What is Trigonometry?

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What is Trigonometry?
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• Trigonometry is the computational component of
  geometry (Three side measure)
• The first work on trigonometric functions related
  to chords of a circle. The first known tables of
  chords was produced by the Greek mathematicians,
  Hipparchus (140BC), Melanus (100AD) and Ptolemy -
  all influenced by the Babylonians using units of
  60.
• The term trigonometry first appears as the
  title of a book Trigonometria B Pitiscus
  published in 1595.

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SINE

• A sine is half of a chord. More accurately, the
  sine of an angle is half the chord of twice the
  angle.

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• The first appearance of the sine of an angle
  appears in work by the Hindu mathematician
  Aryabhata (500) tables of half chords (sine
  tables) using jya for sin. Brahmagupta (628) and
  Bhashara (1150) developed the work further.
• The Sanskrit word for chord-half was jya-ardha,
  which was sometimes shortened to jiva. This was
  brought into Arabic as jiba, and written in
  Arabic simply with two consonants jb, vowels not
  being written. Later, Latin translators selected
  the word sinus to translate jb thinking that the
  word was an arabic word jaib, which meant fold,
  and sinus had fold and bay as two of its
  meanings. In English, sinus was imported as
  "sine".

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• Sine travelled from India, through the Arabic
  language from Baghdad through Spain, into western
  Europe in the Latin language and into modern
  languages such as English.
• Edward Gunter (1624) was the first to use the
  abbreviation sin in 1624.
• The French mathematician, Hérigone, was the first
  to use sin in a book in 1624.

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COSINE

• The cosine of an angle is defined as the sine of
  the complementary angle
• (The complementary angle equals the given angle
  subtracted from a right angle, 90)
• Gunter (1620) suggested co-sinus

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TANGENT

• The tangent is the length of the vertical line
  tangent to the circle from the point of tangency
  to the point where that tangent line cuts the ray
  forming the angle.

tanx
sinx
Tangent comes form the latin tangens to touch
cosx
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• Tangents were first developed for calculating
  heights from the length of the shadow that an
  object cast.
• Thales used the lengths of shadows to calculate
  the heights of pyramids.
• The first known tables of shadows were produced
  by the Arabs around 860.
• The name tangent was first used by Thomas
  Fincke in 1583.
• The abbreviation tan was first used by Albert
  Girard in 1626 (but tan was written over the
  angle tan)
• A

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(No Transcript)
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Angle

• There are several systems of measurement of an
  angle
• The sexagesimal system - degrees
• The circular system - radians
• The centesimal system gradians
• One full rotation in these three measures
  corresponds to 360, 2p rad (or c) or 400 grad
  (or gon).

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Degree

• Babylonians used a base 60 number system. 360
  likely arises from the Babylonian year, which was
  composed of 360 days (12 months of 30 days each).
  The degree is subdivided into 60 minutes per
  degree, and 60 seconds per minute

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Radian
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• It first appeared in print on June 5, 1873, in
  examination questions set by James Thomson at
  Queen's College, Belfast. James Thomson was a
  brother of Lord Kelvin .
• James Muir, a mathematician, and James T.
  Thomson, a physicist, were working independently
  during the late nineteenth century to develop a
  new unit of angle measurement. They met and
  agreed on the name radian, a shortened form of
  the phrase radial angle. Different names were
  used for the new unit until about 1900. Today the
  term radian is in common usage."

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Why?

• 360 per 2p radians or
• 57. 29577951/radian.
• A right angle is p radians.
• 2
• Essential when using trigonometry in calculus.
• (e.g. lim sinx 1 is true only when
• x 0 x x is in radians)

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Gradian

• A unit of angular measure in which the angle of
  an entire circle is 400 gradians. A right angle
  is therefore 100 gradians. A gradian is sometimes
  also called a gon or a grade.
• Circumference of the Earth is 40 000km so 1 gon
  corresponds to 100km or 1 centigon corresponds to
  1km on the equator.
• First introduced as a German engineering unit.
• Used in navigation and surveying (infrequently in
  mathematics!!!)

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• Ideas based on materials to be found _at_
• Daves Short Trig Course
• http//aleph0.clarku.edu/djoyce/java/trig/ri
  ght.html
• www-history.mcs.st-and.ac.uk/history/HistTopics/T
  rigonometric_functions.html
• http//mathworld.wolfram.com/Angle.html