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Early study of triangles can be traced to
Egyptian mathematics Ancient Egyptian mathematics is the mathematics that was developed and used in Ancient Egypt 3000 to c. , from the Old Kingdom of Egypt until roughly the beginning of Hellenistic Egypt. The ancient Egyptians utilized a numeral system for counti ...
(
Rhind Mathematical Papyrus The Rhind Mathematical Papyrus (RMP; also designated as papyrus British Museum 10057, pBM 10058, and Brooklyn Museum 37.1784Ea-b) is one of the best known examples of ancient Egyptian mathematics. It is one of two well-known mathematical papyri ...
) and
Babylonian mathematics Babylonian mathematics (also known as Assyro-Babylonian mathematics) is the mathematics developed or practiced by the people of Mesopotamia, as attested by sources mainly surviving from the Old Babylonian period (1830–1531 BC) to the Seleucid ...
during the 2nd millennium BC. Trigonometry was also prevalent in Kushite mathematics. Systematic study of trigonometric functions began in
Hellenistic mathematics Ancient Greek mathematics refers to the history of mathematical ideas and texts in Ancient Greece during classical and late antiquity, mostly from the 5th century BC to the 6th century AD. Greek mathematicians lived in cities spread around the s ...
, reaching
India India, officially the Republic of India, is a country in South Asia. It is the List of countries and dependencies by area, seventh-largest country by area; the List of countries by population (United Nations), most populous country since ...
as part of Hellenistic astronomy. In
Indian astronomy Astronomy has a long history in the Indian subcontinent, stretching from History of India, pre-historic to History of India (1947–present), modern times. Some of the earliest roots of Indian astronomy can be dated to the period of Indus Valle ...
, the study of trigonometric functions flourished in the
Gupta period The Gupta Empire was an Indian empire during the classical period of the Indian subcontinent which existed from the mid 3rd century to mid 6th century CE. At its zenith, the dynasty ruled over an empire that spanned much of the northern Indian ...
, especially due to
Aryabhata Aryabhata ( ISO: ) or Aryabhata I (476–550 CE) was the first of the major mathematician-astronomers from the classical age of Indian mathematics and Indian astronomy. His works include the '' Āryabhaṭīya'' (which mentions that in 3600 ' ...
(sixth century AD), who discovered the sine function, cosine function, and versine function. During the
Middle Ages In the history of Europe, the Middle Ages or medieval period lasted approximately from the 5th to the late 15th centuries, similarly to the post-classical period of global history. It began with the fall of the Western Roman Empire and ...
, the study of trigonometry continued in Islamic mathematics, by mathematicians such as
al-Khwarizmi Muhammad ibn Musa al-Khwarizmi , or simply al-Khwarizmi, was a mathematician active during the Islamic Golden Age, who produced Arabic-language works in mathematics, astronomy, and geography. Around 820, he worked at the House of Wisdom in B ...
and Abu al-Wafa. The knowledge of trigonometric functions passed to Arabia from the Indian Subcontinent. It became an independent discipline in the
Islamic world The terms Islamic world and Muslim world commonly refer to the Islamic community, which is also known as the Ummah. This consists of all those who adhere to the religious beliefs, politics, and laws of Islam or to societies in which Islam is ...
, where all six
trigonometric functions In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all ...
were known. Translations of Arabic and Greek texts led to trigonometry being adopted as a subject in the Latin West beginning in the
Renaissance The Renaissance ( , ) is a Periodization, period of history and a European cultural movement covering the 15th and 16th centuries. It marked the transition from the Middle Ages to modernity and was characterized by an effort to revive and sur ...
with Regiomontanus. The development of modern trigonometry shifted during the western
Age of Enlightenment The Age of Enlightenment (also the Age of Reason and the Enlightenment) was a Europe, European Intellect, intellectual and Philosophy, philosophical movement active from the late 17th to early 19th century. Chiefly valuing knowledge gained th ...
, beginning with 17th-century mathematics (
Isaac Newton Sir Isaac Newton () was an English polymath active as a mathematician, physicist, astronomer, alchemist, theologian, and author. Newton was a key figure in the Scientific Revolution and the Age of Enlightenment, Enlightenment that followed ...
and James Stirling) and reaching its modern form with
Leonhard Euler Leonhard Euler ( ; ; ; 15 April 170718 September 1783) was a Swiss polymath who was active as a mathematician, physicist, astronomer, logician, geographer, and engineer. He founded the studies of graph theory and topology and made influential ...
(1748).


