Teaching Trigonometry from a Historical Perspective A presentation by Mark Jaffee at the Ohio Council of Teachers of Mathematics Annual Conference Dayton, Ohio October 18,

Mathematics: From the Birth of Numbers Jan Gullberg W.W. Norton and Company, Inc. (New York) W.W. Norton and Company Ltd. (London) 1997 Mathematical Thought from Ancient to Modern Times, Vol I Morris Kline Oxford University Press, 1972 Trigonometric Novelties William R. Ransom J. Weston Walch, 1959

“Trigonometry developed from the study of right-angled triangles by applying their relations of sides and angles to the study of similar triangles.”

The ancient Egyptians used trigonometry only in the context of geometric ratios. They looked upon trigonometric functions as features of similar triangles which were useful in land surveying and when building pyramids Babylonian astronomers related trigonometric functions to arcs of circles and to the lengths of the chords subtending these arcs.

Trigonometry was the creation of Hipparchus, Menelaus, and Ptolemy. They were actually mostly interested in spherical trigonometry that could be used to predict positions of stars and planets, aid the telling of time, calendar-reckoning, and geography. The Greeks developed tables of trigonometric functions from 0 to 180 degrees for every ½ degree.

According to Professor William R. Ransom, who was Emeritus Professor of Mathematics at Tufts University in his book “Trigonometric Novelties”, J. Weston Walch, Publisher, 1959, “As long ago as Hipparchus (180 – 125 BC) and even Euclid (circa 300 BC) it was possible to calculate by algebraic means the lengths of chords of a unit circle subtending arcs with measure of multiples of 3°, as well as halves of the lengths of any known chords.”

But according to Gullberg, Hipparchus is known to have prepared extensive tables of chords at ½ degree intervals for all central angles from 0° to 180°. Ransom maintains that Ptolemy ( circa 140 AD ) was the first to create tables at ½ degree intervals, while Gullberg stated that Ptolemy merely copied the tables of Hipparchus.

They were able to determine exact values of the chords associated with 30° and 60° arcs using what they knew about triangles and they could determine the length of the chord associated with a 45° arc using what they knew about isosceles right triangles. Furthermore, they could evaluate the exact value of a chord associated with an 18° arc using the Golden Triangle. By using methods equivalent to our sin(a – b) formula, they could determine the sin 15° as sin (45° - 30°) and then sin 3° as sin(18° - 15°)

Hipparchus divided the circumference of the circle into 360° and divided the diameter into 120 parts. So, the relationship between his chord measurement and our sine function was

Ptolemy’s Theorem: Given inscribed Quadrilateral ABCD, AC  BD = AB  CD+AD  BC

Proof:

Application of Ptolemy’s Theorem

Ptolemy (circa 139 AD) created a table of chords in increments of ½°. In order to accomplish this he had to determine the chord of 1° first. He did this by observing, then proving that the ratio of two chords of a circle is less than the ratio of the measures of the corresponding arcs. In the diagram the measure of arc A’ is greater than the measure of arc A. Consequently c’ is greater than c. Ptolemy proved that c'/c<A'/A.

Here is how Ptolemy applied these inequalities. Let A’ and A measure 1° and ¾°, respectively. Then, the larger chord will be a 1° chord and the smaller chord will be a ¾° chord. According to the inequalities above, m( 1° chord) < m(¾° chord)∙m1°arc/m3/4°arc =4/3∙ = , and then if the arcs measure 1° and 3/2° m(1° chord)>m(3/2° chord)∙m1°arc/m3/2°arc =2/3∙ = Thus, the measure of a 1° chord is between and So, we can conclude that the measure of a 1° chord is , correct to 6 decimal places.

The Hindus of India were aware of the trigonometry developed by the Alexandrian Greeks. Arybhata the Elder, a prominent astronomer made two significant changes. First, he associated the half-chord with half of the arc of the full chord. This essentially invented the modern notion of the sine function on the unit circle. However, he chose to introduce a radius of 3,438 units.

Aryabhata called this half-chord “ardha-jya”, later abbreviated to “jya”. Years later Arab translators turned this phonetically into “jiba”, a meaningless word in Arabic, merely a transliteration of the Sanskrit “jya”.

Centuries later European mathematics who had no knowledge of the Sanskrit origin, assumed “jb” to be the abbreviation of “jaib” which is good Arabic for “cove”, “bulge”, “bay”, or “bosom”. Consequently, the Arabic to Latin translators used the equivalent term, “sinus”, which eventually became our “sine”

The Arabs used the cosine function and calculated the values of the function using a formula equivalent to sin 2 A + cos 2 A =1

The first table of tangents and cotangents was constructed by the Persian astronomer Ahmad ibn Abdallah Habash al-Hasib al-Marwazi Arab astronomers introduced the tangent and cotangent ratios, defining them as the length of line segments associated with the circle. These two ratios can be found in the work of al-Battânî.(c ). It was through his work that the sine came to Europe.

Abû’l-Wefâ introduced the secant and cosecant as lengths in a work on astronomy.

Medieval European scholars translated Arab and Greek works on trigonometry, but added nothing new of their own until the Renaissance. Johannes Müller von Königsberg, known as Regiomontanus ( ) was the author of the first book entirely devoted to trigonometry. (De Triangulis omnimodus, On Triangles of Every Kind, written in 1464, printed in 1533). It influenced the Prussian mathematician Rheticus to discard the dependence of trigonometric funtions on the circle and make it all about triangles.

“Although made up of Greek phonemes, the term “trigonometry” is actually not a native Greek word.” Bartholomaeus Pitiscus (German mathematician and astronomer) was the first to use the word. He wrote Trigonometria sive de solutione triangularum tractatus brevis et perspicius published in 1595, then revised in 1600 under the title Trigonometria sive de dimensione triangulae.