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# Grade 11 - history of trigonometry project - Louise Keegan - model answers

Module by: Wynberg Girls' High School. E-mail the author

Summary: Model answers for history of trig project

## GRADE 11: MATHEMATICS: History of trigonometry project – Memo and additional information

The timeline should contain all the required information listed in Section A of the project.

### Section A: Timeline

Please note that some the times given are approximations, historians provide slightly different dates for the earlier developments in trig. There is also disagreement about which trig table should be counted as the first. I have included some of the Chinese developments in trig – this happened independently to the other developments – it is not essential that learners include this info.

 Approx. 1600BCE Egyptians & Babylonians: Used ratios of sides of similar triangles, known as trilaterometry Egyptian: Ahmes (1680-1620) – Trig is used in Pyramids Babylonian: Tablet called Plimpton 322 (dated 1800-1600BCE) gives a table of values. It is argued that it could be a table of secants, however some say it is more likely a table of Pythagorean triples. or - which shows that they Pythagoras’ theorem was in use before his time (530BCE) Babylonians introduced the idea of angle measurement 200BCE Greek: Started making use of chords, and noticed ratios/patterns associated 140BCE Greek: Hipparchus of Nicea – wrote 12 books on chords of a circle. He developed the first trigonometric table. This is seen as the first work on trig. It has since been lost, however historians agree that it did exist. Some refer to him as the ‘father of trigonometry’. Greek: As a result of Hipparchus’ tables mathematicians started to work with 360 degrees in a circle (the original idea of breaking a circle into 360 parts is attributed to Hypsicles, a Babylonian Astronomer: 190-120BCE). 100 Greek: Mathematician Ptolemy writes The Almagest, which contains tables of chords and includes explanation on how develop tables and gives examples of solving triangles using chords. He also develops the half-angle theorem. Indian: Idea of sin and cos starts to develop (as opposed to use of chords) C5 Indian: Mathematician Aryabhata (476–550 AD), defined the sine as the relationship between half an angle and half a chord. Also defined cosine, versine, and inverse sine. He developed tables for sine and ‘versine’ values that are still around today. He used the words jya for sine, kojya for cosine. (The words jya and kojya eventually became sine and cosine respectively after a mistranslation.) C6 Indian: Varamihira expands Aryabhata’s work on trigDevelops various formulae C7 Indian: Brahmagupta’s formulas expands Varamihira’s work to include other formulae 718 Chinese: Mathematician Kaiyuan Zhanjing develops table of sines C9 Islamic: Introduction of tan functionMuhammad ibn Mūsā al-Khwārizmī produced sin, cos and tan tables C10 Islamic: Muhammad ibn Jābir al-Harrānī al-Battānī (Albatenius) (853-929) develops formula for tan: Abū al-Wafā' al-Būzjānī develops the following formulae 1031 - 1035 Chinese develop table of chords and arcs C12 Europe: Islamic works are translated and Europeans begin to build on previous work. Regomontanus: treated trigonometry as a distinct discipline Europe: Georg Joachim Rheticus: Defines trig in terms of triangles and ratios, instead of chords and lengths of lines as previously used C18 Europe: Leonhard Euler: used near modern abbreviations for sin, cos and tan, Euler’s formula

### Section B

The first four questions are based on personal reflection, but the answers need to be reasonable and supported.

Question 5: Our modern word “sine” is derived from the Latin word sinus, which means “bay” or “fold”, from a mistranslation of the Sanskrit word jiva. Mathematician Arybhata used the term ardha-jiva, which was shortened to jiva and then transliterated by the Arabs as jiba. European translators in 12th-century confused jiba for jaib, meaning “bay”, probably because jiba and jaib are written the same in the Arabic script if the accents are not shown.

Question 6: References must be provided – see rubric.

### My references:

http://en.wikipedia.org/wiki/History_of_trigonometric_functions(I found this site most helpful, I have attached further information from)

Project idea: with thanks to – http://bangkokmaster.googlepages.com/HistoryofTrigonometryWebQuest.doc

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