Etymology

The term "trigonometry" was derived from
Greek Greek may refer to: Anything of, from, or related to Greece, a country in Southern Europe: *Greeks, an ethnic group *Greek language, a branch of the Indo-European language family **Proto-Greek language, the assumed last common ancestor of all kno ...
τρίγωνον ''trigōnon'', "triangle" and μέτρον ''metron'', "measure". The modern words "sine" and "cosine" are derived from the
Latin Latin ( or ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken by the Latins (Italic tribe), Latins in Latium (now known as Lazio), the lower Tiber area aroun ...
word via mistranslation from
Arabic Arabic (, , or , ) is a Central Semitic languages, Central Semitic language of the Afroasiatic languages, Afroasiatic language family spoken primarily in the Arab world. The International Organization for Standardization (ISO) assigns lang ...
(see ). Particularly
Fibonacci Leonardo Bonacci ( – ), commonly known as Fibonacci, was an Italians, Italian mathematician from the Republic of Pisa, considered to be "the most talented Western mathematician of the Middle Ages". The name he is commonly called, ''Fibonacci ...
's '' sinus rectus arcus'' proved influential in establishing the term. The word ''tangent'' comes from Latin meaning "touching", since the lineSee the figure at Pythagorean identities ''touches'' the circle of unit radius, whereas ''secant'' stems from Latin "cutting" since the line ''cuts'' the circle (see the figure at Pythagorean identities). The prefix " co-" (in "cosine", "cotangent", "cosecant") is found in Edmund Gunter's ''Canon triangulorum'' (1620), which defines the ''cosinus'' as an abbreviation for the ''sinus complementi'' (sine of the complementary angle) and proceeds to define the ''cotangens'' similarly. The words "minute" and "second" are derived from the Latin phrases '' partes minutae primae'' and '' partes minutae secundae''.: "It should be recalled that form the days of Hipparchus until modern times there were no such things as trigonometric ''ratios''. The Greeks, and after them the Hindus and the Arabs, used trigonometric ''lines''. These at first took the form, as we have seen, of chords in a circle, and it became incumbent upon Ptolemy to associate numerical values (or approximations) with the chords. ..It is not unlikely that the 360-degree measure was carried over from astronomy, where the zodiac had been divided into twelve "signs" or 36 "decans". A cycle of the seasons of roughly 360 days could readily be made to correspond to the system of zodiacal signs and decans by subdividing each sign into thirty parts and each decan into ten parts. Our common system of angle measure may stem from this correspondence. Moreover since the Babylonian position system for fractions was so obviously superior to the Egyptians unit fractions and the Greek common fractions, it was natural for Ptolemy to subdivide his degrees into sixty '' partes minutae primae'', each of these latter into sixty '' partes minutae secundae'', and so on. It is from the Latin phrases that translators used in this connection that our words "minute" and "second" have been derived. It undoubtedly was the sexagesimal system that led Ptolemy to subdivide the diameter of his trigonometric circle into 120 parts; each of these he further subdivided into sixty minutes and each minute of length sixty seconds." These roughly translate to "first small parts" and "second small parts".


Ancient


Ancient Near East

The ancient
Egyptians Egyptians (, ; , ; ) are an ethnic group native to the Nile, Nile Valley in Egypt. Egyptian identity is closely tied to Geography of Egypt, geography. The population is concentrated in the Nile Valley, a small strip of cultivable land stretchi ...
and Babylonians had known of theorems on the ratios of the sides of similar triangles for many centuries. However, as pre-Hellenic societies lacked the concept of an angle measure, they were limited to studying the sides of triangles instead.: "Trigonometry, like other branches of mathematics, was not the work of any one man, or nation. Theorems on ratios of the sides of similar triangles had been known to, and used by, the ancient Egyptians and Babylonians. In view of the pre-Hellenic lack of the concept of angle measure, such a study might better be called "trilaterometry", or the measure of three sided polygons (trilaterals), than "trigonometry", the measure of parts of a triangle. With the Greeks we first find a systematic study of relationships between angles (or arcs) in a circle and the lengths of chords subtending these. Properties of chords, as measures of central and inscribed angles in circles, were familiar to the Greeks of Hippocrates' day, and it is likely that Eudoxus had used ratios and angle measures in determining the size of the earth and the relative distances of the sun and the moon. In the works of Euclid there is no trigonometry in the strict sense of the word, but there are theorems equivalent to specific trigonometric laws or formulas. Propositions II.12 and 13 of the ''Elements'', for example, are the laws of cosines for obtuse and acute angles respectively, stated in geometric rather than trigonometric language and proved by a method similar to that used by Euclid in connection with the Pythagorean theorem. Theorems on the lengths of chords are essentially applications of the modern law of sines. We have seen that Archimedes' theorem on the broken chord can readily be translated into trigonometric language analogous to formulas for sines of sums and differences of angles." The Babylonian astronomers kept detailed records on the rising and setting of
star A star is a luminous spheroid of plasma (physics), plasma held together by Self-gravitation, self-gravity. The List of nearest stars and brown dwarfs, nearest star to Earth is the Sun. Many other stars are visible to the naked eye at night sk ...
s, the motion of the
planet A planet is a large, Hydrostatic equilibrium, rounded Astronomical object, astronomical body that is generally required to be in orbit around a star, stellar remnant, or brown dwarf, and is not one itself. The Solar System has eight planets b ...
s, and the solar and lunar
eclipse An eclipse is an astronomical event which occurs when an astronomical object or spacecraft is temporarily obscured, by passing into the shadow of another body or by having another body pass between it and the viewer. This alignment of three ...
s, all of which required familiarity with angular distances measured on the
celestial sphere In astronomy and navigation, the celestial sphere is an abstract sphere that has an arbitrarily large radius and is concentric to Earth. All objects in the sky can be conceived as being projected upon the inner surface of the celestial sphere, ...
. Based on one interpretation of the Plimpton 322
cuneiform Cuneiform is a Logogram, logo-Syllabary, syllabic writing system that was used to write several languages of the Ancient Near East. The script was in active use from the early Bronze Age until the beginning of the Common Era. Cuneiform script ...
tablet (c. 1900 BC), some have even asserted that the ancient Babylonians had a table of secants but does not work in this context as without using circles and angles in the situation modern trigonometric notations will not apply. There is, however, much debate as to whether it is a table of
Pythagorean triple A Pythagorean triple consists of three positive integers , , and , such that . Such a triple is commonly written , a well-known example is . If is a Pythagorean triple, then so is for any positive integer . A triangle whose side lengths are a Py ...
s, a solution of quadratic equations, or a trigonometric table. The Egyptians, on the other hand, used a primitive form of trigonometry for building
pyramid A pyramid () is a structure whose visible surfaces are triangular in broad outline and converge toward the top, making the appearance roughly a pyramid in the geometric sense. The base of a pyramid can be of any polygon shape, such as trian ...
s in the 2nd millennium BC. The
Rhind Mathematical Papyrus The Rhind Mathematical Papyrus (RMP; also designated as papyrus British Museum 10057, pBM 10058, and Brooklyn Museum 37.1784Ea-b) is one of the best known examples of ancient Egyptian mathematics. It is one of two well-known mathematical papyri ...
, written by the Egyptian scribe
Ahmes Ahmes ( “, a common Egyptian name also transliterated Ahmose (disambiguation), Ahmose) was an ancient Egyptian scribe who lived towards the end of the 15th Dynasty, Fifteenth Dynasty (and of the Second Intermediate Period) and the beginning of t ...
(c. 1680–1620 BC), contains the following problem related to trigonometry: Ahmes' solution to the problem is the ratio of half the side of the base of the pyramid to its height, or the run-to-rise ratio of its face. In other words, the quantity he found for the ''seked'' is the cotangent of the angle to the base of the pyramid and its face.


