**Trigonometry**

Its History and Facts

By

Reo Watanabe

**A. Timeline of Trigs.**

**Sources**

**http://en.wikipedia.org/wiki/Trigonometry**

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

Babylonian Times

The Babylonians established the measurement of angles in degrees, minutes, and seconds. Their high-level in mathematics can be seen from Plimpton 322, a clay tablet from ca.1800BC that contains the Pythagorean Triple.

Greco-Roman Times

Hipparchus compiled the trigonometric triangle for solving triangles back in the 2nd century BC

Also devised the sine table.

The idea of 360° in a circle came up by ca. 260 BC when Aristarchus of Samos composed On the Sizes and Distances of the Sun and Moon

Ptolemy, showing the same Babylonian influence as Hipparchus, divided the circle into 360° and the diameter into 120 parts.

Menaleus provided the way one could observe the ratios/lengths with a circle, in the late 1st Century AD

The term "sine" is derived from the Latin word of

sinus,

which means "bay", "bosom" or "fold", translating the Arabic term

jayb

.

jayb,

in origin, is a corruption of Sanskrit

jīvā

, or "chord". Sanskrit

jīvā

in learned usage was a synonym of

jyā

"chord", originally the term for "bow-string".

**http://www.cartage.org.lb/en/themes/sciences/Mathematics/Trigonometry/history/History%20.html**

**http://en.wikipedia.org/wiki/Plimpton_322**

Muslim Times

Muslim scientists also produced tables of great precision. For example, their tables of the sine and tangent, constructed for steps of 1/60 of a degree, were accurate for better than one part in 700 million.

The idea of tangents and cotangents came by 860 which used two measures translated into Latin as

umbra recta

and

umbra versa

.

The term "tangent" was 1st used in 1583 by Thomas Fincke.

By 980 Abu'l-Wafa knew that sin 2x = 2 sin x cos x

Also was the person who established sine in its modern form.

**http://www-history.mcs.st-andrews.ac.uk/HistTopics/Trigonometric_functions.html**

European Times

In the 13th century, German astronomer Georges Joachim, known as Rheticus introduced the modern conception of trigonometric functions as ratios instead of as the lengths of certain lines.

The French mathematician François Viète introduced the polar triangle into spherical trigonometry, and stated the multiple-angle formulas for sin(nq) and cos(nq) in terms of the powers of sin(q) and cos(q).

The term "sine" was used more practically after Fibonacci used the term

sinus rectus arcus

.

Edmund Gunter was the first to use the abbreviation sin in 1624 in a drawing, and was used 10 years later in a book written by French mathematician Hérigone while Cavalieri used Si and Oughtred S

Versine, or versed sines, were used during this time

Trigonometry developed greatly in this time period, with the term first appearing as a title of a book written in 1595 by Bartholomeo Pitiscus.

18th century was when trig. really took off, with Bernoulli finding the relation between sin-1z and log z in 1702 while Cotes showed that ix = log(cos x + i sin x ), and so on.

John Napier invented the Napier's analogies, which are memory aids for ten laws for solving spherical triangles, and some proportions for solving oblique spherical triangles.

B. Significant Persons

Egyptian

Greco-Roman

Indian

Islamic

European

**http://en.wikipedia.org/wiki/Ab%C5%AB_al-Waf%C4%81%27_B%C5%ABzj%C4%81n%C4%AB**

Abū al-Wafā' Būzjānī

: Persian mathematician/ astronomer in Baghdad who established several trigonometric identities such as sin(a±b) in their modern form.

**http://www-history.mcs.st-and.ac.uk/history/Indexes/HistoryTopics.html**

Ahmes:

Scribe who wrote the Rhind Papyrus, which contains the chief source of information on Egyptian mathematics

Arybhata I

: Author of

Aryabhatiya

which summarized the development of Indian mathematics up to that point.

Bartholomeo Pitiscus

: Polish theologian who first coined the word Trigonometry in his book

Trigonometria

.

Brahmagupta

: mathematician who made advances in astronomy and most importantly in number systems including algorithms for square roots and the solution of quadratic equations.

Leonhard Euler:

Swiss mathematician who contributed greatly to mathematics through various findings including formulas such as eix = cos x + i sin x.

Georg Joachim Rheticus:

Austrian mathematician and astronomer who published the trigonometrical sections of Copernicus's

De Revolutionibus

.

Hipparchus

: Greek mathematician who compiled an early example of trigonometric tables and gave methods for solving spherical triangles.

Menelaus

： Greek geometer who applied spherical geometry to astronomy. He is best known for creating what we know as Menelaus's theorem.

Albatenius

: Astronomer of Mesopotamia who first applied trigonometry into his field of science instead of geometry.

Abu Ja'far Muhammad ibn Musa Al-Khwarizmi

: Islamic mathematician who wrote on Hindu-Arabic numerals and was among the first to use zero as a place holder in positional base notation. The word algorithm derives from his name, and his algebra treatise

Hisab al-jabr w'al-muqabala

gives us the word "algebra" and can be considered as the first book to be written on algebra.

Ptolemy

: The most influential Greek Astronomer/Mathematician of his time and his works influenced these fields for thousands of years, most notably dividing the circle into 360° and the diameter into 120 parts.

Regiomontanus

: German scholar who completed the

Epitome of the Almagest

, a translation and comment of

Ptolemy's Almagest.

Thales

: First known Greek philosopher, scientist and mathematician who brought geometry from Egypt.

Varahamihira

: Mathematician who found trigonometric formulas which translate into what we now know as sin x = cos(π/2 - x), sin2x + cos2x = 1, and (1 - cos 2x)/2 = sin2x.

C. Questions

1. What was the very first recorded step in the development of trigonometry?

The Pythagorean Triple being used on Babylonian tablets were the oldest, although if what Ahmes said were true the things he wrote would be even older.

2. Who do you think should be called the father of trigonometry?

No one, because trigonometry developed individually in each and every civilizations throughout history, and not by a single person in a certain time frame.

3. Find three mathematicians from three different eras and state their contributions; why were they so important?

Hipparchus- He was the person who compiled an early example of trigonometric tables also methods for solving spherical triangles.

Brahmagupta - He helped the number system be structured as it is today.

Pitiscus - The term trigonometry would not have existed had he not coined it.

4. Where did the word "sine" come from?

The term "sine" is derived from the Latin word of

sinus,

which means "bay", "bosom" or "fold", translating the Arabic term

jayb

.

Jayb,

in origin, is a corruption of Sanskrit

jīvā

, or "chord". Sanskrit

jīvā

in learned usage was a synonym of

jyā

"chord", originally the term for "bow-string".

5. On the timeline of the history of trig functions, what is the most significant fact?

To me, the most significant fact is probably how Rheticus structured trigonometry into how it is in our world today.

6. What is the most obscure fact in the history of trig functions?

To me, the most obscure fact is Ahnes' recordings on the papyrus, which does not really seem much.

7. What was the most interesting/surprising discovery that you made in this project?

The most interesting find was how each civilization had come to similar discoveries even though they were not exactly connected with each other and how they were also using different methods.

**The most helpful site was the one on the very last on this list was the last one, as the site gave me a lot of information about the historical figures.**