Did you know . . .?

The complex origins of trigonometry are embedded in the history of the simple word "sine," a mistranslation of an Arabic transliteration of a Sanskrit mathematical term! The complex etymology of "sine" reveals trigonometry's roots in Babylonian, Greek, Hellenistic, Indian, and Arabic mathematics and astronomy.

Although trigonometry now is usually taught beginning with plane triangles, its origins lie in the world of astronomy and spherical triangles. Before the sixteenth century, astronomy was based on the notion that the earth stood at the center of a series of nested spheres. To calculate the positions of stars or planets, one needed to use concepts we now refer to as trigonometry.

The earliest uses of trigonometric functions were related to the chords of a circle, and the recognition that the length of the chord subtended by a given angle x was (in modern terms) 2sin(x/2). The Greek astronomer and mathematician Hipparchus produced the first known table of chords in 140 BC. His work was further developed by astronomers Menelaus (ca. AD 100) and Ptolemy (ca. AD 100), who relied on Babylonian observations and traditions.

(illustration source: http://www.hps.cam.ac.uk/starry/sacroarmill.html)

Babylonian and Greek influences mingled with rich native mathematical developments in India around AD 500 to produce a trigonometry closer to its modern form. Hindu mathematical works such as that of Aryabhata give tables of half chords, known by the term jya-ardha or simply jya, which bears the following relationship to our modern concept of sine: jya x = r sinx, as illustrated below.

Jya here represents the half chord AM.

From India the sine function was introduced to the Arab world in the 8th century, where the term jya was transliterated into jiba or jyb. Early Latin translations of Arabic mathematical treatises mistook jiba for the Arabic word jaib, which can mean the opening of a woman's garment at the neck. Accordingly, jaib was translated into the Latin sinus, which can mean "fold" (in a garment), "bosom," "bay," or even "curve." Hence our word "sine."

[illustration source: George Gheverghese Joseph, The Crest of the Peacock: The Non-European Roots of Mathematics, new ed. (Princeton, NJ: Princeton University Press, 1991, 2000), p. 282.]

Another set of trigonometric functions, tangent and cotangent, developed from the study of the lengths of shadows cast by objects of various heights. Thales of Miletus used shadow lenghts to calculate the heights of the pyramids in around 600 BC. Both Indian and Arabic mathematics developed a trigonometric tradition based on shadow lengths, a tradition that, in turn, influenced European mathematics.

Click here for an applet illustrating sticks and shadows. (You can follow the links to move through illustrations of basic trigonometry functions, including the use of the tangent function for surveying.)

The functions secant and cosecant derive from tables first used by navigators in the fifteenth century.

As for the word "trigonometry," it first appeared as the title of a book Trigonometria (literally, the measuring of triangles), published by Bartholomeo Pitiscus in 1595.

In their own words:

Well do I know that I am mortal, a creature of one day.
But if my mind follows the winding paths of the stars
Then my feet no longer rest on earth, but standing by
Zeus himself I take my fill of ambrosia, the divine dish.
--Ptolemy, Almagest, bk .1. (quotation source: http://www-groups.dcs.st-andrews.ac.uk/~history/Mathematicians/Ptolemy.html)

Aryabhata is the master who, after reaching the furthest shores and plumbing the inmost depths of the sea of ultimate knowledge of mathematics, kinematics and spherics, handed over the three sciences to the learned world. --Bhaskara I, commentary on Aryabhatiya. (quotation source: http://www-groups.dcs.st-andrews.ac.uk/~history/Mathematicians/Aryabhata_I.html)

Hieronymus says that [Thales] even succeeded in measuring the pyramids by observation of the length of their shadow at the moment when our shadows are equal to our own height. --Diogenes Laertius, quoting Hieronymus, a pupil of Aristotle. (quotation source: http://www-groups.dcs.st-andrews.ac.uk/~history/Mathematicians/Thales.html)

You, who wish to study great and wonderful things, who wonder about the movement of the stars, must read these theorems about triangles. Knowing these ideas will open the door to all of astronomy and to certain geometric problems.--Johann Regiomontanus,
De Tringulis Omnimodis (quotation source: http://www-groups.dcs.st-and.ac.uk/~history/Quotations/Regiomontanus.html)

A real groaner version of the Chief Sohcahtoa story: http://forum.swarthmore.edu/dr.math/problems/burd5.15.98.html

Applet illustrating sines of angles of a triangle that you can manipulate: http://www.univie.ac.at/future.media/moe/galerie/trig/trig.html

Applet illustrating trig functions on a triangle you can manipulate: http://www.univie.ac.at/future.media/moe/galerie/wfun/wfun.html#winkelf

Applet illustrating graphs of sin, cos, and tan functions: http://www.univie.ac.at/future.media/moe/galerie/fun2/fun2.html#sincostan

http://www-groups.dcs.st-andrews.ac.uk/~history/HistTopics/Trigonometric_functions.html

http://members.aol.com/bbyars1/trig.html

http://www.britannica.com/eb/article?eu=115495&tocid=12231

S.O.S math's trigonometry pages: http://www.sosmath.com/trig/trig.html

Real world applications of trigonometry: http://aleph0.clarku.edu/~djoyce/java/trig/apps.html

Interactive trigonometry tutor: http://www.syvum.com/math/trigonometry.html

Trigonometry in a nutshell from Dr. Math: http://forum.swarthmore.edu/dr.math/problems/jimmy.04.11.01.html

By Laura Smoller, UALR Department of History.

May 2001.