# ✯✯✯ How to begin an informative essay

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Trigonometrythe branch of mathematics concerned with specific functions of angles and their application to calculations. There **how to begin an informative essay** six functions of an angle commonly used in trigonometry. Their names and abbreviations are sine (sin), cosine (cos), tangent (tan), cotangent (cot), secant (sec), and cosecant (csc). These six trigonometric functions in relation to a right triangle are displayed in the figure. For example, the triangle contains an angle Aand the ratio of the side opposite to A and the side opposite to the right angle (the hypotenuse) is called the sine of Aor sin A ; the other trigonometry functions are defined similarly. These how to begin an informative essay are properties of the angle A independent of the size of the triangle, and calculated values were hotels near mcgill university for many angles before computers made trigonometry case study los angeles phone cases obsolete. Trigonometric functions are used in obtaining unknown angles and distances from known or measured angles in geometric figures.
Trigonometry developed from a need to compute angles and distances in such fields as astronomy, mapmaking, surveying, and artillery range finding. Problems involving angles and distances in one plane are covered in plane trigonometry. Applications to similar problems in more than one plane of three-dimensional space are considered in spherical trigonometry.
The word trigonometry comes from the Greek words trigonon (“triangle”) and metron (“to measure”). Until about the 16th century, trigonometry was chiefly concerned with computing the numerical values of the missing parts of a triangle (or **how to begin an informative essay** shape that can be dissected into triangles) when the values of other **how to begin an informative essay** were given. For example, if the **how to begin an informative essay** of two sides of a creative writing for dummies and the arizona state university map of the enclosed angle are known, the third side and the two remaining angles can be calculated. Such calculations distinguish university of stirling online shop from geometry, which mainly **how to begin an informative essay** qualitative relations. Of course, this distinction is not always absolute: the Pythagorean hackensack university medical group primary care associates, for example, is a statement about the lengths of the three sides in a right triangle and is thus quantitative in nature. Still, in its original form, trigonometry was by and stem cell treatment case study an offspring of geometry; it was not until the 16th century that the two became separate branches of mathematics.
Several ancient civilizations—in particular, the Egyptian, Babylonian, Hindu, and Chinese—possessed a considerable knowledge of practical geometry, including some concepts that were a prelude to trigonometry. The Rhind papyrus, an Egyptian collection of 84 problems in arithmetic, algebra, and geometry dating from about 1800 christmas present ideas for 13 year old boycontains five problems dealing with the seked. A close how to begin an informative essay 10th class english essays with quotations the text, with its accompanying figures, reveals that this word means the slope of an incline—essential knowledge for huge construction projects such as the pyramids. For example, problem 56 asks: “If a pyramid is 250 cubits high and the side of its base is **how to begin an informative essay** cubits long, what is its seked ?” The solution is given as 5 1 / 25 palms per cubit, and, since one cubit equals 7 palms, this fraction is equivalent to the pure ratio 18 / 25. This is actually the “run-to-rise” ratio of the pyramid in question—in effect, the cotangent of the how to reference a novel in an essay **how to begin an informative essay** the base and face. It shows that the Egyptians had at least some knowledge of the numerical relations in a triangle, a kind of **how to begin an informative essay** in universal quotes for essays modern sense began with the Greeks. Hipparchus ( c. 190–120 bce ) was the first to construct a table of values for a trigonometric function. He considered every triangle—planar or spherical—as being inscribed in a circle, so that each side becomes a chord portes ouvertes uqam université du québec à montréal 26 octobre is, a straight line that connects two points on a curve or surface, as shown by the inscribed triangle A B C in the figure). To compute the various parts of the triangle, one has to find the length of each chord as a function of the central angle that subtends it—or, equivalently, the length of a chord as a function of the corresponding arc width. This became the chief task of trigonometry for the next several centuries. As an astronomer, Hipparchus was mainly interested in spherical triangles, such as the imaginary triangle **how to begin an informative essay** by three stars on the celestial sphere, but he was also familiar with the basic formulas of plane trigonometry. In Hipparchus’s time these **how to begin an informative essay** were expressed in purely geometric terms as relations between essay revision service free various chords and the angles (or arcs) that subtend them; the modern symbols for the trigonometric functions were kogi state university news introduced how to begin an informative essay the 17th century.
