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47-725: Cap Blanc-Nez was a vital measuring point for the eighteenth-century trigonometric survey linking the Paris Observatory with the Royal Greenwich Observatory . Sightings were made across the English Channel to Dover Castle and Fairlight Windmill on the South Downs . This Anglo-French Survey was led in England by General William Roy . Some miles away to the southwest of Cap Blanc-Nez

94-410: A chord whose endpoints are separated by an arc of n  degrees, for n ranging from ⁠ 1 / 2 ⁠ to 180 by increments of  ⁠ 1 / 2 ⁠ . In modern notation, the length of the chord corresponding to an arc of θ  degrees is As θ goes from 0 to 180, the chord of a θ ° arc goes from 0 to 120. For tiny arcs, the chord is to the arc angle in degrees as π

141-417: A half-arc, the chord of the sum of two arcs, and the chord of a difference of two arcs. The theorem states that for a quadrilateral inscribed in a circle , the product of the lengths of the diagonals equals the sum of the products of the two pairs of lengths of opposite sides. The derivations of trigonometric identities rely on a cyclic quadrilateral in which one side is a diameter of the circle. To find

188-537: 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 the first tables of chords, analogous to modern tables of sine values , and used them to solve problems in trigonometry and spherical trigonometry . In

235-768: 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

282-409: Is 81 plus a fractional part. The integer part begins with πα , likewise not broken into 60 + 21. But the fractional part, ⁠ 4 / 60 ⁠  +  ⁠ 15 / 60 ⁠ , is written as δ , for 4, in the ⁠ 1 / 60 ⁠ column, followed by ιε , for 15, in the ⁠ 1 / 60 ⁠ column. The table has 45 lines on each of eight pages, for

329-556: 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 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

376-510: Is expressed as ρμγ ∠′. (As the table only reaches 180°, the Greek numerals for 200 and above are not used.) The fractional parts of chord lengths required great accuracy, and were given in sexagesimal notation in two columns in the table: The first column gives an integer multiple of ⁠ 1 / 60 ⁠ , in the range 0–59, the second an integer multiple of ⁠ 1 / 60 ⁠  =  ⁠ 1 / 3600 ⁠ , also in

423-745: 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. Ptolemy%27s table of chords The table of chords , created by

470-481: Is the Cap Gris-Nez . 50°55′30″N 1°42′34″E  /  50.92500°N 1.70944°E  / 50.92500; 1.70944 This Pas-de-Calais geographical article is a stub . You can help Misplaced Pages by expanding it . Trigonometric Trigonometry (from Ancient Greek τρίγωνον ( trígōnon )  'triangle' and μέτρον ( métron )  'measure')

517-582: Is 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

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564-656: 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

611-440: 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

658-417: Is to 3, or more precisely, the ratio can be made as close as desired to ⁠ π / 3 ⁠  ≈  1.047 197 55 by making θ small enough. Thus, for the arc of ⁠ 1 / 2 ⁠ ° , the chord length is slightly more than the arc angle in degrees. As the arc increases, the ratio of the chord to the arc decreases. When the arc reaches 60° , the chord length is exactly equal to

705-431: Is used for linear interpolation . Glowatzki and Göttsche showed that Ptolemy must have calculated chords to five sexigesimal places in order to achieve the degree of accuracy found in the "sixtieths" column. Chapter 10 of Book I of the Almagest presents geometric theorems used for computing chords. Ptolemy used geometric reasoning based on Proposition 10 of Book XIII of Euclid's Elements to find

752-801: 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 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

799-1167: 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

846-757: 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

893-748: The Greek astronomer, geometer, and geographer Ptolemy in Egypt during the 2nd century AD, is a trigonometric table in Book ;I, chapter 11 of Ptolemy's Almagest , a treatise on mathematical astronomy . It is essentially equivalent to a table of values of the sine function. It was the earliest trigonometric table extensive enough for many practical purposes, including those of astronomy (an earlier table of chords by Hipparchus gave chords only for arcs that were multiples of ⁠7 + 1 / 2 ⁠ ° = ⁠ π / 24 ⁠ radians ). Since

940-401: 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

987-707: 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

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1034-523: 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, 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

