| The branch of
mathematics that is related to the study of the
triangle is called Trigonometry. A triangle is
a close region that is constructed with the help
of three straight lines that finally form its
structure. Trigonometry is associated with the
study of the relationships that are found between
the angles and the sides of the triangle.
Trigonometry has got the pride to be one of the
most ancient subjects that were extremely famous
all over the world and scholars from all over
the world studied those ancient subjects. (Brown,
1990)
The invention of trigonometry is associated with
the geometric school of Alexandria that was well-known
for the studies of astronomy. Quantitative geometry
is necessary for quantitative astronomy. Quantitative
geometry is the geometry of globe. The trigonometry
was invented by the astronomers. Those astronomers
had applied the trigonometry to the study of sky.
Hypparchus of Rhodes (II century. B.C.) is called
the founder of trigonometry although spherical
geometry had been discussed by Eudoxus of Cnidos
and Euclid from Alexandria. Theodosius of Tripoli
and Menelaus of Alexandria had contributed fundamentally
towards spherical trigonometry. Both of them had
written under the title ‘Sphaerica’.
But Ptolemy Claude is known to provide the biggest
information about trignometrical methods from
Alexandria. (Foerster, 1990)
The oldest known work on trigonometric tables
is the Syntaxis Mathematica written by Ptolemy
of Alexandria about 140 AD. We know almost nothing
about Ptolemy himself, but his work (of thirteen
books) surpassed all previous similar works and
became known as the greatest, "almagest,"
which we call it to this day. In the Almagest,
Ptolemy describes (and proves) a method for deriving
a table of chords subtended by arcs of 1 degree
to 180 degrees. This approximates today's table
of sines (and cosines). Ptolemy extended the work
of Hipparchus (died 125 B.C.) and followed his
method. (Brown, 1990)He started with the circle
divided into 360 parts and a diameter of 120 units.
Using basic constructions for the sides of regular
pentagon, hexagon, and decagon, he computed the
chords for arcs of 36 degrees, 60 degrees, and
72 degrees. He also computed chords of supplement
arcs, proving the theorem which bears his name
and which allows him to compute chords of difference
of arcs. He computed chords of half arcs and eventually
found the chord for 1/2 degrees; from this one
can compute his table. In this process, Ptolemy
uses several of the relationships we will develop
below involving chords. (The sine function, as
we know it, was devised in the sixth century by
Arabs). (Toomer, 1973)
Alexandrian trigonometry uses chords while modern
trigonometry uses sine. The semi circumference
was divided into degrees, 180 equal parts in the
Babylonian tradition and the diameter was divided
into 120 equal parts. As a result, goniometer
was obtained. Goniometer consists of two parts,
the round part and the flat part. Arcs are measured
with the help of the round part and the relative
cords are measured with the help of the flat part.
The arc is measured in degrees, which is a unit
that measures the cicumfeference 360. the cord
is measured in units so that the radius measures
60. (Foerster, 1990)
The Theorem of Ptolemy
According to the theorem of Ptolemy, if a quadrilateral
is inscribed in a circle, the sum of the products
of the two opposed sides is equal to the product
of the multiplication of the diagonals.
According to the quadrilateral PQRS, we have the
following formulae.
PR x QS = PQ x RS + PS x QR.
The diameter of a circle is PS. Tomoleo has used
this theorem for such a case.
If we suppose that a = arc PQ and ß = arc
PR, then
Arc QR = ß – a arc QS = 180 –
a and arc RS = 180 – ß
So it is followed from the theorem of Ptolemy
that
c (ß) c (180-a) = c (a) c (180 - ß)
+ 120 c (ß – a). …………..
(A)
The triangles ABD and ACD are inscribed in a circumference
and they are right-angled triangles. When we apply
the Pythagoras theorem, we get,
barPR2 + barRS2 = bar PS2 = 1202
barPQ2 + barQS2 = bar PS2 = 1202
c2 (180-ß) = 1202 – c2 (ß)
c2 (180-a) = 1202 – c2 (a).
The chord of ß – a can be obtained
as all the terms given in the expression (A) are
known, so
C (ß – a) = c (ß) c (180 –
a) – c (a) c (180-ß) ……………
(B)
12
if c (a) = 120 sin a / 2
the expression (B) can be written as
120sin ß –a = 120 sin ß /
2 x 120 sin180-a / 2 – 120 sin a / 2 x 120
sin180 – ß / 2
2 120
By simplifying,
sin ß – a = sin ß / 2 sin (90-a
/ 2) – sin a / 2 sin (90 – ß
/ 2)
2
As sin (90 – a / 2) = cos a / 2, the above
expression can be more simplified into:
sin ß – a = sin ß / 2 cos a
/ 2 – sin a / 2 cos ß / 2.
2
By putting ß = 2 a in expression (A), we
can get the formula of the bisection
c (2a) c (180 – a) = c (a) c (180 –
2 a) + 120 c (2a – a)
= c (a) c (180-2a) + 120 c (a)
= c (a) (120 + c (180 – 2 a)).
