Trigonometry
Ryan
Mr. de Groof
Grade 10 Composition
01/10/13
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Have you ever thought about why we were learning “trigonometry” in middle
and high school? People may think that it isn’t useful at all, and not everyone is going
to be an engineer. In fact, “trigonometry is helpful for developing students’ knowledge
of both pure and applied mathematics. In college, a higher level of trigonometry is
required for several different career majors” (McLynn). It is not simply used to build a
pyramid now. Today it is applied to astronomy, geography, navigation, construction,
engineering, and several other fields. From ancient Egypt to modern life, people
improved their knowledge of trigonometry. The concept of trigonometry has existed
throughout history and its applications can still be seen today. In our daily lives, it
became an essential knowledge.
The word “trigonometry” is derived from Greek language, which is split to three
parts. “Tri” meaning “three”, “gon” meaning “sides”, and “metron” meaning
“measure”. In other words, trigonometry is a category of mathematics that tells us
how the angles and sides of a triangle are related (James). The calculations based on
the angles of triangles and the solid figures derived from it. Trigonometry was first
developed by Hipparchus. Before him, basic studies of triangles can be traced back to
the Egyptians and the Babylonians. In addition, there were also multiple nations using
trigonometry for a long period. Pythagoras invented mathematical functions, which
made a big impact in the history of trigonometry. Pursing this further, some famous
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mathematicians developed the trigonometry functions that we are recently using today.
Though it's not an interesting subject, trigonometry is used in several career fields.
The Greek mathematician Hipparchus was the first person whose systematic use
of trigonometry who we have documentary evidence. He developed the first
trigonometric table, which is known as a table of chords for angles in a circle of large
fixed radius (“Hipparchus”). He also used trigonometry to develop lunar and solar
theories and to study the motion of the sun and moon (McLynn). He established the
solar theory first, purely from studies of solstices and equinoxes. Next he worked out
the lunar theory using the solar theory, then the star catalogue and theory of
precession using the solar and lunar theories. Finally, he discovered the planetary
theory by using all the previous work (Jones, pp. 445). Hipparchus measured the
distance from the Earth to the Moon during a total solar eclipse at Syene and a partial
eclipse at Alexandria (“Hipparchus (190 - 120 B.C.)”).
Before Hipparchus, the study of triangles can be traced back to the Egyptians and
Babylonians. The Egyptians divided the 360 degrees of the ecliptic into 36 sections of
10 degrees each. Each section is called a decan. And each decan contained a
constellation of stars lined up along the ecliptic. Furthermore, Stars in a decan will
rise above the horizon about every 40 minutes because the rotation of 24 hours is
divided into 36 sections. This system was used to determine the night hours and the
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seasons (Katz).
The Babylonians were famous for their examinations of celestial bodies. They
developed the division of the circle into 360 degrees by 1800 B.C.E. And about 500
B.C.E, Babylonian astronomers divided the heaven into twelve regions of 30 degrees
each, called the 12 Houses of the Zodiac (Katz). The houses were discovered when
the Babylonians started to survey the natural phenomena associated with rising and
setting of the heavenly bodies (“Twelve”). Additionally, the zodiac is divided evenly
into four quadrants. Ancestors believed that each quadrant has a particular meaning.
From nature to society, it represented a specific area of life (“12”).
Besides the Egyptians and Babylonians, there were also multiple nations using
trigonometry for hundreds of years. For example, trigonometry was developed from
the work of Hipparchus in India. Indian mathematicians such as Bhaskara, Madhava,
and Jyesthadeva improved the functions. Furthermore, the formulas were
rediscovered in Europe as part of the development of calculus in the 17th century.
About the Chinese trigonometry, a Chinese mathematician called Liu Hui wrote a
collection of nine problems to compute lengths indirectly. Another expert called Yi
Xing constructed a table for the shadow length of a gnomon at various latitudes and
times; it seems to have been adopted from the Indian trigonometric functions. In the
field of Muslim trigonometry, it was brought to Baghdad from the Indians, which was
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based on Hipparchus. Therefore, Muslims were introduced the trigonometric
functions and the Pythagorean identities (Kochman).
