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Trigonometry
History of Trigonometry
Principles of Trigonometry
Introduction
TRIGONOMETRY
•Trigonometry is the branch of
mathematics concerned with specific
functions of angles and their application
to calculations.
•There are six functions of an angle
commonly used in trigonometry. Their
names and abbreviations are sine (sin),
cosine (cos), tangent (tan), cotangent
(cot), secant (sec), and cosecant (csc).
Introduction
TRIGONOMETRY
• Six trigonometric functions in relation
to a right triangle.
Introduction
TRIGONOMETRY
• Based on the definitions, various simple
relationships exist among the functions.
For example, csc A = 1/sin A, sec A =
1/cos A, cot A = 1/tan A, and tan A = sin
A/cos A.
Introduction
TRIGONOMETRY
• Trigonometric functions are used in
obtaining unknown angles and
distances from known or measured
angles in geometric figures.
• Trigonometry developed from a need
to compute angles and distances in
such fields as astronomy, map making,
surveying, and artillery range finding.
Introduction
TRIGONOMETRY
• Problems involving angles and
distances in one plane are covered in
plane trigonometry.
• Applications to similar problems in
more than one plane of threedimensional space are considered in
spherical trigonometry.
History of Trigonometry
Classical Trigonometry
Modern Trigonometry
HISTORY OF TRIGONOMETRY
Classical Trigonometry
• The word trigonometry comes from the
Greek words trigonon (“triangle”) and
metron (“to measure”).
• Until about the 16th century,
trigonometry was chiefly concerned
with computing the numerical values
of the missing parts of a triangle when
the values of other parts were given.
HISTORY OF TRIGONOMETRY
Classical Trigonometry
• For example, if the lengths of two sides
of a triangle and the measure of the
enclosed angle are known, the third
side and the two remaining angles can
be calculated.
• Such calculations distinguish
trigonometry from geometry, which
mainly investigates qualitative
relations.
HISTORY OF TRIGONOMETRY
Classical Trigonometry
• Of course, this distinction is not always
absolute: the Pythagorean theorem, for
example, is a statement about the
lengths of the three sides in a right
triangle and is thus quantitative in
nature.
HISTORY OF TRIGONOMETRY
Classical Trigonometry
• Still, in its original form, trigonometry
was by and large an offspring of
geometry; it was not until the 16th
century that the two became separate
branches of mathematics.
HISTORY OF TRIGONOMETRY
Classical Trigonometry
• Ancient Egypt and the
Mediterranean world
• India and the Islamic world
• Passage to Europe
HISTORY OF TRIGONOMETRY
CLASSICAL TRIGONOMETRY
Ancient Egypt and the Mediterranean world
• Several ancient civilizations—in
particular, the Egyptian, Babylonian,
Hindu, and Chinese—possessed a
considerable knowledge of practical
geometry, including some concepts that
were a prelude to trigonometry.
HISTORY OF TRIGONOMETRY
CLASSICAL TRIGONOMETRY
Ancient Egypt and the Mediterranean world
• The Rhind papyrus, an Egyptian
collection of 84 problems in arithmetic,
algebra, and geometry dating from
about 1800 BC, contains five problems
dealing with the seked.
HISTORY OF TRIGONOMETRY
CLASSICAL TRIGONOMETRY
Ancient Egypt and the Mediterranean world
• For example, problem 56 asks: “If a
pyramid is 250 cubits high and the side
of its base is 360 cubits long, what is its
seked?” The solution is given as 51/25
palms per cubit; and since one cubit
equals 7 palms, this fraction is
equivalent to the pure ratio 18/25.
HISTORY OF TRIGONOMETRY
CLASSICAL TRIGONOMETRY
Ancient Egypt and the Mediterranean world
• This is actually the “run-to-rise” ratio of
the pyramid in question—in effect, the
cotangent of the angle between the
base and face (see the figure).
• It shows that the Egyptians had at least
some knowledge of the numerical
relations in a triangle, a kind of “prototrigonometry.”
