Historical reflections on teacHing trigonometry photo.com K Karl Dolenc/iStoc Copyright © 2010 The National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved. This material may not be copied or distributed electronically or in any other format without written permission from NCTM. Historical reflections on teacHing trigonometry Returning to the beginnings of trigonometry— the circle—has implications for how we teach it. David m. Bressoud he study of trigonometry suffers from a Trigonometry—circle trigonometry—arose basic dichotomy that presents a serious from the study of the heavens by the classical obstacle to many students. On the one Greeks. It took more than a thousand years from hand, we have triangle trigonometry, in the initial development of trigonometry by Hip- which angles are commonly measured parchus in the early second century BCE before Tin degrees and trigonometric functions are defined triangle trigonometry was developed in earnest. as ratios of sides of a right-angled triangle. On the An emphasis on triangles rather than circles is other hand, we have circle trigonometry, in which implicit in Al-Khwarizmi’s work on shadows in angles are commonly measured in radians and trig- the early ninth century CE, but this was not fully onometric functions are expressed in terms of the developed until Al-Biruni’s Exhaustive Treatise on coordinates of a point on the unit circle centered at Shadows of 1021. Even so, applications of triangle the origin. Faced with two such distinct conceptual trigonometry and, especially, the interpretation of approaches to trigonometry, is it any wonder that trigonometric functions as ratios of sides of right- so many of our students get confused? angled triangles did not achieve prominence until Once students begin to use the sine and cosine the sixteenth century. The switch in instructional as examples of periodic functions, circle trigo- emphasis from circle trigonometry to triangle nometry dominates, but there is a tradition that trigonometry did not occur until the mid- to late- triangle trigonometry is the simpler and more basic nineteenth century. form and that students need to be grounded in this This historical overview of the development of before being introduced to circle trigonometry. In trigonometry will present an argument for begin- fact, the historical evidence points in exactly the ning the study of trigonometry with the circle opposite direction. definitions of the trigonometric functions and angle Vol. 104, no. 2 • September 2010 | mathematicS teacher 107 measurement. The historical information is based Given an arc of a circle, find the length of the on Van Brummelenarc (2009) with additional details chord that connects the endpoints of the arc from Katz (2009) and Heath (1981); additional (see fig. 1). information on the history of trigonometry and chordclassroom ideas can be found in Baumgart (1989). Perhaps the first problem that used what today we understand as trigonometry was one solved by Hip- THE FUNDAMENTAL PROBLEM parchus of Rhodes (ca. 190–120 BCE): The seasons OF TRIGONOMETRY are of unequal length. Winter, at eighty-nine days, is The fundamental problem from which trigonom- the shortest; summer, at ninety-three and five-eighths etry emerged is one that students seldom if ever see: days, is the longest. Hipparchus explained why. Hipparchus used the observed lengths of the seasons to determine the length of the arc traveled arc arc by the sun in its orbit during each season. He then found the lengths of the chords that connect the chord chord sun’s position at the ends of the seasons. These lengths enabled him to determine how far the earth is from the center of the sun’s orbit (see fig. 2). (For a full description of this problem and its solu- tion, see Hipparchus.pdf at www.nctm.org/mt.) Arc lengths were usually measured in degrees, 360° being the full circumference of the circle. The length of the chord (denoted as crd) would depend Winter on the radius, denoted by R. Some chords are easy solstice to find (see fig. 3): Fig. 1 Students in trigonometry are rarely7 asked to find the 89 /8 length of a chord intercepting an arc of a given length. crd 180° = 2R 89 days crd 90° = 12 R ≈ 1.414R days Winter crd 60° = R solstice Winter solstice Autumnal7 Euclid showed how to calculate other chord Earth 89 /8 equinox Spring days 7 lengths when he determined the lengths of the 89 89 /8 equinox days center days sides of regular inscribed pentagons and decagons 89 2 days (see fig. 4): Earth Autumnal equinox Spring 55− equinox centerEarth Autumnal equinox