Classical antiquity

Ancient Greek and Hellenistic mathematicians made use of the chord. Given a circle and an arc on the circle, the chord is the line that subtends the arc. A chord's perpendicular bisector passes through the center of the circle and bisects the angle. One half of the bisected chord is the sine of one half the bisected angle, that is, : \mathrm\ \theta = 2 r\sin \frac, and consequently the sine function is also known as the ''half-chord''. Due to this relationship, a number of trigonometric identities and theorems that are known today were also known to
Hellenistic In classical antiquity, the Hellenistic period covers the time in Greek history after Classical Greece, between the death of Alexander the Great in 323 BC and the death of Cleopatra VII in 30 BC, which was followed by the ascendancy of the R ...
mathematicians, but in their equivalent chord form. Although there is no trigonometry in the works of
Euclid Euclid (; ; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the '' Elements'' treatise, which established the foundations of geometry that largely domina ...
and
Archimedes Archimedes of Syracuse ( ; ) was an Ancient Greece, Ancient Greek Greek mathematics, mathematician, physicist, engineer, astronomer, and Invention, inventor from the ancient city of Syracuse, Sicily, Syracuse in History of Greek and Hellenis ...
, in the strict sense of the word, there are theorems presented in a geometric way (rather than a trigonometric way) that are equivalent to specific trigonometric laws or formulas. For instance, propositions twelve and thirteen of book two of the ''Elements'' are the laws of cosines for obtuse and acute angles, respectively. Theorems on the lengths of chords are applications of the
law of sines In trigonometry, the law of sines (sometimes called the sine formula or sine rule) is a mathematical equation relating the lengths of the sides of any triangle to the sines of its angles. According to the law, \frac \,=\, \frac \,=\, \frac \,=\ ...
. And Archimedes' theorem on broken chords is equivalent to formulas for sines of sums and differences of angles. To compensate for the lack of a table of chords, mathematicians of Aristarchus' time would sometimes use the statement that, in modern notation, sin ''α''/sin ''β'' < ''α''/''β'' < tan ''α''/tan ''β'' whenever 0° < β < α < 90°, now known as Aristarchus's inequality. The first trigonometric table was apparently compiled by
Hipparchus Hipparchus (; , ;  BC) was a Ancient Greek astronomy, Greek astronomer, geographer, and mathematician. He is considered the founder of trigonometry, but is most famous for his incidental discovery of the precession of the equinoxes. Hippar ...
of
Nicaea Nicaea (also spelled Nicæa or Nicea, ; ), also known as Nikaia (, Attic: , Koine: ), was an ancient Greek city in the north-western Anatolian region of Bithynia. It was the site of the First and Second Councils of Nicaea (the first and seve ...
(180 – 125 BC), who is now consequently known as "the father of trigonometry.": "For some two and a half centuries, from Hippocrates to Eratosthenes, Greek mathematicians had studied relationships between lines and circles and had applied these in a variety of astronomical problems, but no systematic trigonometry had resulted. Then, presumably during the second half of the 2nd century BC, the first trigonometric table apparently was compiled by the astronomer Hipparchus of Nicaea (ca. 180–ca. 125 BC), who thus earned the right to be known as "the father of trigonometry". Aristarchus had known that in a given circle the ratio of arc to chord decreases as the arc decreases from 180° to 0°, tending toward a limit of 1. However, it appears that not until Hipparchus undertook the task had anyone tabulated corresponding values of arc and chord for a whole series of angles." Hipparchus was the first to tabulate the corresponding values of arc and chord for a series of angles. Although it is not known when the systematic use of the 360° circle came into mathematics, it is known that the systematic introduction of the 360° circle came a little after
Aristarchus of Samos Aristarchus of Samos (; , ; ) was an ancient Greek astronomer and mathematician who presented the first known heliocentric model that placed the Sun at the center of the universe, with the Earth revolving around the Sun once a year and rotati ...
composed '' On the Sizes and Distances of the Sun and Moon'' (c. 260 BC), since he measured an angle in terms of a fraction of a quadrant.: "Instead we have an treatise, perhaps composed earlier (ca. 260 BC), ''On the Sizes and Distances of the Sun and Moon'', which assumes a geocentric universe. In this work Aristarchus made the observation that when the moon is just half-full, the angle between the lines of sight to the sun and the moon is less than a right angle by one thirtieth of a quadrant. (The systematic introduction of the 360° circle came a little later. In trigonometric language of today this would mean that the ratio of the distance of the moon to that of the sun (the ration ME to SE in Fig. 10.1) is sin(3°). Trigonometric tables not having been developed yet, Aristarchus fell back upon a well-known geometric theorem of the time which now would be expressed in the inequalities sin α/ sin β < α/β < tan α/ tan β, for 0° < β < α < 90°.)" It seems that the systematic use of the 360° circle is largely due to Hipparchus and his table of chords. Hipparchus may have taken the idea of this division from Hypsicles who had earlier divided the day into 360 parts, a division of the day that may have been suggested by Babylonian astronomy. In ancient astronomy, the zodiac had been divided into twelve "signs" or thirty-six "decans". A seasonal cycle of roughly 360 days could have corresponded to the signs and decans of the zodiac by dividing each sign into thirty parts and each decan into ten parts. It is due to the Babylonian
sexagesimal Sexagesimal, also known as base 60, is a numeral system with 60 (number), sixty as its radix, base. It originated with the ancient Sumerians in the 3rd millennium BC, was passed down to the ancient Babylonians, and is still used—in a modified fo ...