If we consider that c2 (180-a) = 1202 –
c2 (a) and raise both the terms to the second
power, we get
C2 (2a) (1202 – c2 (a)) = c2 (a) (120 +
c (180 – 2a)) 2
And solving
C2 (a) = 1202 c2 (2a) / [120 + c (180 –
2 a)] 2 + c2 (2a)
This is formula is already a formula of bisection
but a better formula can be obtained by evolving
the denominator:
(120 + c (180 – 2 a)) 2 + c2 (2 a) = 1202
+ c2 (180 – 2 a) + 2 x 120 x c (180 –
2 a) + c2 (2 a)
= 1202 + 120 2 - c2 (2 a) + 2 x 120 x c (180 –
2 a) – c2 (2 a)
= 2 x 120 (120 + c (180 – 2 a))
and it can be concluded
c2 (a) = 60 c2 (2 a) / 120 + c (180 – 2
a) ……………(C)
With the help of expression (C), the chords that
correspond to the smaller and smaller angels can
be calculated. (Toomer, 1996)
Development of Trigonometry by Arabs and Indians
Mathematical sciences continued to develop during
the Roman period although Romans did not play
any role in the development of mathematical sciences
but they did not hinder its progress. The Arabs
were the natural successors of the Greek geometers.
The Arabs had faced different traditions and they
had assimilated most of them very quickly. The
Arabs were standing on such cross roads that had
a variety of mathematical traditions. At one side,
the Babylonian and Egyptian cultures were merging
with the classic Greek geometry and on the other
side; they were facing the innovations of Indian
mathematicians. The Arab influence encouraged
some fundamental discoveries that include both
on paper and technological to reach the west because
those fundamental discoveries were very crucial
in the development of science and in the diffusion
of culture. Those discoveries include both positional
and scientific notations such as the use of numeric
characters that were called Arab. (Blitzer, 2003)
India is responsible for the innovation that
was related to Alexandrine trigonometry. That
innovation was the use of sine instead of the
use of chord. Work was done on the implementation
of sine and the first table of sine was developed
that was known as Surya Siddhanta. This table
was developed around IV or V centuries ago. That
table is very important because it contains that
calculus of sine of the multiple of 3o 45’,
until 90o. Indian astronomers are also responsible
for the addition of the cosine to the sine, the
cotangent and the tangent. During the eighth century
A.D., the translation of the sine table into Arab
took place. The Arab astronomers were very genius
and they put their efforts in the field of circular
functions and then they realized that those circular
functions need some changes as well as improvements.
(Foerster, 1990)
The sine of the complementary arc was known as
cosine: cos a = sin (900 – a)
As the cosine were to find directly in the tables
of sine so there was not any need of developing
the table of cosine. Gnomonics is called the science
of sundials and the cotangent and the tangent
were related to this science. The hypotenuse of
the triangles that contain gnomon and its shades
represented the cosecant and the secant. So it
can be said that the construction vertical and
horizontal sundials is connected with the cotangent
(and cosecant) and tangent (and secant)
respectively. Among tangent, cotangent, secant
and cosecant, only the table of tangent has been
formed because it was realized that the cotangent
is complementary tangent just the same as the
cosine. The original term for tangent was zill,
that is umbra recta in Latin and the original
term for cotangent was zill makus, which is umbra
versa in Latin. T. Fink (1561-1656) is responsible
for the introduction of the term tangent in 1583.
While E. Gunter (1581-1626) introduced the term
cotangent in 1620. After the introduction of those
functions, it was realized that the tables of
those functions should be prepared while it was
also felt that the already existing tables require
some improvement. Initially Arab mathematicians
and then the Europeans put their efforts in the
formation of the tables as well as in the improvement
of the older one. (Blitzer, 2003)
Development of Trigonometry in Europe
Arabs are responsible to bring the trigonometry
into the West. No significant contributions in
the field of trigonometry were found before the
fifteenth century. In the fifteenth century, attention
was again given to trigonometrical studies as
a requirement for astronomy. Tables that are more
specific in two directions are required for a
higher precision of instruments. Among those two
directions of tables are sines that have a bigger
number of decimals and angles that have smaller
intervals. George Peurbach (1423-1461) put his
attention towards the interval between the arcs.
He calculated the table of sine that contained
the intervals of 10’. Johann Muller (1436-1476)
work harder on the same field and composed such
a table in which the intervals were only of one
prime. There was a significant increase in exactitude.
(Foerster, 1990)This one was given from the dimension
of the radius of the goniometric circle. Today
this is given by the number of the decimal figures.
The values of R sin a were reported in integer
numbers, that had the range from 0 to 10000 that
could correspond to four decimal figures, when
the radius was taken as R = 10000. That radius
was also called toto sine. Then the value of radius
was taken as R = 600000 in the table composed
by Peurbach. Then the value of radius was taken
first as R = 6000000 and then R = 10000000 that
could correspond to seven decimals in the table
composed by Regiomontanus. That time is considered
as the first time when the base 10 was definitely
adopted and the liberation had taken place from
the use of sexagesimal system of sine. (Blitzer,
2003)
Typography is also responsible for the development
of trigonometry. Typography is totally based on
rectilinear trigonometry and this feature of typography
is totally different from astronomy. Efforts are
done for the study of triangles and their solutions
because they are required in the topographical
survey. Regiomontanus wrote the first treatise
of trigonometry in 1464 that is called De trianglulis
omnimodis. Nicolaus Copernicus was the one who
included De revolutionibus orbium caelestium in
his work. G. J. Rheticus then shed light on Copernicus’s
work in his work. Rheticus is known for the preparation
of a monumental series of tables that contain
the six circular functions. Those functions possess
the intervals of 10” and they work for a
radius whose value is R = 10000000. (Blitzer,
2003)
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