crd72°= RR≈ 1.176 3 Spring 2 92 /4 equinox 5 center days 93 /8 51− days crd 36°= RR≈ 0.618 3 2 92 /4 Summer 5 days 93 /8 days solstice 3 92 /4 (For Euclid’s proofs, see Euclid.pdf at www.nctm Summer 5 93 /8 dayssolstice .org/mt.) days Summer The general problem for astronomical work Fig. 2 hipparchus used chord lengthssolstice to determine the requires determining the chord length for any arc distance from the earth to the center of the sun’s orbit. length. This is almost certainly the first example in mathematics history of a functional relation- ship without an explicit formula for calculating 60° the output value for each input value. We are told R R60° R that Hipparchus constructed a table of approximate R 90° R R values. The earliest such table that still survives 12R 60° 90° R R wasR built by Ptolemy of Alexandria (90–168 CE) 12R 90° in his great work on astronomy, the Mathematical 2R 1 Treatise, better known by the Latinized version of its Arabic name, the Almagest (literally, The Great Book). Ptolemy constructed a table of chord lengths for a circle of radius 60 in half-degree increments. Ptolemy’s work includes full proofs and requires Fig. 3 the lengths of some chords are easily determined. some very impressive use of Euclidean geometry. 108 mathematicS teacher | Vol. 104, no. 2 • September 2010 72° 36° R R (For a description of Ptolemy’s work, including proofs, see Ptolemy.pdf. at www.nctm.org/mt.) 55− 51− 36° What chord lengths have to do with trigonom- R 72° R 36° 2R 72° R2 etry may not be immediately clear. If we rotate R R the circle so that the chord is vertical and insert a 55− 51− few radial lines, we can see how to translate chord R R lengths into the sine function: If the chord subtends 2 55− 2 51− R R an arc length of 2q and the radius of the circle is 2 2 R, then half the chord length is Rsinq. The chord of arc length 2q is 2Rsinq (see fig. 5), or crd(2q) = 2Rsinq. Ptolemy’s table is equivalent to a table of sines in quarter-degree increments. His calculations Fig. 4 not until euclid could geometers determine the side lengths of regular were carried out to seven-digit accuracy. pentagons and decagons. ptolemy used these measures to construct his table of E The shift to the half-chord, or sine, was made chords. See the Web materials for how the golden ratio is involved. by Indian astronomers in the third, fourth, or fifth R Rsin2q century CE. They also provided the source for our E word sine. The Sanskrit word for chord was jya, 36° and what we would refer to as the sine they named ����� R the ardha-jya, meaning “half-chord.” In time, as Rsin q astronomers adopted the practice of working only E with the half-chord, the prefix was dropped, and 36° ����� just jya or jiva came to refer to the half-chord, or sine. When Arab astronomers learned Indian trigo- R G Z nometry, they transliterated jiva as jyba, with a Rsin2q spelling equivalent to jyb because the vowel a was not written. But jyba is not an Arabic word, and 36° ����� Z the word jayb is also spelled with the same three G letters, jyb. By the time Arab trigonometric texts were translated into Latin, the pronunciation had Fig. 5 When the chord intercepts an arc of 2q, then its shifted. Jayb can mean a fold or a bay; so European length is 2Rsinq, where R is the radius of the circle. astronomers translated this word into the Latin sinus, which encompasses that meaning. From sec sinus, we get the English sine. But the term’s true cot G Z meaning is “half-chord.” cos sec tan THE DEVELOPMENT OF ANGLE csccot MEASUREMENT cos sin Today we define degrees as a fraction of a complete tan revolution, a characterization that is neither pre- csc radius cise nor particularly clear. Until the late nineteenth century, degrees were considered a measure of arc sin length: 1° equals 1/360th of the circumference of the circle. One could speak of degrees as the measure of radius an angle between two intersecting line segments, but sec one would measure the angle by centering a circle Fig. 6 Until the late 1800s, angles were measured by cen- cot at the intersection and determining the length of the tering a circle at the intersection of two line segments. cos arc between these segments (see fig. 6). The size of a degree—because it is a fixed fraction of 60ths of 60ths. Georg Rheticus (1514–74), the first tan of the circumference—depends on the radius of the European to publish tables of all six trigonometric csc circle. So does the length of the chord. Those who functions, chose a radius of 1,000,000,000,000,000 = created tables of sines would pick a radius convenient 1015.