Meanings please !!!!?

Question

I Have learnt a whole chapter on terms like
sin cos tan cot sec and cosec

they all are greek words , can any one please tell me what do they mean ?

Answer 1

Not sure if you want to know what they translate to or what they are. I can tell you that they're all trigonometric functions (sin = sine, cos = cosine, tan = tangent, cot = cotangent, sec = secant and cosec = cosecant) and you can probably find all the information you're looking for in my source.

Answer 2

Doesn't really sound like you 'learnt' much.

A whole chapter huh?

They are names for various ratios of sides of triangles.

Answer 3

From the web site "Online Etymology Dictionary"

sine:
trigonometric function, 1593 (in Thomas Fale's "Horologiographia, the Art of Dialling"), from L. sinus "fold in a garment, bend, curve." Used by Gherardo of Cremona c.1150 in M.L. translation of Arabic geometrical text to render Arabic jiba "chord of an arc, sine" (from Skt. jiva "bowstring"), which he confused with jaib "bundle, bosom, fold in a garment."

tangent (adj.)
1594, "meeting at a point without intersecting," from L. tangentem (nom. tangens), prp. of tangere "to touch," from PIE base *tag- "to touch, to handle" (cf. L. tactus "touch," Gk. tetagon "having seized," O.E. þaccian "stroke, strike gently"). First used by Dan. mathematician Thomas Fincke in "Geomietria Rotundi" (1583). The noun also is attested from 1594; extended sense of "slightly connected with a subject" is first recorded 1825. Tangential is recorded from 1630; fig. sense of "divergent, erratic" is from 1787.

secant
1593, from L. secantem (nom. secans) "cutting," prp. of secare "to cut" (see section). First used by Dan. mathematician Thomas Fincke in Geometria Rotundi (1583).

Answer 4

They are ratios that relate an angle to the length of the sides of a triangle.
sin(z)=y/r
cos(z)=x/r
tan(z)=y/x
where z = the angle between 0degrees and r

Answer 5

sine, its inverse is cosine, tangent, its inverse cotangent, secant, and inverse cosecant. They're all trigonomentric functions especially for triangles.

Answer 6

In mathematics, the trigonometric functions are functions of an angle; they are important when studying triangles and modeling periodic phenomena, among many other applications. They are commonly defined as ratios of two sides of a right triangle containing the angle, and can equivalently be defined as the lengths of various line segments from a unit circle. More modern definitions express them as infinite series or as solutions of certain differential equations, allowing their extension to positive and negative values and even to complex numbers. All of these approaches will be presented below.

The study of trigonometric functions dates back to Babylonian times, and a considerable amount of fundamental work was done by ancient Greek, Indian and Arab mathematicians.

In modern usage, there are six basic trigonometric functions, which are tabulated below along with equations relating them to one another. Especially in the case of the last four, these relations are often taken as the definitions of those functions, but one can define them equally well geometrically or by other means and then derive these relations. A few other functions were common historically (and appeared in the earliest tables), but are now seldom used, such as the versine (1 − cos θ) and the exsecant (sec θ − 1). Many more relations between these functions are listed in the article about trigonometric identities.

History
There is evidence that the Babylonians first used trigonometric functions, based on a table of numbers written on a Babylonian cuneiform tablet, Plimpton 322 (circa 1900 BC), which can be interpreted as a table of secants.[1] There is, however, much debate on whether it was a trigonometric table. The earliest use of sine appears in the Sulba Sutras written in ancient India from the 8th century BC to the 6th century BC, which correctly computes the sine of π/4 (45°) as 1/√2 in a procedure for circling the square (the opposite of squaring the circle), though they hadn't yet developed the notion of a sine in a general sense.[2]


An artist's rendition of Claudius PtolemyTrigonometric functions were later studied by Hipparchus of Nicaea (180-125 BC), who tabulated the lengths of circle arcs (angle A times radius r) with the lengths of the subtending chords (2r sin(A/2)). Later, Claudius Ptolemy of Egypt (2nd century) expanded upon this work in his Almagest, deriving addition/subtraction formulas for the equivalent of sin(A + B) and cos(A + B). Ptolemy derived the equivalent of the half-angle formula sin2(A/2) = (1 − cos(A))/2, and created a table of his results. 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.