The application of trigonometry is related by the work begun by Pythagoras.
According to the legend, when one day Pythagoras walked passes a blacksmith, he
heard beautiful and harmonious pounding sounds made by a hammer. He then decided
to discover how the mathematical equations were applied to music (Christoph).
Meanwhile, he claimed that “the sounds made by plucking two strings of different
lengths are consonant if both lengths are small integer multiples of a common length”
(“Uses”).
The trigonometry functions that are recently used today were developed by Sir
Isaac Newton and James Stirling. About three centuries ago, Isaac Newton invented
an interpolation formula for equal-interval data. The word “interpolation” is
introduced as "to alter or enlarge texts by insertion of new matter". In 1675, he
finished the foundation of classical interpolation theory. In his forward interpolation
formula, it is used to find the value of a particular item in a range, and the extremes
are known. Later on in 1719, Stirling integrated Newton’s work in the Methodus
Differentialis, which is now known well as the Newton-Stirling formula (Meijering).
The functions of trigonometry are used in a variety of career fields. By using
algebraic manipulation and basic geometry, we may calculate the ratio between two
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side lengths. Moreover, we may find the angle if the sides are given. In astronomy,
trigonometry is used to find the distance between two celestial bodies. To calculate
that distance, we have to first measure its apparent brightness. Then we may be able
to determine how far away it is (“How”). In geography, it is used to find the height of
mountains. In the same way in oceanography, it is used to find the height of tides in
oceans. In navigation, it is used to find the distance of the shore from a point of the
sea. Likewise, we may locate one’s position on the globe by measuring the angle
between celestial bodies and the visible horizon in celestial navigation (“Celestial”).
In construction, engineers of various types use the fundamentals of trigonometry to
design bridges, build structures and solve scientific problems (Lingarajaprabhu).
As a result, I’d say that trigonometry is an important invention. To those who
might never approach this field, trigonometry isn’t quite serious. Textbooks usually
provide special angles for us easy to solve, but in real life we do not see examples
with those specific angles so often. Fortunately, the calculator is invented before us. It
made mathematical operations easier. Much great architecture is built and our society
and economy is relatively improving.
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References
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Meaning. N.p., n.d. Web. 24 Dec. 2012.
<http://www.astrotheme.com/astrology_houses.php>.
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<http://www2.stetson.edu/~efriedma/periodictable/html/Hs.html>.
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<http://www.astro.cornell.edu/academics/courses/astro201/hipparchus.htm>.
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23 Dec. 2012. <http://www.astrology.com.au/astrology/houses/>.
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Christoph Riedweg, Pythagoras: His Life, Teaching and Influence, Cornell: Cornell
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With Examples. 30 Jul. 2012 EzineArticles.com. 10 Dec. 2012.
<http://ezinearticles.com/?Understand-Trigonometry-With-Examples&id=7204867>.
Jones, Alexander (Sep., 1991). The Adaptation of Babylonian Methods in Greek
Numerical Astronomy. Isis, Vol. 82, No. 3, pp. 440-453.
Katz, V. (Ed.) (2007) The Mathematicians of Egypt, Mesopotamia, China, India, and
Islam. Princeton. Princeton University Press.
Kochman. Trigonometry. N.p.: Department of Mathematics and Statistics. York
University, n.d. PDF.
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Lingarajaprabhu, Malini "Applications of Trigonometry in Real Life." FASCINATING
FACTS OF MATHEMATICS. N.p., n.d. Web. 10 Dec. 2012.
<http://malini-math.blogspot.tw/2011/08/applications-of-trigonometry-in-real.html>.
Meijering, Erik. "A Chronology of Interpolation." A Chronology of Interpolation. N.p.,
n.d. Web. 24 Dec. 2012.
<http://www.imagescience.org/meijering/research/chronology/>.
McLynn, Dennis "Discovering Trigonometry." Discovering Trigonometry. 29 Aug.
2011 EzineArticles.com. 10 Dec. 2012. <http://ezinearticles.com/?Discovering-Trigonometry&id=6527237>.