numeral system A numeral system is a writing system for expressing numbers; that is, a mathematical notation for representing numbers of a given set, using digits or other symbols in a consistent manner. The same sequence of symbols may represent differe ...
that each degree is divided into sixty minutes and each minute is divided into sixty seconds. Menelaus of Alexandria (c. 100 AD) wrote in three books his ''Sphaerica''. In Book I, he established a basis for spherical triangles analogous to the Euclidean basis for plane triangles. He established a theorem that is without Euclidean analogue, that two spherical triangles are congruent if corresponding angles are equal, but he did not distinguish between congruent and symmetric spherical triangles. Another theorem that he establishes is that the sum of the angles of a spherical triangle is greater than 180°. Book II of ''Sphaerica'' applies spherical geometry to astronomy. And Book III contains the "theorem of Menelaus".: "In Book I of this treatise Menelaus establishes a basis for spherical triangles analogous to that of Euclid I for plane triangles. Included is a theorem without Euclidean analogue – that two spherical triangles are congruent if corresponding angles are equal (Menelaus did not distinguish between congruent and symmetric spherical triangles); and the theorem ''A'' + ''B'' + ''C'' > 180° is established. The second book of the ''Sphaerica'' describes the application of spherical geometry to astronomical phenomena and is of little mathematical interest. Book III, the last, contains the well known "theorem of Menelaus" as part of what is essentially spherical trigonometry in the typical Greek form – a geometry or trigonometry of chords in a circle. In the circle in Fig. 10.4 we should write that chord AB is twice the sine of half the central angle AOB (multiplied by the radius of the circle). Menelaus and his Greek successors instead referred to AB simply as the chord corresponding to the arc AB. If BOB' is a diameter of the circle, then chord A' is twice the cosine of half the angle AOB (multiplied by the radius of the circle)." He further gave his famous "rule of six quantities". Later, Claudius Ptolemy (c. 90 – c. 168 AD) expanded upon Hipparchus' ''Chords in a Circle'' in his ''
Almagest The ''Almagest'' ( ) is a 2nd-century Greek mathematics, mathematical and Greek astronomy, astronomical treatise on the apparent motions of the stars and planetary paths, written by Ptolemy, Claudius Ptolemy ( ) in Koine Greek. One of the most i ...
'', or the ''Mathematical Syntaxis''. The Almagest is primarily a work on astronomy, and astronomy relies on trigonometry.
Ptolemy's table of chords The table of chords, created by the Greece, 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 astron ...
gives the lengths of chords of a circle of diameter 120 as a function of the number of degrees ''n'' in the corresponding arc of the circle, for ''n'' ranging from 1/2 to 180 by increments of 1/2. The thirteen books of the ''Almagest'' are the most influential and significant trigonometric work of all antiquity. A theorem that was central to Ptolemy's calculation of chords was what is still known today as Ptolemy's theorem, that the sum of the products of the opposite sides of a
cyclic quadrilateral In geometry, a cyclic quadrilateral or inscribed quadrilateral is a quadrilateral (four-sided polygon) whose vertex (geometry), vertices all lie on a single circle, making the sides Chord (geometry), chords of the circle. This circle is called ...
is equal to the product of the diagonals. A special case of Ptolemy's theorem appeared as proposition 93 in Euclid's ''Data''. Ptolemy's theorem leads to the equivalent of the four sum-and-difference formulas for sine and cosine that are today known as Ptolemy's formulas, although Ptolemy himself used chords instead of sine and cosine. Ptolemy further derived the equivalent of the half-angle formula : \sin^2\left(\frac\right) = \frac. Ptolemy used these results to create his trigonometric tables, but whether these tables were derived from Hipparchus' work cannot be determined.: "The theorem of Menelaus played a fundamental role in spherical trigonometry and astronomy, but by far the most influential and significant trigonometric work of all antiquity was composed by Ptolemy of Alexandria about half a century after Menelaus. ..Of the life of the author we are as little informed as we are of that of the author of the Elements. We do not know when or where Euclid and Ptolemy were born. We know that Ptolemy made observations at Alexandria from AD. 127 to 151 and, therefore, assume that he was born at the end of the 1st century. Suidas, a writer who lived in the 10th century, reported that Ptolemy was alive under Marcus Aurelius (emperor from AD 161 to 180).
Ptolemy's ''Almagest'' is presumed to be heavily indebted for its methods to the ''Chords in a Circle'' of Hipparchus, but the extent of the indebtedness cannot be reliably assessed. It is clear that in astronomy Ptolemy made use of the catalog of star positions bequeathed by Hipparchus, but whether or not Ptolemy's trigonometric tables were derived in large part from his distinguished predecessor cannot be determined. ..Central to the calculation of Ptolemy's chords was a geometric proposition still known as "Ptolemy's theorem": ..that is, the sum of the products of the opposite sides of a cyclic quadrilateral is equal to the product of the diagonals. ..A special case of Ptolemy's theorem had appeared in Euclid's ''Data'' (Proposition 93): ..Ptolemy's theorem, therefore, leads to the result sin(''α'' − ''β'') = sin ''α'' cos ''β'' − cos ''α'' sin ''Β''. Similar reasoning leads to the formula ..These four sum-and-difference formulas consequently are often known today as Ptolemy's formulas.
It was the formula for sine of the difference – or, more accurately, chord of the difference – that Ptolemy found especially useful in building up his tables. Another formula that served him effectively was the equivalent of our half-angle formula."
Neither the tables of Hipparchus nor those of Ptolemy have survived to the present day, although descriptions by other ancient authors leave little doubt that they once existed.


