Trigonometry developed from the study of right-angled triangles byapplying their relations of sides and angles to the study of similar triangles. Theword trigonometry comes from the Greek words "trigonon" which means triangle, and "metria" which means measure. The term trigonometry was first invented by the Germanmathematician Bartholomaeus Pitiscus, in his work, Trigonometria sive dedimensione triangulea, and first published 1595. This is the branch of mathematics that deals with the ratios betweenthe sides of right triangles with reference to either of its acute angles andenables you to use this information to find unknown sides or angles of anytriangle.
The primary use of trigonometry is for operation,cartography, astronomy and navigation, but modernmathematicians has extended the uses of trigonometricfunctions far beyond a simple study of triangles to maketrigonometry indispensable in many other areas. Especially astronomy was very tightly connected withtrigonometry, and the first presentation of trigonometry as ascience independent of astronomy is credited to the PersianNasir ad-Din in the 13 century.
EARLY TRIGONOMETRY Trigonometric functions have a varied history. The oldEgyptians looked upon trigonometric functions as features ofsimilar triangles, which were useful in land surveying andwhen building pyramids. The old Babylonian astronomers related trigonometricfunctions to arcs of circles and to the lengths of the chordssubtending the arcs. They kept detailed records on the risingand setting of stars, the motion of the planets, and the solarand lunar eclipses, all of which required familiarity withangular distances measured on the celestial sphere.
The first trigonometric tablewas apparently compiled byHipparchus of Nicaea (180 -125 BC), who is nowconsequently known as "thefather of trigonometry."Hipparchus was the first totabulate the correspondingvalues of arc and chord for aseries of angles.
Menelaus of Alexandria (ca. 100A.D.) wrote in three books hisSphaerica. In Book I, he establisheda basis for spherical trianglesanalogous to the Euclidean basis forplane triangles. Book II of Sphaericaapplies spherical geometry toastronomy. And Book III contains the"theorem of Menelaus". He furthergave his famous "rule of sixquantities".
One of his most important theorems state that ifthe three lines forming a triangle are cut by a transversal,the product of the length of three segments which haveno common extremity is equal to the products of theother three. This appears as a lemma to a similar propositionrelating to spherical triangle, “the chords of threesegments doubled” replacing “three segments.” Theproposition was often known in the Middle Ages as theregula sex quantitatum or rule of six quantities because ofthe six segments involved.
Claudius Ptolemy (ca. 90 - ca. 168A.D.) expanded upon HipparchusChords in a Circle in his Almagest, orthe Mathematical Syntaxes . Thethirteen books of the Almagest arethe most influential and significanttrigonometric work of all antiquity. Atheorem that was central toPtolemys calculation of chords waswhat is still known today as Ptolemystheorem.
Ptolemy’s theoremPtolemys theorem is a relation inEuclidean geometry between the foursides and two diagonals of a cyclicquadrilateral (a quadrilateral whosevertices lie on a common circle). Thetheorem is named after the Greekastronomer and mathematicianPtolemy (Claudius Ptolemaeus). lACl · lBDl = lABl · lCDl + lBCl · lADlThis relation may be verbally expressedas follows:If a quadrilateral is inscribed in a circlethen the sum of the products of its twopairs of opposite sides is equal to theproduct of its diagonals.
Madhava’s workMadhavas sine table is the table oftrigonometric sines of various anglesconstructed by the 14th centuryKerala mathematician-astronomerMadhava of Sangamagrama. Thetable lists the trigonometric sines ofthe twenty-four angles 3.75°, 7.50°,11.25°, ... , and 90.00° (angles thatare integral multiples of 3.75°, i.e.1/24 of a right angle, beginning with3.75 and ending with 90.00). Thetable is encoded in the letters ofDevanagari using the Katapayadisystem.
ISLAMIC MATHEMATICS • In the early 9th century, Muhammad ibn Musa al-Khwarizmi produced accurate sine and cosine tables, and the first table of tangents. He was also a pioneer in spherical trigonometry.• In 830, Habash al-Hasib al-Marwazi produced the first table of cotangents.
• By the 10th century, in the work of Abu al-Wafa al- Buzjani, Muslim mathematicians were using all six trigonometric functions. He also developed the following trigonometric formula:• sin (2x) = 2 sin (x) cos (x).
CHINESE MATHEMATICS• The polymath Chinese scientist, mathematician and official, Shen Kuo (1031–1095) used trigonometric functions to solve mathematical problems of chords and arcs.• Sal Restivo writes that Shens work in the lengths of arcs of circles provided the basis for spherical trigonometry developed in the 13th century by the mathematician and astronomer Guo Shoujing (1231–1316).• Guo Shoujing used spherical trigonometry in his calculations to improve the calendar system and Chinese astronomy.• Despite the achievements of Shen and Guos work in trigonometry, another substantial work in Chinese trigonometry would not be published again until 1607, with the dual publication of Euclids Elements by Chinese official and astronomer Xu Guangqi (1562–1633) and the Italian Jesuit Matteo Ricci (1552–1610).
EUROPEAN MATHEMATICS• 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.• 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 Valentin Otho in 1596.• In the 17th century, Isaac Newton and James Stirling developed the general Newton-Stirling interpolation formula for trigonometric functions.
• In the 18th century, Leonhard Eulers 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 "Eulers formula" eix = cosx + isinx.• 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 in the 17th century and Colin Maclaurin in the 18th century were also very influential in the development of trigonometric series.
Proof of Euler’s formulaStep 1: Step 3:For several number x: Because i2 = -1 by definition, we let y = cos x + isin x have -sin x + icos x = i(isin x + cos x)i is the unreal element Step 6: Step 7: Step 8: Now, if we get the Now place e the power of Now combining the steps, integral of all side we equal sides, we get: we get: get: y = eix [Because e ln y = y] eix = cos x + isin x ln y = ix
From the diagram, we can see that the ratios sin θ and cos θ are defined as: andNow, we use these results tofind an important definitionfor tan θ:Now, also so we can conclude Now, also so we can concludethat: that:
Also, for the values in the diagram, we can use Pythagoras Theorem and obtain: y2 + x2 = r2 Dividing through by r2 gives us: so we obtain the important result: sin2 θ + cos2 θ = 1sin2θ + cos2 θ = 1 through bycos2θ gives us: sin2θ + cos2 θ = 1 through by sin2θ gives us: SoSo 1 + cot2 θ = csc2 θtan2 θ + 1 = sec2 θ
CONCLUSION• Trigonometry is the branch of mathematics that deals with the ratios between the sides of right triangles with reference to either of its acute angles and enables you to use this information to find unknown sides or angles of any triangle.• The father of trigonometry is Hipparchus, an Greek mathematician who is first to tabulate the corresponding values of arc and chord for a series of angles.• Trigonometry is not the work of any one man or nation. Its history spans thousands of years and has touched every major civilization. It should be noted that from the time of Hipparchus until modern times there was no such thing as a trigonometric ratio . Instead, the Greeks and after them the Hindus and the Muslims used trigonometric lines . These lines first took the form of chords and later half chords, or sines. These chord and sine lines would then be associated with numerical values, possibly approximations, and listed in trigonometric tables.