HISTORY OF TRIGONOMETRY
CLASSICAL TRIGONOMETRY
Ancient Egypt and the Mediterranean world
• Trigonometry in the modern sense
began with the Greeks.
• Hipparchus (c. 190–120 BC) was the
first to construct a table of values for a
trigonometric function.
HISTORY OF TRIGONOMETRY
CLASSICAL TRIGONOMETRY
Ancient Egypt and the Mediterranean world
• He considered every triangle—planar
or spherical—as being inscribed in a
circle, so that each side becomes a
chord (that is, a straight line that
connects two points on a curve or
surface.
• as shown by the inscribed triangle ABC
in the figure).
HISTORY OF TRIGONOMETRY
CLASSICAL TRIGONOMETRY
chord: inscribed triangle
• This figure illustrates the
relationship between a
central angle θ (an angle
formed by two radii in a
circle) and its chord AB
(equal to one side of an
inscribed triangle) .
Triangle inscribed
in a circle
HISTORY OF TRIGONOMETRY
CLASSICAL TRIGONOMETRY
Ancient Egypt and the Mediterranean world
• To compute the various parts of the
triangle, one has to find the length of
each chord as a function of the central
angle that subtends it—or, equivalently,
the length of a chord as a function of the
corresponding arc width.
• This became the chief task of
trigonometry for the next several
centuries.
HISTORY OF TRIGONOMETRY
CLASSICAL TRIGONOMETRY
Ancient Egypt and the Mediterranean world
• As an astronomer, Hipparchus was
mainly interested in spherical triangles,
such as the imaginary triangle formed
by three stars on the celestial sphere,
but he was also familiar with the basic
formulas of plane trigonometry.
HISTORY OF TRIGONOMETRY
CLASSICAL TRIGONOMETRY
Ancient Egypt and the Mediterranean world
• In Hipparchus's time these formulas
were expressed in purely geometric
terms as relations between the various
chords and the angles (or arcs) that
subtend them; the modern symbols for
the trigonometric functions were not
introduced until the 17th century.
• (See the table of common
trigonometry formulas.)
HISTORY OF TRIGONOMETRY
CLASSICAL TRIGONOMETRY
Ancient Egypt and the Mediterranean world
• The first major ancient work on
trigonometry to reach Europe intact
after the Dark Ages was the Almagest
by Ptolemy (c. AD 100–170).
• He lived in Alexandria, the intellectual
centre of the Hellenistic world, but little
else is known about him.
HISTORY OF TRIGONOMETRY
CLASSICAL TRIGONOMETRY
Ancient Egypt and the Mediterranean world
• Although Ptolemy wrote works on
mathematics, geography, and optics,
he is chiefly known for the Almagest, a
13-book compendium on astronomy
that became the basis for mankind's
world picture until the heliocentric
system of Nicolaus Copernicus began to
supplant Ptolemy's geocentric system in
the mid-16th century.
HISTORY OF TRIGONOMETRY
CLASSICAL TRIGONOMETRY
Ancient Egypt and the Mediterranean world
• In order to develop this world picture—
the essence of which was a stationary
Earth around which the Sun, Moon,
and the five known planets move in
circular orbits—Ptolemy had to use
some elementary trigonometry.
HISTORY OF TRIGONOMETRY
CLASSICAL TRIGONOMETRY
Ancient Egypt and the Mediterranean world
• Chapters 10 and 11 of the first book of
the Almagest deal with the
construction of a table of chords, in
which the length of a chord in a circle is
given as a function of the central angle
that subtends it, for angles ranging
from 0° to 180° at intervals of one-half
degree.
HISTORY OF TRIGONOMETRY
CLASSICAL TRIGONOMETRY
Ancient Egypt and the Mediterranean world
••Constructing
a table a table of sines, which
This is essentially
ofcan
chords
be seen by denoting the radius r,
• c = 2r sin (A/2).
the
arc
A
,
and
the
length
of
the
• Hence, a table of
subtended
chord
values
for chords
in a c (see the figure), to
obtain
c = 2radius
r sin A/2.
circle
of fixed
is also a table of
values for the sine of
angles (by doubling
the arc).
