Half Is BetterSine 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 1/2o to 180o

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

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

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

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

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.

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...............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 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

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.