Sine and Cosine by L74A0D8s


									Sine and Cosine
    Eric Wishnie
   Seth Coldsmith
             What is sine?
• Without a calculator, determine the sine of
  the central of the given circle with radius
  2.5 inches.
               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
                The Indians
• Half chords
  – Half the chord of twice the angle = sine of
  – Table of half-chords
  – Half-chord = Rsin(a), R = radius, a = angle
          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
• 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
• 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.
        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
      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
            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.
• 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-
• 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.
                   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
• 1561 – 1613 A.D. Bartholomeo Pitiscus invented the word
• 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.
• Berlinghoff, William. Gouvea, Fernando. A
  Gentle History for Teachers and Others.
  Farmington, Maine Oxton House Publishers
• Katz, Victor J. A History of Mathematics: Brief
  Edition. New York, New York. Pearson/Wesley
• NCTM 31st Yearbook. Historical Topics for the
  Mathematics Classroom: The History of
  Trigonometry. Washington D.C. NCTM 1969.

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