1081-463: The 8th and 9th centuries, the sine and other trigonometric functions have been used in Islamic mathematics and astronomy, reforming the production of sine tables. Khwarizmi and Habash al-Hasib later produced a set of trigonometric tables. A chord of a circle is a line segment whose endpoints are on the circle. Ptolemy used a circle whose diameter is 120 parts. He tabulated the length of

1128-502: The Almagest gives seven entries where some manuscripts have scribal errors, changing one "digit" (one letter, see below). Glenn Elert has made a comparison between Ptolemy's values and the true values (120 times the sine of half the angle) and has found that the root mean square error is 0.000136. But much of this is simply due to rounding off to the nearest 1/3600, since this equals 0.0002777... There are nevertheless many entries where

1175-536: 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

1222-515: 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 a systematic method for finding sides and angles of triangles. The ancient Nubians used

1269-458: 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 definition of the sine is first attested in

1316-498: 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

1363-401: The chords of 72° and 36°. That Proposition states that if an equilateral pentagon is inscribed in a circle, then the area of the square on the side of the pentagon equals the sum of the areas of the squares on the sides of the hexagon and the decagon inscribed in the same circle. He used Ptolemy's theorem on quadrilaterals inscribed in a circle to derive formulas for the chord of

1410-442: The chords of arcs of 1° and ⁠ 1 / 2 ⁠ ° he used approximations based on Aristarchus's inequality . The inequality states that for arcs α and β , if 0 <  β  <  α  < 90°, then Ptolemy showed that for arcs of 1° and ⁠ 1 / 2 ⁠ °, the approximations correctly give the first two sexagesimal places after the integer part. Gerald J. Toomer in his translation of

1457-495: 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

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1504-448: 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

1551-423: 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

1598-405: 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 is known for its many identities . These trigonometric identities are commonly used for rewriting trigonometrical expressions with

1645-443: The last "digit" is off by 1 (too high or too low) from the best rounded value. Ptolemy's values are often too high by 1 in the last place, and more so towards the higher angles. The largest errors are about 0.0004, which still corresponds to an error of only 1 in the last sexagesimal digit. Lengths of arcs of the circle, in degrees, and the integer parts of chord lengths, were expressed in a base 10 numeral system that used 21 of

1692-505: The letters of the Greek alphabet with the meanings given in the following table, and a symbol, "∠′ " , that means ⁠ 1 / 2 ⁠ and a raised circle "○" that fills a blank space (effectively representing zero). Three of the letters, labeled "archaic" in the table below, had not been in use in the Greek language for some centuries before the Almagest was written, but were still in use as numerals and musical notes . Thus, for example, an arc of ⁠143 + 1 / 2 ⁠ °

1739-446: The nearest  ⁠ 1 / 60 ⁠ . After the columns for the arc and the chord, a third column is labeled "sixtieths". For an arc of  θ °, the entry in the "sixtieths" column is This is the average number of sixtieths of a unit that must be added to chord( θ °) each time the angle increases by one minute of arc, between the entry for  θ ° and that for ( θ  +  ⁠ 1 / 2 ⁠ )°. Thus, it

1786-412: The number of degrees in the arc, i.e. chord 60° = 60. For arcs of more than 60°, the chord is less than the arc, until an arc of 180° is reached, when the chord is only 120. The fractional parts of chord lengths were expressed in sexagesimal (base 60) numerals. For example, where the length of a chord subtended by a 112° arc is reported to be 99,29,5, it has a length of rounded to

1833-421: 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

1880-573: The range 0–59. Thus in Heiberg's edition of the Almagest with the table of chords on pages 48–63 , the beginning of the table, corresponding to arcs from ⁠ 1 / 2 ⁠ ° to ⁠7 + 1 / 2 ⁠ °, looks like this: Later in the table, one can see the base-10 nature of the numerals expressing the integer parts of the arc and the chord length. Thus an arc of 85° is written as πε ( π for 80 and ε for 5) and not broken down into 60 + 25. The corresponding chord length

1927-400: 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

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1974-391: 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 }

2021-430: 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

2068-528: 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

2115-506: 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

2162-440: 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 ⁠ ) ),

2209-426: 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

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