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 dont 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
  wasnt 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 cont

• Occasionally needed to use the sine of the
  complementary angle (cosine)
• Wasnt 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 Cont

• 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.