Sine and Cosine

1. Sine and Cosine Eric Wishnie Seth Coldsmith

2. What is sine? • Without a calculator, determine the sine of the central of the given circle with radius 2.5 inches.

3. The Greeks • Hipparchus of Rhodes • Modeling the night sky • Chords over angles • Circle, radius 3438, table of chords • Claudius Ptolmey • Almagest, proved basic theorems of chords • Spherical triangles • New table of chords

4. The Indians • Half chords • Half the chord of twice the angle = sine of angle • Table of half-chords • Half-chord = Rsin(a), R = radius, a = angle

5. Arab to European • Additions to Indian mathematics • Expansion on spherical triangles • “Shadow” function ≈ tangent • Mistranslation to European • Arab words written without vowels • The word jiba which was written jb was misread as jaib. • Jaib means a “cove” or “bay” so this translated to Sinus

6. Sinus • This word was obviously created by a man. • Why???? • Originally meant bosom. • Later came to mean the fold of a garment at the bosom (cleavage). • Also was applied to a cove or bay that took the shape of the bosom. (If you don’t believe us, take a closer look at the picture on page 149 of the book.) • Latin root of our word Sinuous from which sine originated

7. Regiomontanus • Real name was Johannes Muller • Wrote On All Sorts of Triangles in 1463 but wasn’t published until decades later. • Only used Sine • Used it as the length of a line segment and not as a ratio as we know today • Computed a large table of sines to a circle with a radius of 60,000 (known as the “total sine”) from which these calculations were based upon.

8. Regiomontanus con’t • Occasionally needed to use the sine of the complementary angle (cosine) • Wasn’t originally recognized as cosine in the form of its own name, quantity, or ratio • Went from sinus complimenti to co. sinus to cosine in a century • Could be considered the Father of Trigonometry

9. Other Trig Contributions • Joachim Rheticus (1514 – 1574 A.D. ) shows how to define sine and cosine using a right triangle without referencing a circle, • Thomas Fincke (1561 – 1656 A.D.) invented the words tangent and secant • Bartholomeo Pitiscus (1561 – 1613 A.D) invented the word “trigonometry” as part of the title to his book. • Gilles de Roberval (1602 – 1675 A.D.) sketches the sine curve while trying to find the area of a cycloid.

10. Leonhard Euler • Euler thinks of Sine as a function. • Invented calculus and more specifically the use of functions. • Shows sine to be a function of the measure of the arc of an angle in the unit circle measured in radians. • Makes the sine curve make sense.

11. Timeline • 190 – 120 B.C. Hipparchus of Rhodes wants to model the night sky and uses chords, angles, a circle of radius 3438 to create his table of chords. • 85 – 165 A.D. Claudius Ptolemy writes Almagest, proved basic theorems of chords, used these to create spherical triangles, and created a new table of chords. • 400 – 499 A.D. First written evidence in India of the use of half-chords. • 6th Century A.D. (Aryabhata) through the 12th Century A.D. (Bhaskara) there are more and more sophisitcated methods developed to approximate the length of half-chords • 12th Century A.D. through the 15th Century A.D. the Europeans get their hands on the Indian manuscripts that describe the calculations of half chords and after translating them they further the development of the calculations of half-chords.

12. Timeline Con’t • 1463 A.D. Johannes Muller (Regiomontanus) writes On All Sorts of Triangles uses sine and cosine even though he does not define them as we do today. (Father of Trigonometry) • 15th Century A.D. trigonometry finally becomes an object of interest outside of astrology. • 1514 – 1574 A.D. Joachim Rheticus defines sine and cosine in term of right triangles. • 1561 – 1656 A.D. Thomas Fincke invented the words tangent and secant. • 1561 – 1613 A.D. Bartholomeo Pitiscus invented the word trigonometry. • 1602 – 1675 A.D. Gilles de Roberval sketched a sine curve while computing the area of a cycloid. • 17th Century A.D. Trigonometry is used to solve certain types of algebraic problems. • 1707 – 1783 A.D. Leonhard Euler introduced sine as a function of the unit circle.

13. References • Berlinghoff, William. Gouvea, Fernando. A Gentle History for Teachers and Others. Farmington, Maine Oxton House Publishers 2002. • Katz, Victor J. A History of Mathematics: Brief Edition. New York, New York. Pearson/Wesley 2004. • NCTM 31st Yearbook. Historical Topics for the Mathematics Classroom: The History of Trigonometry. Washington D.C. NCTM 1969.