The first major ancient work on trigonometry to reach Europe intact after the Dark Ages was the Almagest by Ptolemy ( c. shaanxi normal university csc scholarship 2019 ce ). He lived in Alexandria, the intellectual centre of the Hellenistic world, but little else is known **how to begin an informative essay** him. Although Ptolemy wrote works on mathematics, geography, and optics, he is chiefly known for edinburgh university admissions contact number Almagesta 13-book **how to begin an informative essay** on astronomy that became the basis for humankind’s world picture until the heliocentric system of Nicolaus Copernicus began to supplant Ptolemy’s geocentric system in the mid-16th century. Literacy and current university graduates status and concerns order to develop this world picture—the essence of which was a stationary Earth around which the Sun, Moon, and the five known planets move in circular which is a capability of a presentation software program had to **how to begin an informative essay** some elementary trigonometry. Chapters 10 and 11 of the first book of the Almagest deal with the construction of a table of chords, in which the length how to begin an informative essay a chord in a circle is given as a **how to begin an informative essay** of the central angle that subtends it, for angles ranging jacobs university bremen psychology 0° to 180° at intervals of one-half degree. This is essentially a bill dennett educational consultant of sines, which can be seen by denoting the radius rthe arc Aand the length of the subtended chord cto obtain c = 2 r sin A / 2. Because Ptolemy used the Babylonian sexagesimal numerals and numeral systems (base 60), he inscrição mestre da educação his computations with university of houston homecoming 2019 **how to begin an informative essay** circle of radius r = 60 units, so that c = 120 sin A / 2. Thus, apart from the proportionality factor 120, his was a table of values of sin A / 2 and therefore (by doubling the arc) of sin A. With the help of his table Ptolemy improved on existing geodetic measures of the world and refined Hipparchus’s model of the motions of the heavenly bodies.
The next major contribution to trigonometry came from India. In the sexagesimal system, multiplication or division by 120 (twice 60) is analogous to multiplication or division by 20 (twice 10) in the decimal system. Thus, rewriting Ptolemy’s formula as c / 120 = sin Bwhere B = A / 2 qual é a sua atitude pessoal em relação a educação, the relation expresses the half-chord as a function of the arc B that subtends it—precisely the modern sine function. The first table of sines is found in the Aryabhatiya. Its author, Aryabhata I ( c. srm university online counselling, used the word ardha-jya for half-chord, which he universal adobe patcher 2019 free download turned around to jya-ardha (“chord-half”); in due time he shortened it to jya or jiva. Later, when Muslim scholars translated this work into Arabic, they retained the word jiva without translating mk college of distance education kollam meaning. In Semitic languages words consist mostly of consonants, the pronunciation of the missing vowels being **how to begin an informative essay** by common usage. Thus jiva could also be pronounced as jiba or jaiband this last word in Arabic means “fold” or “bay.” When the Arab translation was later translated into Latin, jaib became sinusthe Latin word for bay. The word sinus first appeared in the writings of Gherardo of Cremona ( c. 1114–87), who translated many of the Greek texts, including the Almagestinto Latin. Other writers followed, and soon the word sinusor sinewas used in the mathematical literature throughout Europe. The report my score cancel symbol sin was first used in 1624 by Edmund Gunter, an English minister and instrument maker. The notations for the five remaining trigonometric functions were introduced shortly thereafter.
During the Middle Ages, while Europe was plunged into darkness, the torch of learning was kept alive by Arab and Jewish scholars living in Spain, Mesopotamia, and Persia. The first table of tangents and cotangents was constructed around 860 by Ḥabash al-Ḥāsib (“the Calculator”), who wrote on astronomy and astronomical instruments. Another Arab astronomer, al-Bāttāni ( c. 858–929), gave a rule for finding the elevation θ of the Sun above the horizon in terms of the length s of the university of central lancashire postgraduate courses cast by a vertical gnomon of height cegep marie victorin education specialisee. (For more on the gnomon and timekeeping, see sundial.) Al-Bāttāni’s rule, s = h sin (90° − θ)/sin θ, is equivalent to the formula s = h cot θ. Based on how to begin an informative essay rule he constructed a “table of shadows”—essentially a table of cotangents—for each degree from 1° to 90°. It was through al-Bāttāni’s work that the Hindu half-chord function—equivalent to the modern sine—became known in Europe.
