Trig and Astronomy 1
The Way Trigonometry is used in Astronomy
By: Joanna Matthews
Practical Applications of Advanced Mathematics
Mrs. Amy Goodrum
July 15, 2003
Trig and Astronomy 2
Abstract
This report is about how trigonometry is used in Astronomy. Even though trigonometry is
applied in many areas, such as engineering, chemistry, surveying, and physics, it is
mainly used in astronomy Trigonometry is used to find the distance of stars, the distance
from one planet to another and from one plant to the sun. It is possible to find the radius
of the Earth also. This report will basically give more insight in the way trigonometry and
astronomy goes hand in hand.
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Background
Trigonometry comes from a Greek word "trigonometria" put together from these
3 words: Tri (three) gonia (angle) metro (measure). Trigonometry has been around for
many centuries, but in 140 BC a man named Hipparchus apparently wrote 12 books on
the table of chords and became the founder of trigonometry. He was the first Greek
mathematician to study triangular geometry. This study led him to write the 12 books.
Other people have added to Hipparchus’ work, but the two people that stands out
the most are were Menelaus (ca. AD 100) and Ptolemy (ca. AD 100). Menelaus was a
Greek mathematician that created six books on tables of chords. He created a couple of
triangle properties. Menelaus had a big hand in spherical trigonometry also. He was like
the one that was after Hipparchus’ work the most.
Ptolemy was a Greek astronomer who was highly respected in his city because of
his work. He was the first mathematician to complete the tables of chords, which were
13 books. Although his work had respect, there was controversy behind it. People said
that he stole ideas and inventions to further his work. There was no proof of these
accusations and his is still respected and appreciated.
The Muslims, Chinese, Indians, and Babylonians had their own information that
aided to trigonometry. The Muslims introduced the tangent function. The Chinese tried to
present the tangent function, but complications came up and they couldn’t complete the
advancements. The Indians made the cosine tables. Their techniques that explain the
cosine tables were introduce after the tables themselves were published. The Babylonians
helped out Menelaus and Hipparchus in their studies. They provide shelter when
Menelaus and Hipparchus did studies there and gave them ideas in tight situations.
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Research
The three main functions of trigonometry are sine, cosine, and tangent. Astronomy as
been around for basically ever, but just a couple thousand centuries ago these two studies
has been put together. And now the two is like inseparable.
Before, people thought that the
Earth was the center of the universe
instead of the Sun. This thought, or
hypothesis, was because of the
method parallax. Parallax means the
apparent shift of a nearby star
against the fixed background that
can be seen as the Earth goes around
the Sun. The shift of a star happens
in six-month intervals. This means
that if someone looks at a set of stars
in January then if that person looks again in six months, they will be looking at that same
set of stars. The thing that makes it confusing is that one star has moved to and different
place. It may look like a few inches on Earth but it can be a few million of miles in space.
Parallax is used to help astronomers find the distance of certain stars and constellations.
Also, the method of parallax helped prove the fact that the Sun is the center.
In 240 B.C., a mathematician named Eratosthenes discovered the radius of the Earth
using trigonometry and geometry. He used the Sun and two cities that were within a few
thousand miles of each other. The cities he used were Syene and Alexandria that was
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5,000 shades apart (559 feet=1 shade). Eratosthenes used a stick that was perpendicular
to the ground in Syene at noon, and the stick produced no shadow. He did the same in
Alexandria, and the stick made a shadow that showed that the Sun’s rays were one-
fiftieth (1/50th) of a regular circle. This problem can be solved by either law of sine or
parallel lines. The law of sines states that side A divided by sine of angle A equals side B
divided by sine of angle B equals side C divided by sine of angle C. Either way that it is
done your answer should be 4212.48 miles.
In 2001, a group of European astronomers did an experiment that started in 1997
about the distance of Venus from the Sun. The Sun, The Earth, and Venus makes a
triangle because the Sun is stationary and the Venus rotates opposite of the Earth. With
the Earth rotating 15 degrees
every hour, it was possible to
find the angle of Venus to the
Earth and the Sun to the Earth.
This research lasted for four
years to get accurate
measurement. By the time they
got All the measurement, they
calculate the that Venus was
about 105,000,000 kilometers away from the Sun and the Earth was about 150,000,000
kilometers away. (See Appendix)
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Without the help of historians, astronomers, and mathematicians it possible to
prove that the study of astronomy would not accurate. All of these groups of people have
an important hand in how astronomy has become today.
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References
AOL 1997-2001 Venus project. (2001). Retrieved July 16, 2003 from
http://www.amtsgym-sdbg.dk/as/venus/ven-dist.htm
Ahmed, J. (n.d.) “Trigonometry and astronomy.” Retrieved July 12, 2003 from
http://www.geocities.com/Hollywood/Academy/8245/main.html
Claudius Ptolemy. (n.d.) Retrieved July 13, 2003 from http://www-gap.dcs.st-
and.ac.uk/~history/Mathematicians/Ptolemy.html
History. (n.d.). Retrieved July 12,2003 from
http://www.uaschools.org/high%20school%20web%20site/hpdm/trig/Graphs_and
_Identity
Joyce, D. (1996). “Applications of trigonometry.” Retrieved July 12, 2003 from
http://aleph0.clarku.edu/~djoyce/java/trig/apps.html
Masters, K. (April 2002). “Curious about astronomy: how is astronomy impacted by
trigonometry?” Retreived July 12, 2003 from
http://curious.astro.cornell.edu/question.php?number=475
O’Connor, J. and Robertson, E. (June 1996) “Trigonometric functions.” Retrieved July
12, 2003 from http://www-gap.dcs.st-
and.ac.uk/~history/HistTopics/Trigonometric_functions.html
Pietruschka, P. (September 1995). “ History and uses of trigonometry.” Retrieved July
12, 2003 from http://mathforum.org/library/drmath/view/53934.html
Smoller,L. (May 2001) “The birth of trigonometry.” Retrieved July 12, 2003 from
http://www.ualr.edu/~lasmoller/trig.html
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The math forum trig/calc problem of the week archive. (n.d.) Retrieved July 12, 2003
from http://mathforum.org/calcpow/solutions/solrtion.ehtml?puzzle=20
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Appendix
Concerning angular size - 1 arcsecond = 1/3600 degree
Sun - Venus Data 2001
Angular Diameter
Day / Month Solar Data Venus Data
Jan 16 Sun 12h11m Venus 15h16m 24.1 arcseconds
Jan 23 Sun 12h13m Venus 15h14m 25.9 arcseconds
Jan 30 Sun 12h14m Venus 15h09m 28.1 arcseconds
Feb 06 Sun 12h15m Venus 15h02m 31.0 arcseconds
Feb 13 Sun 12h15m Venus 14h52m 34.2 arcseconds
Feb 20 Sun 12h14m Venus 14h39m 37.8 arcseconds
Feb 27 Sun 12h13m Venus 14h21m 42.1 arcseconds
Mar 06 Sun 12h12m Venus 13h58m 47.2 arcseconds
Mar 13 Sun 12h10m Venus 13h28m 52.2 arcseconds
Mar 20 Sun 12h08m Venus 12h51m 56.5 arcseconds
Apr 24 - Morning
Sun 11h59m Venus 09h53m 45.7 arcseconds
Object
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