Indian mathematics

Some of the early and very significant developments of trigonometry were in
India India, officially the Republic of India, is a country in South Asia. It is the List of countries and dependencies by area, seventh-largest country by area; the List of countries by population (United Nations), most populous country since ...
. Influential works from the 4th–5th century AD, known as the Siddhantas (of which there were five, the most important of which is the Surya Siddhanta) first defined the sine as the modern relationship between half an angle and half a chord, while also defining the cosine, versine, and inverse sine. Soon afterwards, another Indian mathematician and
astronomer An astronomer is a scientist in the field of astronomy who focuses on a specific question or field outside the scope of Earth. Astronomers observe astronomical objects, such as stars, planets, natural satellite, moons, comets and galaxy, galax ...
,
Aryabhata Aryabhata ( ISO: ) or Aryabhata I (476–550 CE) was the first of the major mathematician-astronomers from the classical age of Indian mathematics and Indian astronomy. His works include the '' Āryabhaṭīya'' (which mentions that in 3600 ' ...
(476–550 AD), collected and expanded upon the developments of the Siddhantas in an important work called the '' Aryabhatiya''. The ''Siddhantas'' and the ''Aryabhatiya'' contain the earliest surviving tables of sine values and versine (1 − cosine) values, in 3.75° intervals from 0° to 90°, to an accuracy of 4 decimal places. They used the words '' jya'' for sine, '' kojya'' for cosine, '' utkrama-jya'' for versine, and ''otkram jya'' for inverse sine. The words '' jya'' and '' kojya'' eventually became ''sine'' and ''cosine'' respectively after a mistranslation described above. In the 7th century, Bhaskara I produced a
formula In science, a formula is a concise way of expressing information symbolically, as in a mathematical formula or a ''chemical formula''. The informal use of the term ''formula'' in science refers to the general construct of a relationship betwe ...
for calculating the sine of an acute angle without the use of a table. He also gave the following approximation formula for sin(''x''), which had a relative error of less than 1.9%: : \sin x \approx \frac, \qquad \left(0\leq x\leq\pi\right). Later in the 7th century,
Brahmagupta Brahmagupta ( – ) was an Indian Indian mathematics, mathematician and Indian astronomy, astronomer. He is the author of two early works on mathematics and astronomy: the ''Brāhmasphuṭasiddhānta'' (BSS, "correctly established Siddhanta, do ...
redeveloped the formula : \ 1 - \sin^2(x) = \cos^2(x) = \sin^2\left (\frac - x\right ) (also derived earlier, as mentioned above) and the Brahmagupta interpolation formula for computing sine values. Madhava (c. 1400) made early strides in the
analysis Analysis (: analyses) is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (38 ...
of trigonometric functions and their
infinite series In mathematics, a series is, roughly speaking, an addition of infinitely many terms, one after the other. The study of series is a major part of calculus and its generalization, mathematical analysis. Series are used in most areas of mathemati ...
expansions. He developed the concepts of the
power series In mathematics, a power series (in one variable) is an infinite series of the form \sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots where ''a_n'' represents the coefficient of the ''n''th term and ''c'' is a co ...
and
Taylor series In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor ser ...
, and produced the
power series In mathematics, a power series (in one variable) is an infinite series of the form \sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots where ''a_n'' represents the coefficient of the ''n''th term and ''c'' is a co ...
expansions of sine, cosine, tangent, and arctangent. Using the Taylor series approximations of sine and cosine, he produced a sine table to 12 decimal places of accuracy and a cosine table to 9 decimal places of accuracy. He also gave the power series of π and the
angle In Euclidean geometry, an angle can refer to a number of concepts relating to the intersection of two straight Line (geometry), lines at a Point (geometry), point. Formally, an angle is a figure lying in a Euclidean plane, plane formed by two R ...
,
radius In classical geometry, a radius (: radii or radiuses) of a circle or sphere is any of the line segments from its Centre (geometry), center to its perimeter, and in more modern usage, it is also their length. The radius of a regular polygon is th ...
,
diameter In geometry, a diameter of a circle is any straight line segment that passes through the centre of the circle and whose endpoints lie on the circle. It can also be defined as the longest Chord (geometry), chord of the circle. Both definitions a ...
, and
circumference In geometry, the circumference () is the perimeter of a circle or ellipse. The circumference is the arc length of the circle, as if it were opened up and straightened out to a line segment. More generally, the perimeter is the curve length arou ...
of a circle in terms of trigonometric functions. His works were expanded by his followers at the Kerala School up to the 16th century. The Indian text the Yuktibhāṣā contains proof for the expansion of the
sine In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side opposite th ...
and
cosine In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side opposite that ...
functions and the derivation and proof of the
power series In mathematics, a power series (in one variable) is an infinite series of the form \sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots where ''a_n'' represents the coefficient of the ''n''th term and ''c'' is a co ...
for inverse tangent, discovered by Madhava. The Yuktibhāṣā also contains rules for finding the sines and the cosines of the sum and difference of two angles.