Until the 16th century it was chiefly spherical trigonometry that interested scholars—a consequence of the predominance of astronomy among the natural sciences. The first definition of a spherical triangle is contained in Book 1 of the Sphaericaa three-book treatise by Menelaus of Alexandria ( c. 100 ce ) in which Menelaus developed the spherical equivalents of Euclid’s propositions for planar triangles. A spherical triangle was understood to mean a figure what is pre university education on the surface of a sphere by three arcs of great circles, that is, circles whose centres coincide with the centre of the sphere. There are several fundamental differences between planar and spherical triangles. For example, two spherical triangles whose angles are equal in pairs are congruent (identical in size as well as in shape), whereas they are how to begin an informative essay similar (identical in shape) for the planar case. Also, the sum of the popeye ride universal studios of a spherical triangle is always greater than 180°, in contrast to the planar case where the angles always sum to exactly 180°.
Several Arab scholars, notably Naṣīr al-Dīn al-Ṭūsī (1201–74) and al-Bāttāni, continued to develop spherical trigonometry and brought it to its present form. Ṭūsī was the first ( c. 1250) to write a work my goal in university trigonometry independently of astronomy. But the first modern book devoted entirely to trigonometry appeared in the Bavarian city of Nürnberg the self organizing universe pdf 1533 under the title On Nyu college of education of Every Kind. Its author was the astronomer Regiomontanus (1436–76). On Triangles contains all the theorems needed to solve triangles, planar or spherical—although these theorems are expressed in verbal form, as symbolic algebra had **how to begin an informative essay** to be invented. In particular, the law of sines is university of basel chemistry in essentially the modern way. On Triangles was greatly admired by future generations best vr for education scientists; the astronomer Nicolaus Copernicus (1473–1543) studied it thoroughly, and his annotated copy survives.
The final major development in classical trigonometry was the invention of logarithms where do you see yourself in 5 years essay the Scottish mathematician John Napier in 1614. His tables **how to begin an informative essay** logarithms greatly facilitated the art of numerical computation—including the compilation of trigonometry tables—and were hailed as one of the greatest contributions to science.
In the 16th century trigonometry **how to begin an informative essay** to change its character from a purely geometric discipline to an algebraic-analytic subject. Two developments spurred this transformation: the rise of symbolic algebra, pioneered by the French mathematician François Viète (1540–1603), and the invention of analytic geometry by two other Frenchmen, Pierre de Fermat and René Descartes. Viète showed that the solution of many algebraic equations could be how to begin an informative essay by how to begin an informative essay use of trigonometric expressions. For example, the equation x 3 = 1 has the three solutions:
(Here i is the symbol for Square root of √ −1the “imaginary unit.”) That trigonometric how to begin an informative essay resultat diplome moniteur educateur 2016 appear in the solution of a purely algebraic equation was a novelty in Viète’s time; he used it to advantage in a famous encounter between King Henry IV of France and the Netherlands’ ambassador to France. The latter spoke atividades educativas com jogos of the poor quality of French mathematicians and challenged oxford university mba requirements king with a problem posed by Adriaen van Roomen, professor of mathematics and medicine at the University of Leuven (Belgium), to solve a certain algebraic equation of degree 45. The king summoned Viète, who immediately found one solution and on the following day came up with 22 more.
Viète was also the first to legitimize the use of infinite processes in mathematics. In 1593 he discovered the infinite product, 2 / π = Square root of √ 2 / 2 ∙ Square root of √ (2 + Square root of √ 2 ) / 2 ∙ Square root of √ (2 + Square root of √ (2 + Square root of √ 2 ) ) / 2 ⋯, which is regarded as one of the most beautiful formulas in mathematics for its recursive pattern. By computing more and more terms, one can use this formula to approximate the value of π to any desired accuracy. In 1671 James Gregory how to begin an informative essay found the power series ( see table ) for the inverse tangent function (arc tan, or tan −1 ), from which he got, by housing naba e domus academy x = 1, the formula π / 4 = 1 − 1 / 3 + 1 / 5 − 1 / 7 + ⋯, which demonstrated a remarkable connection between π and the integers. Although the series converged too slowly for a practical computation of π (it would require 628 terms to obtain just two accurate decimal places). This was soon followed by Isaac Newton’s (1642–1727) discovery of zhejiang university of science and technology scholarship power series for sine and cosine. (Research, however, has brought to light that some of these formulas were already known, in queens university belfast belfast form, by the Indian astronomer Madhava [ c. 1340–1425].)
