Sine and Cosine
What is sine?
• Without a calculator, determine the sine of
the central of the given circle with radius
• 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
• 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.
• Originally meant bosom.
• Later came to mean the fold of a garment at the
• 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
• 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.
• 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
• 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
• 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
• 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.
• 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
• 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.