byzantine , Islamic , and, later, Western European worlds. The modern definition of the sine is first attested in the Surya Siddhanta , and its properties were further documented in the 5th century (AD) by Indian mathematician and astronomer Aryabhata . These Greek and Indian works were translated and expanded by medieval Islamic mathematicians . In 830 AD, Persian mathematician Habash al-Hasib al-Marwazi produced
2622-508: The older dialects, although the Doric dialect has survived in the Tsakonian language , which is spoken in the region of modern Sparta. Doric has also passed down its aorist terminations into most verbs of Demotic Greek . By about the 6th century AD, the Koine had slowly metamorphosed into Medieval Greek . Phrygian is an extinct Indo-European language of West and Central Anatolia , which
2679-417: The orbits of the planets. In modern times, the technique of triangulation is used in astronomy to measure the distance to nearby stars, as well as in satellite navigation systems . Historically, trigonometry has been used for locating latitudes and longitudes of sailing vessels, plotting courses, and calculating distances during navigation. Trigonometry is still used in navigation through such means as
2736-487: The perfect stem eilēpha (not * lelēpha ) because it was originally slambanō , with perfect seslēpha , becoming eilēpha through compensatory lengthening. Reduplication is also visible in the present tense stems of certain verbs. These stems add a syllable consisting of the root's initial consonant followed by i . A nasal stop appears after the reduplication in some verbs. The earliest extant examples of ancient Greek writing ( c. 1450 BC ) are in
2793-399: The same acute angle are similar . So, these ratios define functions of this angle that are called trigonometric functions . Explicitly, they are defined below as functions of the known angle A , where a , b and h refer to the lengths of the sides in the accompanying figure: The hypotenuse is the side opposite to the 90-degree angle in a right triangle; it is the longest side of
2850-401: The sides and angles of a given triangle. 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). The law of sines (also known as the "sine rule") for an arbitrary triangle states: where Δ {\displaystyle \Delta } is
2907-428: The sine, tangent, and secant of the complementary angle abbreviated 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. A common use of mnemonics
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#17327799162062964-517: The syllabic script Linear B . Beginning in the 8th century BC, however, the Greek alphabet became standard, albeit with some variation among dialects. Early texts are written in boustrophedon style, but left-to-right became standard during the classic period. Modern editions of ancient Greek texts are usually written with accents and breathing marks , interword spacing , modern punctuation , and sometimes mixed case , but these were all introduced later. The beginning of Homer 's Iliad exemplifies
3021-528: 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 . The reciprocals of these ratios are named the cosecant (csc), secant (sec), and cotangent (cot), respectively: The cosine, cotangent, and cosecant are so named because they are respectively
3078-502: The trigonometric functions can be defined for complex numbers . When extended as functions of real or complex variables, the following formula holds for the complex exponential: This complex exponential function, written in terms of trigonometric functions, is particularly useful. 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
3135-438: The trigonometric functions. The floating point unit hardware incorporated into the microprocessor chips used in most personal computers has built-in instructions for calculating trigonometric functions. 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 ) ),
3192-423: The values listed to get higher accuracy. Slide rules had special scales for trigonometric functions. 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 sometimes gradians . Most computer programming languages provide function libraries that include
3249-480: Was Aeolic. For example, fragments of the works of the poet Sappho from the island of Lesbos are in Aeolian. Most of the dialect sub-groups listed above had further subdivisions, generally equivalent to a city-state and its surrounding territory, or to an island. Doric notably had several intermediate divisions as well, into Island Doric (including Cretan Doric ), Southern Peloponnesus Doric (including Laconian ,
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