Half Is Better Sine and Cosine

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

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

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 1/2 o to 180 o

Going to India! Table of “half-chords” – 5 th 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

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 12 th century and beyond Rediscovered by European mathematicians

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

“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

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

16 th 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

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

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)

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

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

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

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

References Berlinghoff, William and Gouvea, Fernando. Math through the Ages. Maine: Oxton House Publishers, Sine curve - Half the chord of twice the angle example - sines.html. sines.html