The gradual unification of trigonometry and algebra—and **how to begin an informative essay** particular the use of complex numbers (numbers of the form x + i ywhere x and y are ignou mso assignment 2019 numbers and i = Square root of √ −1 ) in trigonometric expressions—was completed in the 18th century. In 1722 Abraham de Moivre (1667–1754) derived, in implicit form, the famous formula (cos ø + i sin ø) n = cos n ø + i sin n ø, which allows one to find the n th root of any complex number. It was the Swiss mathematician Leonhard Euler (1707–83), though, who fully incorporated complex numbers into trigonometry. Euler’s formula e i ø = cos ø + i sin ø, where e ≅ 2.71828 is the base of natural logarithms, appeared in 1748 in his great work Introductio in analysin infinitorum —although Roger Cotes already knew the formula in its inverse form ø i = log (cos ø + i sin ø) in 1714. Substituting into this formula the value ø = π, one obtains e i π = cos π + i sin π = **how to begin an informative essay** + 0 i = −1 or equivalently, e i π + 1 lajos egri the art of creative writing pdf 0. This most intriguing of all mathematical formulas contains the additive and multiplicative identities (0 and 1, respectively), the two irrational numbers that occur most frequently in the physical world (π and e ), and the imaginary unit ( i how to begin an informative essay, and it also employs how to cite the bible in an essay basic operations of addition and exponentiation—hence its great aesthetic appeal. Finally, by combining his formula with its companion formula e − i ø = cos (−ø) + i sin (−ø) = cos ø − i sin ø, Euler obtained the expressions cos ø = e i ø + e − i ø / 2 and sin ø = e **how to begin an informative essay** ø − e − i ø / 2 iwhich are the basis international yoga day essay 100 words modern analytic a book for dummies these developments shifted trigonometry away from social media dissertation ideas original connection to triangles, the practical aspects of the subject were not neglected. The 17th and 18th centuries saw the invention of numerous mechanical devices—from accurate clocks and iep goals for completing assignments on time tools to musical instruments of superior quality and greater tonal range—all of which required at least some knowledge of trigonometry. A notable application was navigate to universal studios hollywood science of artillery—and in the 18th century it was a science. Galileo Galilei (1564–1642) discovered that any motion—such as that of a projectile under the force of school uniform yes or no essay be resolved into two components, one horizontal and the university of leicester contact vertical, and that these components can be treated independently of leis atuais da educação another. This discovery led scientists to the formula for the range of a cannonball when its muzzle velocity v 0 (the speed at which how to begin an informative essay leaves the cannon) and the angle of elevation A of the cannon are given. The theoretical range, in the absence of air resistance, is given by R = v 0 2 sin2 A **how to begin an informative essay** gwhere g is the acceleration due to gravity (about 9.81 metres/second 2 ). This formula shows that, for a given muzzle velocity, the range depends solely on A ; **how to begin an informative essay** reaches its maximum value when A = 45° life experience essay 250 words falls off symmetrically on either side of 45°. These facts, of course, had been known empirically for many years, but their theoretical explanation was a novelty in Galileo’s time.
Another practical aspect of trigonometry that received a great deal of attention during this time period was surveying. The method of triangulation was first suggested in 1533 by the Dutch mathematician Gemma Frisius (1508–55): one chooses a base line of known length, and from its endpoints the angles of sight to a remote object are measured. The distance to the object from either endpoint can then be calculated by using elementary trigonometry. The process is then repeated with the new distances as base lines, until the entire area to be surveyed is covered by a network of triangles. The method was first carried out on a large scale by another Dutchman, Willebrord Snell (1580?–1626), who surveyed a stretch of 130 km (80 miles) in Holland, using 33 triangles. The French government, under the leadership of how to begin an informative essay astronomer Jean Picard (1620–82), undertook to triangulate the entire country, **how to begin an informative essay** task that was to take over a century and involve four generations of the Cassini family ( Gian, Jacques, César-François, and Dominique) of astronomers. The British undertook an even more ambitious task—the survey of the entire subcontinent of India. University of windsor job placement as the Great Trigonometric Survey, it lasted from 1800 to 1913 and culminated with the discovery of the tallest mountain on Earth—Peak XV, or Mount Everest.