HISTORY OF TRIGONOMETRY
CLASSICAL TRIGONOMETRY
Ancient Egypt and the Mediterranean world
• Because Ptolemy used the Babylonian
sexagesimal numerals and numeral
systems (base 60), he did his
computations with a standard circle of
radius r = 60 units, so that
c = 120 sin A/2.
HISTORY OF TRIGONOMETRY
CLASSICAL TRIGONOMETRY
Ancient Egypt and the Mediterranean world
• Thus, apart from the proportionality
factor 120, his was a table of values of
sin A/2 and therefore (by doubling the
arc) of sin A.
• With the help of his table Ptolemy
improved on existing geodetic
measures of the world and refined
Hipparchus' model of the motions of
the heavenly bodies.
HISTORY OF TRIGONOMETRY
Classical Trigonometry
• Ancient Egypt and the
Mediterranean world
• India and the Islamic world
• Passage to Europe
HISTORY OF TRIGONOMETRY
CLASSICAL TRIGONOMETRY
Indian and Islamic World
• The next major contribution to
trigonometry came from India.
• In the sexagesimal system,
multiplication or division by 120 (twice
60) is analogous to multiplication or
division by 20 (twice 10) in the decimal
system.
HISTORY OF TRIGONOMETRY
CLASSICAL TRIGONOMETRY
Indian and Islamic World
• Thus, rewriting Ptolemy's formula as
c/
A/ , the relation
=
sin
B
,
where
B
=
120
2
expresses the half-chord as a function
of the arc B that subtends it—precisely
the modern sine function.
• The first table of sines is found in the
Āryabhaṭīya.
HISTORY OF TRIGONOMETRY
CLASSICAL TRIGONOMETRY
Indian and Islamic World
• Its author, Āryabhaṭa I (c. 475–550),
used the word ardha-jya for half-
chord, which he sometimes turned
around to jya-ardha (“chord-half”); in
due time he shortened it to jya or jiva.
• Later, when Muslim scholars translated
this work into Arabic, they retained the
word jiva without translating its
meaning.
HISTORY OF TRIGONOMETRY
CLASSICAL TRIGONOMETRY
Indian and Islamic World
• In Semitic languages words consist
mostly of consonants, the pronunciation
of the missing vowels being understood
by common usage.
• Thus jiva could also be pronounced as
jiba or jaib, and this last word in Arabic
means “fold” or “bay.”
HISTORY OF TRIGONOMETRY
CLASSICAL TRIGONOMETRY
Indian and Islamic World
• When the Arab translation was later
translated into Latin, jaib became
sinus, the Latin word for bay.
• The word sinus first appeared in the
writings of Gherardo of Cremona (c.
1114–87), who translated many of the
Greek texts, including the Almagest,
into Latin.
HISTORY OF TRIGONOMETRY
CLASSICAL TRIGONOMETRY
Indian and Islamic World
• Other writers followed, and soon the
word sinus, or sine, was used in the
mathematical literature throughout
Europe.
• The abbreviated symbol sin was first
used in 1624 by Edmund Gunter, an
English minister and instrument maker.
• The notations for the five remaining
trigonometric functions were introduced
shortly thereafter.
HISTORY OF TRIGONOMETRY
CLASSICAL TRIGONOMETRY
Indian and Islamic World
• During the Middle Ages, while Europe
was plunged into darkness, the torch of
learning was kept alive by Arab and
Jewish scholars living in Spain,
Mesopotamia, and Persia.
• The first table of tangents and
cotangents was constructed around
860 by Ḥabash al-Ḥāsib (“the
Calculator”), who wrote on astronomy
and astronomical instruments.
HISTORY OF TRIGONOMETRY
CLASSICAL TRIGONOMETRY
Indian and Islamic World
• Another Arab astronomer, al-Bāttāni (c.
858–929), gave a rule for finding the
elevation θ of the Sun above the horizon
in terms of the length s of the shadow
cast by a vertical gnomon of height h.