Chinese mathematics

In
China China, officially the People's Republic of China (PRC), is a country in East Asia. With population of China, a population exceeding 1.4 billion, it is the list of countries by population (United Nations), second-most populous country after ...
,
Aryabhata Aryabhata ( ISO: ) or Aryabhata I (476–550 CE) was the first of the major mathematician-astronomers from the classical age of Indian mathematics and Indian astronomy. His works include the '' Āryabhaṭīya'' (which mentions that in 3600 ' ...
's table of sines were translated into the Chinese mathematical book of the '' Kaiyuan Zhanjing'', compiled in 718 AD during the
Tang dynasty The Tang dynasty (, ; zh, c=唐朝), or the Tang Empire, was an Dynasties of China, imperial dynasty of China that ruled from 618 to 907, with an Wu Zhou, interregnum between 690 and 705. It was preceded by the Sui dynasty and followed ...
. Although the Chinese excelled in other fields of mathematics such as solid geometry,
binomial theorem In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial. According to the theorem, the power expands into a polynomial with terms of the form , where the exponents and a ...
, and complex algebraic formulas, early forms of trigonometry were not as widely appreciated as in the earlier Greek, Hellenistic, Indian and Islamic worlds. Instead, the early Chinese used an empirical substitute known as ''chong cha'', while practical use of plane trigonometry in using the sine, the tangent, and the secant were known. However, this embryonic state of trigonometry in China slowly began to change and advance during the
Song dynasty The Song dynasty ( ) was an Dynasties of China, imperial dynasty of China that ruled from 960 to 1279. The dynasty was founded by Emperor Taizu of Song, who usurped the throne of the Later Zhou dynasty and went on to conquer the rest of the Fiv ...
(960–1279), where Chinese mathematicians began to express greater emphasis for the need of spherical trigonometry in calendrical science and astronomical calculations. The
polymath A polymath or polyhistor is an individual whose knowledge spans many different subjects, known to draw on complex bodies of knowledge to solve specific problems. Polymaths often prefer a specific context in which to explain their knowledge, ...
Chinese scientist, mathematician and official
Shen Kuo Shen Kuo (; 1031–1095) or Shen Gua, courtesy name Cunzhong (存中) and Art name#China, pseudonym Mengqi (now usually given as Mengxi) Weng (夢溪翁),Yao (2003), 544. was a Chinese polymath, scientist, and statesman of the Song dynasty (960� ...
(1031–1095) used trigonometric functions to solve mathematical problems of chords and arcs. Victor J. Katz writes that in Shen's formula "technique of intersecting circles", he created an approximation of the arc ''s'' of a circle given the diameter ''d'',
sagitta Sagitta is a dim but distinctive constellation in the northern sky. Its name is Latin for 'arrow', not to be confused with the significantly larger constellation Sagittarius 'the archer'. It was included among the 48 constellations listed by t ...
 ''v'', and length ''c'' of the chord subtending the arc, the length of which he approximated as : s = c + \frac. Sal Restivo writes that Shen's work in the lengths of arcs of circles provided the basis for
spherical trigonometry Spherical trigonometry is the branch of spherical geometry that deals with the metrical relationships between the edge (geometry), sides and angles of spherical triangles, traditionally expressed using trigonometric functions. On the sphere, ge ...
developed in the 13th century by the mathematician and astronomer Guo Shoujing (1231–1316). As the historians L. Gauchet and Joseph Needham state, Guo Shoujing used spherical trigonometry in his calculations to improve the Chinese calendar, calendar system and Chinese astronomy. Along with a later 17th-century Chinese illustration of Guo's mathematical proofs, Needham states that:
Guo used a quadrangular spherical pyramid, the basal quadrilateral of which consisted of one equatorial and one ecliptic arc, together with two meridian arcs, one of which passed through the summer solstice point...By such methods he was able to obtain the du lü (degrees of equator corresponding to degrees of ecliptic), the ji cha (values of chords for given ecliptic arcs), and the cha lü (difference between chords of arcs differing by 1 degree).
Despite the achievements of Shen and Guo's work in trigonometry, another substantial work in Chinese trigonometry would not be published again until 1607, with the dual publication of ''Euclid's Elements'' by Chinese official and astronomer Xu Guangqi (1562–1633) and the Italian Jesuit Matteo Ricci (1552–1610).


