NURINA AYUNINGTYAS (093174003)

ANDDWITYA BUDI ANGGRAENY (093174046)

Present:

HISTORY OF

TRIGONOMETRY

INTRODUCTION

DEVELOPME

NTEarly

trigonometr

y

Greek Indian Islamic

Chines

e

Europea

n

REFERENCE

s

conclusion

INTRODUCTION

Trigonometry developed from the study of right-angled triangles by applying their relations of sides and angles to the study of similar triangles. The word trigonometry comes from the Greek words

"trigonon" which means triangle, and "metria" which means measure.

The term trigonometry was first invented by the German mathematician Bartholomaeus Pitiscus, in his work, Trigonometria sive de dimensione triangulea, and first published 1595.

This 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 primary use of trigonometry is for operation, cartography, astronomy and navigation, but modern mathematicians has extended the uses of trigonometric functions far beyond a simple study of triangles to make trigonometry indispensable in many other areas.

Especially astronomy was very tightly connected with trigonometry, and the first presentation of trigonometry as a science independent of astronomy is credited to the Persian Nasir ad-Din in the 13 century.

development

EARLY TRIGONOMETRY

Trigonometric functions have a varied history. The old Egyptians looked upon trigonometric functions as features of similar triangles, which were useful in land surveying and when building pyramids.

The old Babylonian astronomers related trigonometric functions to arcs of circles and to the lengths of the chords subtending the arcs. They kept detailed records on the rising and setting of stars, the motion of the planets, and the solar and lunar eclipses, all of which required familiarity with angular distances measured on the celestial sphere.

GREEK MATHEMATICS

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 the bisected angle, that is, crd , 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 mathematicians, but in their equivalent chord form.

The first trigonometric table was apparently compiled by Hipparchus of Nicaea (180 - 125 BC), who is now consequently known as "the father of trigonometry." Hipparchus was the first to tabulate the corresponding values of arc and chord for a series of angles.

Menelaus of Alexandria (ca. 100 A.D.) wrote in three books his Sphaerica. In Book I, he established a basis for spherical triangles analogous to the Euclidean basis for plane triangles. Book II of Sphaerica applies spherical geometry to astronomy. And Book III contains the "theorem of Menelaus". He further gave his famous "rule of six quantities".

One of his most important theorems state that if the three lines forming a triangle are cut by a transversal, the product of the length of three segments which have no common extremity is equal to the products of the other three.

This appears as a lemma to a similar proposition relating to spherical triangle, “the chords of three segments doubled” replacing “three segments.” The proposition was often known in the Middle Ages as the regula sex quantitatum or rule of six quantities because of the six segments involved.

Theorem of Menelaus

Claudius Ptolemy (ca. 90 - ca. 168 A.D.) expanded upon Hipparchus' Chords in a Circle in his Almagest, or the Mathematical Syntaxes . 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.

Ptolemy’s theoremPtolemy's theorem is a relation in Euclidean geometry between the four sides and two diagonals of a cyclic quadrilateral (a quadrilateral whose vertices lie on a common circle). The theorem is named after the Greek astronomer and mathematician Ptolemy (Claudius Ptolemaeus).

lACl · lBDl = lABl · lCDl + lBCl · lADl

This relation may be verbally expressed as follows:If a quadrilateral is inscribed in a circle then the sum of the products of its two pairs of opposite sides is equal to the product of its diagonals.

INDIAN MATHEMATICS

• The next sifnificant development of trigonometry were in India.

• Influential works from the 4th–5th century, known as the Siddhantas, first

defined the sine as the modern relationship between half an angle and half a

chord, while also defining the cosine, versine (1 – cosine), and inverse sine.

• Aryabhata (476–550 AD), collected and expanded upon the developments of

the Siddhantas in an important work called the Aryabhatiya.

• In the 6th century, Varahamihira used the formulas

sin2 (x) + cos2 (x) = 1, sin (x) cos ,

and = sin2 (x).

• In the 7th century, Bhaskara I produced a formula for calculating

the sine of an acute angle without the use of a table.

• Later in the 7th century, Brahmagupta redeveloped the formula

1 – sin2 (x) = cos2 (x) = sin2

• Madhava (c. 1400) made early strides in the analysis of

trigonometric functions and their infinite series expansions. He

developed the concepts of the power series and Taylor series,

and produced the power series expansions of sine, cosine,

tangent, and arctangent.

Madhava’s work

Madhava's sine table is the table of trigonometric sines of various angles constructed by the 14th century Kerala mathematician-astronomer Madhava of Sangamagrama. The table lists the trigonometric sines of the twenty-four angles 3.75°, 7.50°, 11.25°, ... , and 90.00° (angles that are integral multiples of 3.75°, i.e. 1/24 of a right angle, beginning with 3.75 and ending with 90.00). The table is encoded in the letters of Devanagari using the Katapayadi system.

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.

• Muhammad ibn Jabir al-Harrani al-Battani (Albatenius) (853-929)

discovered the reciprocal functions of secant and cosecant, and

produced the first table of cosecants for each degree from 1° to 90°.

He was also responsible for establishing a number of important

trigometrical relationships, such as:

tan a = and sec a =

• 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).

• Also in the late 10th and early 11th centuries, the

Egyptian astronomer Ibnu Yunus performed many careful

trigonometric calculations and demonstrated the

following trigonometric identity:

cosAcosB =

• 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" in its modern form.

note:cos (A+B) = cosAcosB - sinAsinBcos (A-B) = cosAcosB + sinAsinB

so

cos (A+B) + cos (A- B) = cosAcosB – sinAsinB + cosAcosB + sinAsinB = 2cosAcosB

cosAcosB =

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 Shen's 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 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).

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 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 = 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 formula

Step 1:For several number x: let y = cos x + isin x

i is the unreal element

Step 2:Taking the first derivative of equal sides gives us: = -sin x + icos x

Step 3:Because i2 = -1 by definition, we have -sin x + icos x = i(isin x + cos x)

Step 4:Join step 1, 2, and 3, we get: = iy

Step 5: If we multiply ( to both side, we get:

() dy = i (dx)

Step 6:Now, if we get the integral of all side we get: ln y = ix

Step 7:Now place e the power of equal sides, we get: y = eix [Because e ln y = y]

Step 8:Now combining the steps, we get: eix = cos x + isin x

From the diagram, we can see that the ratios sin θ and cos θ are defined as:

and

Now, we use these results to find an important definition for tan θ:Now, also so we can conclude that:

Now, also so we can conclude 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 = 1

sin2θ + cos2 θ = 1 through by cos2θ gives us:

So

tan2 θ + 1 = sec2 θ

sin2θ + cos2 θ = 1 through by sin2θ gives us:

So

1 + cot2 θ = csc2 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.

REFERENCES

• http://aleph0.clarku.edu/~djoyce/ma105/

trighist.html

• http://home.c2i.net/greaker/comenius/9899/

historytrigonometry/Trigonometry1.html

• http://en.wikipedia.org/wiki/History_of_trigonometry

• http://www.bookrags.com/research/fundamental-

trigonometric-functions-wom/

THANK YOU

VERY MUCH