The next significant developments of trigonometry were in India. Mathematician-astronomer Aryabhata (476–550), in his work Aryabhata-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. His works also 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. He used the words jya for sine, kojya for cosine, ukramajya for versine, and otkram jya for inverse sine. The words jya and kojya eventually became sine and cosine respectively after a mistranslation.

Our modern word sine is derived from the Latin word sinus, which means "bay" or "fold", from a mistranslation (via Arabic) of the Sanskrit word jiva, alternatively called jya.[3] Aryabhata used the term ardha-jiva ("half-chord"), which was shortened to jiva and then transliterated by the Arabs as jiba (جب). European translators like Robert of Chester and Gherardo of Cremona in 12th-century Toledo confused jiba for jaib (جب), meaning "bay", probably because jiba (جب) and jaib (جب) are written the same in the Arabic script (this writing system, in one of its forms, does not provide the reader with complete information about the vowels).

Other Indian mathematicians later expanded Aryabhata's works on trigonometry. Varahamihira developed the formulas sin2x + cos2x = 1, sin x = cos(π/2 − x), and (1 − cos(2x))/2 = sin2x. Bhaskara I produced a formula for calculating the sine of an acute angle without the use of a table. Brahmagupta developed the formula 1 − sin2x = cos2x = sin2(π/2 − x), and the Brahmagupta interpolation formula for computing sine values, which is a special case of the Newton–Stirling interpolation formula up to second order.

The Indian works were later translated and expanded by Muslim mathematicians. Persian mathematician Muḥammad ibn Mūsā al-Ḵwārizmī produced tables of sines and tangents, and also contributed to spherical trigonometry. By the 10th century, in the work of Abu'l-Wafa, Muslim mathematicians were using all six trigonometric functions, and had sine tables in 0.25° increments, to 8 decimal places of accuracy, as well as tables of tangent values. Abu'l-Wafa also developed the trigonometric formula sin 2x = 2 sin x cos x. Persian mathematician Omar Khayyam solved cubic equations using approximate numerical solutions found by interpolation in trigonometric tables.

All of these earlier works on trigonometry treated it mainly as an adjunct to astronomy; perhaps the first treatment as a subject in its own right was by Indian mathematician Bhaskara II and Persian mathematician Nasir al-Din Tusi, who also gave the formula a/sin A = b/sin B = c/sin C and listed the six distinct cases of a right angled triangle in spherical trigonometry. Regiomontanus was perhaps the first mathematician in Europe to treat trigonometry as a distinct mathematical discipline, in his De triangulis omnimodus written in 1464, as well as his later Tabulae directionum which included the tangent function, unnamed.

In the 13th century, Persian mathematician Nasir al-Din Tusi states the law of sines and provides a proof for it. In the work of Persian mathematician Ghiyath al-Kashi (14th century), there are trigonometric tables giving values of the sine function to four sexagesimal digits (equivalent to 8 decimal places) for each 1° of argument with differences to be added for each 1/60 of 1°. Timurid mathematician Ulugh Beg's (14th century) accurate tables of sines and tangents were correct to 8 decimal places.

Madhava (c. 1400) in South India made early strides in the mathematical analysis of trigonometric functions and their infinite series expansions. He developed the concepts of the power series and Taylor series, and produced the trigonometric series 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, π/4, the radius, diameter, circumference and angle θ in terms of trigonometric functions. His works were expanded by his followers at the Kerala School upto the 16th century.

The Opus palatinum de triangulis of Rheticus, a student of Copernicus, was probably the first 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 Valentin Otho in 1596.

Leonhard Euler's Introductio in analysin infinitorum (1748) was mostly responsible for establishing the analytic treatment of trigonometric functions in Europe, defining them as infinite series and presenting "Euler's formula" eix = cos(x) + i sin(x). Euler used the near-modern abbreviations sin., cos., tang., cot., sec., and cosec. Brook Taylor defined the general Taylor series and gave the series expansions and approximations for all six trigonometric functions. The works of James Gregory and Colin Maclaurin were also very influential in the development of trigonometric series.


For more info see http://en.wikipedia.org/wiki/Cosine