• Al-Bāttāni's rule, s = h sin (90° − θ)/sin θ,
is equivalent to the formula s = h cot θ.
HISTORY OF TRIGONOMETRY
CLASSICAL TRIGONOMETRY
Indian and Islamic World
• Based on this rule he constructed a
“table of shadows”—essentially a table
of cotangents—for each degree from 1°
to 90°.
• It was through al-Bāttāni's work that
the Hindu half-chord function—
equivalent to the modern sine—
became known in Europe.
HISTORY OF TRIGONOMETRY
Classical Trigonometry
• Ancient Egypt and the
Mediterranean world
• India and the Islamic world
• Passage to Europe
HISTORY OF TRIGONOMETRY
CLASSICAL TRIGONOMETRY
Passage to Europe
• Until the 16th century it was chiefly
spherical trigonometry that interested
scholars—a consequence of the
predominance of astronomy among
the natural sciences.
HISTORY OF TRIGONOMETRY
CLASSICAL TRIGONOMETRY
Passage to Europe
• The first definition of a spherical
triangle is contained in Book 1 of the
Sphaerica, a three-book treatise by
Menelaus of Alexandria (c. AD 100) in
which Menelaus developed the
spherical equivalents of Euclid's
propositions for planar triangles.
HISTORY OF TRIGONOMETRY
CLASSICAL TRIGONOMETRY
Passage to Europe
• A spherical triangle was understood to
mean a figure formed on the surface of
a sphere by three arcs of great circles,
that is, circles whose centres coincide
with the centre of the sphere (as shown
in the animation).
HISTORY OF TRIGONOMETRY
CLASSICAL TRIGONOMETRY
Passage to Europe
• There are several fundamental
differences between planar and
spherical triangles; for example, two
spherical triangles whose angles are
equal in pairs are congruent (identical
in size as well as in shape), whereas
they are only similar (identical in
shape) for the planar case.
HISTORY OF TRIGONOMETRY
CLASSICAL TRIGONOMETRY
Passage to Europe
• Also, the sum of the angles of a
spherical triangle is always greater than
180°, in contrast to the planar case
where the angles always sum to exactly
180°.
HISTORY OF TRIGONOMETRY
CLASSICAL TRIGONOMETRY
Passage to Europe
• Several Arab scholars, notably Naṣīr alDīn al-Ṭūsī (1201–74) and al-Bāttāni,
continued to develop spherical
trigonometry and brought it to its
present form.
• Ṭūsī was the first (c. 1250) to write a
work on trigonometry independently of
astronomy.
HISTORY OF TRIGONOMETRY
CLASSICAL TRIGONOMETRY
Passage to Europe
• But the first modern book devoted
entirely to trigonometry appeared in
the Bavarian city of Nürnberg in 1533
under the title On Triangles of Every
Kind.
• Its author was the astronomer
Regiomontanus (1436–76).
HISTORY OF TRIGONOMETRY
CLASSICAL TRIGONOMETRY
Passage to Europe
• On Triangles contains all the theorems
needed to solve triangles, planar or
spherical—although these theorems are
expressed in verbal form, as symbolic
algebra had yet to be invented.
• In particular, the law of sines (see the
table) is stated in essentially the
modern way.
HISTORY OF TRIGONOMETRY
CLASSICAL TRIGONOMETRY
Passage to Europe
• On Triangles was greatly admired by
future generations of scientists; the
astronomer Nicolaus Copernicus (1473–
1543) studied it thoroughly, and his
annotated copy survives.
• The final major development in
classical trigonometry was the invention
of logarithms by the Scottish
mathematician John Napier in 1614.
HISTORY OF TRIGONOMETRY
CLASSICAL TRIGONOMETRY
Passage to Europe
• His tables of logarithms greatly
facilitated the art of numerical
computation—including the
compilation of trigonometry tables—
and were hailed as one of the greatest
contributions to science.
PREPARED BY :DEEPANSHU SINGHAL