Medieval


Islamic world

Previous works from India and Greece were later translated and expanded in the Islamic Golden Age, medieval Islamic world by Mathematics in medieval Islam, Muslim mathematicians of mostly List of Iranian scientists and scholars, Persian and List of Arab scientists and scholars, Arab descent, who enunciated a large number of theorems which freed the subject of trigonometry from dependence upon the complete quadrilateral, as was the case in Hellenistic mathematics due to the application of Menelaus' theorem. According to E. S. Kennedy, it was after this development in Islamic mathematics that "the first real trigonometry emerged, in the sense that only then did the object of study become the Spherical trigonometry, spherical or plane triangle, its sides and
angle In Euclidean geometry, an angle can refer to a number of concepts relating to the intersection of two straight Line (geometry), lines at a Point (geometry), point. Formally, an angle is a figure lying in a Euclidean plane, plane formed by two R ...
s." Methods dealing with spherical triangles were also known, particularly the method of Menelaus of Alexandria, who developed "Menelaus' theorem" to deal with spherical problems. However, E. S. Kennedy points out that while it was possible in pre-Islamic mathematics to compute the magnitudes of a spherical figure, in principle, by use of the table of chords and Menelaus' theorem, the application of the theorem to spherical problems was very difficult in practice. In order to observe holy days on the Islamic calendar in which timings were determined by phases of the moon, astronomers initially used Menelaus' method to calculate the place of the moon and
star A star is a luminous spheroid of plasma (physics), plasma held together by Self-gravitation, self-gravity. The List of nearest stars and brown dwarfs, nearest star to Earth is the Sun. Many other stars are visible to the naked eye at night sk ...
s, though this method proved to be clumsy and difficult. It involved setting up two intersecting right triangles; by applying Menelaus' theorem it was possible to solve one of the six sides, but only if the other five sides were known. To tell the time from the sun's altitude, for instance, repeated applications of Menelaus' theorem were required. For medieval Astronomy in medieval Islam, Islamic astronomers, there was an obvious challenge to find a simpler trigonometric method. In the early 9th century AD, Muhammad ibn Mūsā al-Khwārizmī produced accurate sine and cosine tables. He was also a pioneer in
spherical trigonometry Spherical trigonometry is the branch of spherical geometry that deals with the metrical relationships between the edge (geometry), sides and angles of spherical triangles, traditionally expressed using trigonometric functions. On the sphere, ge ...
. In 830 AD, Habash al-Hasib al-Marwazi discovered the tangent and the cotangent and produced the first table of these Trigonometric functions, trigonometric functions. Muhammad ibn Jābir al-Harrānī al-Battānī (Albatenius) (853–929 AD) discovered the secant and the cosecant, and produced the first table of cosecants for each degree from 1° to 90°. By the 10th century AD, in the work of Abū al-Wafā' al-Būzjānī, all six
trigonometric functions In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all ...
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 developed the following trigonometric formula: :\ \sin(2x) = 2 \sin(x) \cos(x) (a special case of Ptolemy's angle-addition formula; see above) In his original text, Abū al-Wafā' states: "If we want that, we multiply the given sine by the cosine minute of arc, minutes, and the result is half the sine of the double". Abū al-Wafā also established the angle addition and difference identities presented with complete proofs: :\sin(\alpha \pm \beta) = \sqrt \pm \sqrt :\sin(\alpha \pm \beta) = \sin \alpha \cos \beta \pm \cos \alpha \sin \beta For the second one, the text states: "We multiply the sine of each of the two arcs by the cosine of the other ''minutes''. If we want the sine of the sum, we add the products, if we want the sine of the difference, we take their difference". He also discovered the Law of sines#Curvature, law of sines for spherical trigonometry:Jacques Sesiano, "Islamic mathematics", p. 157, in :\frac = \frac = \frac. Also in the late 10th and early 11th centuries AD, the Egyptian astronomer Ibn Yunus performed many careful trigonometric calculations and demonstrated the following trigonometric identity: :\cos a \cos b = \frac Al-Jayyani (989–1079) of al-Andalus wrote ''The book of unknown arcs of a sphere'', which is considered "the first treatise on
spherical trigonometry Spherical trigonometry is the branch of spherical geometry that deals with the metrical relationships between the edge (geometry), sides and angles of spherical triangles, traditionally expressed using trigonometric functions. On the sphere, ge ...
". It "contains formulae for Special right triangles, right-handed triangles, the general law of sines, and the solution of a spherical triangle by means of the polar triangle." This treatise later had a "strong influence on European mathematics", and his "definition of ratios as numbers" and "method of solving a spherical triangle when all sides are unknown" are likely to have influenced Regiomontanus. The method of triangulation was first developed by Muslim mathematicians, who applied it to practical uses such as surveying and Islamic geography, as described by Abu Rayhan Biruni in the early 11th century. Biruni himself introduced triangulation techniques to Abu Rayhan Biruni#Geodesy and geography, measure the size of the Earth and the distances between various places. In the late 11th century, Omar Khayyám (1048–1131) solved cubic equations using approximate numerical solutions found by interpolation in trigonometric tables. In the 13th century, Naṣīr al-Dīn al-Ṭūsī 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 ''Book on the Complete Quadrilateral'', 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. Nasir al-Din al-Tusi has been described as the creator of trigonometry as a mathematical discipline in its own right. The law of cosines, in geometric form, can be found as propositions II.12–13 in Euclid's Elements, Euclid's ''Elements'' (c. 300 BC), but was not used for the solution of triangles per se. Medieval Islamic mathematicians developed a method for finding the third side of an arbitrary triangle given two sides and the included angle based on the same concept but more similar to the modern formulation of the law of cosines. A sketch of the method can be found in Naṣīr al-Dīn al-Ṭūsī's ''Book on the Complete Quadrilateral'' (c. 1250), and the same method is described in more detail in Jamshīd al-Kāshī's ''Key of Arithmetic'' (1427). Al-Kāshī also computed the sine of 1° accurate to 8
sexagesimal Sexagesimal, also known as base 60, is a numeral system with 60 (number), sixty as its radix, base. It originated with the ancient Sumerians in the 3rd millennium BC, was passed down to the ancient Babylonians, and is still used—in a modified fo ...
digits, and constructed the most accurate trigonometric tables to date, accurate to four
sexagesimal Sexagesimal, also known as base 60, is a numeral system with 60 (number), sixty as its radix, base. It originated with the ancient Sumerians in the 3rd millennium BC, was passed down to the ancient Babylonians, and is still used—in a modified fo ...
places (equivalent to 8 decimal places) for each 1° of arc. Al-Kāshī presumably worked on Ulugh Beg's even more comprehensive trigonometric tables, with five-place (sexagesimal) entries for each minute of arc.


