Half Is Better

1. Half Is Better Sine and Cosine

2. Hipparchus of Rhodes(190 – 120 B.C.) • Planetary motion • Celestial sphere • Position of stars were specified by angles • Relate angles to a straight line segment • Chords • Future positions of stars and planets

3. Hipparchus of Rhodes • Table of chords • Worked with a circle with radius 3438 • Why 3438? • Didn’t survive • Referenced by other mathematicians

4. Claudius Ptolemy (85 – 165 A.D.) • The greatest ancient Greek astronomer • Almagest – theory of chords • “Spherical triangles” • Explained how to construct a table of chords • Devised a method to compute approx. to chords from 1/2o to 180o

5. Going to India! • Table of “half-chords” – 5th century • Many situations require one to use half the chord of twice the angle • Indian astronomers understood this • Called them jyā-ardha – “half-chords” • Shortened to jyā

6. Still in India • Earliest tables used circles with radius 3438 (Hipparchus of Rhodes) • No way to exactly find the length of a chord of an arbitrary angle • Many Indian mathematicians found approximations through the 12th century and beyond • Rediscovered by European mathematicians

7. Arabs • Indian mathematics came to Europe by way of Arabic mathematicians • Arabs learned astronomy from jyā tables • Instead of translating jyā, they invented the word jiba • Discovered connections between trigonometry and algebra

8. “Trigonometry” • Computing sines of arbitrary angles and solving cubic equations • Expanded understanding of spherical triangles • Added a “shadow” function (tangent) • Improved methods for computing “half-chord” and “shadow” tables

9. The Mistake • Europeans discovered Arabic material • Translating jiba • jb → jaib – “cove” or “bay” • Chose sinus – “Something is sinuous if it has lots of coves and hollows.” • This turned into our modern word sine

10. 16th Century • Our “trigonometry” was a part of astronomy until this time • Began to break apart as a topic of interest itself • Johannes Müller (1436 – 1476) • On All Sorts of Triangles (1463) • Not published for several decades • Knows of tangent but only uses sine • Applications of both plane and spherical triangles

11. Cosine? • Needed to use the sine of the complementary angle • sin(90o - ) • No special name yet • By the 17th century, sinus complementi had become co. sinus, then cosinus.

12. The Next Few Decades… • Works influenced by On All Sorts of Triangles by Müller • Re-workings of On All Sorts of Triangles • George Joachim Rheticus (1514 – 1574) • Sines and other functions of right triangles • No reference to circles • Thomas Finche (1561 – 1656) • Invented the words tangent and secant • Bartholomeo Pitiscus (1561 – 1613) • Invented the word trigonometry for his book title (1595)

13. After Calculus • Leonhard Euler (1707 – 1783) • Thought of sine as a ratio instead of a line segment • Used sine as a function, the way we now use functions • Sine is a function of the arc in a unit circle

14. Sine Curve • Gilles de Roberval (1602 – 1675) • Sketched the sine curve • He was computing the area under a cycloid • Not clear if he understood what he did

15. Hipparchus of Rhodes...............190 – 120 B.C. Claudius Ptolemy……………......85 – 165 A.D. Almagest – theory of chords Table of “Half Chords”…………....5th century Indian mathematics came to Europe….........…..……......… ~12th century Mistranslation of jiba….... ~12 – 16th centuries Timeline

16. Timeline Continued • Johannes Müller……………………1436 – 1476 • On All Sorts of Triangles………………………….1463 • Cosine…………………………..……17th century • Bartholomeo Pitiscus………………1561 – 1613 • Invented “trigonometry”………………………..1595 • Gilles de Roberval…………….……1602 – 1675 • Sketched sine curve • Leonhard Euler………………….…..1707 – 1783

17. References • Berlinghoff, William and Gouvea, Fernando. Math through the Ages. Maine: Oxton House Publishers, 2002. • Sine curve - http://edgrenweb.se/math/. • Half the chord of twice the angle example - http://aleph0.clarku.edu/~djoyce/java/trig/sines.html.