European renaissance

In 1342, Levi ben Gershon, known as Gersonides, wrote ''On Sines, Chords and Arcs'', in particular proving the sine law for plane triangles and giving five-figure sine tables. A simplified trigonometric table, the "''toleta de marteloio''", was used by sailors in the Mediterranean Sea during the 14th-15th Centuries to calculate navigation courses. It is described by Ramon Llull of Majorca in 1295, and laid out in the 1436 atlas of Venice, Venetian captain Andrea Bianco. Regiomontanus was perhaps the first mathematician in Europe to treat trigonometry as a distinct mathematical discipline, in his ''De triangulis omnimodis'' written in 1464, as well as his later ''Tabulae directionum'' which included the tangent function, unnamed.


Modern

The ''Opus palatinum de triangulis'' of Georg Joachim Rheticus, a student of Copernicus, was probably the first in Europe to define trigonometric functions directly in terms of right triangles instead of circles, with tables for all six trigonometric functions; this work was finished by Rheticus' student Valentinus Otho, Valentin Otho in 1596. In the 17th century,
Isaac Newton Sir Isaac Newton () was an English polymath active as a mathematician, physicist, astronomer, alchemist, theologian, and author. Newton was a key figure in the Scientific Revolution and the Age of Enlightenment, Enlightenment that followed ...
and James Stirling developed the general Newton–Stirling interpolation formula for trigonometric functions. In the 18th century,
Leonhard Euler Leonhard Euler ( ; ; ; 15 April 170718 September 1783) was a Swiss polymath who was active as a mathematician, physicist, astronomer, logician, geographer, and engineer. He founded the studies of graph theory and topology and made influential ...
's ''Introduction in analysin infinitorum'' (1748) was mostly responsible for establishing the analytic treatment of trigonometric functions in Europe, deriving their infinite series and presenting "Euler's formula" ''e''''ix'' = cos ''x'' + ''i'' sin ''x''. Euler used the near-modern abbreviations ''sin.'', ''cos.'', ''tang.'', ''cot.'', ''sec.'', and ''cosec.'' Prior to this, Roger Cotes had computed the derivative of sine in his ''Harmonia Mensurarum'' (1722).. The proof of Cotes is mentioned on p. 315. Also in the 18th century, Brook Taylor defined the general Taylor series and gave the series expansions and approximations for all six trigonometric functions. The works of James Gregory (astronomer and mathematician), James Gregory in the 17th century and Colin Maclaurin in the 18th century were also very influential in the development of trigonometric series.


See also

* Greek mathematics * History of mathematics * Trigonometric functions * Trigonometry *
Ptolemy's table of chords The table of chords, created by the Greece, 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 astron ...
* Aryabhata's sine table * Rational trigonometry


Citations and footnotes


References

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Further reading

* * * * * * {{Islamic mathematics History of mathematics, Trigonometry History of geometry, Trigonometry Trigonometry