Trigonometry is the branch of mathematics concerned with specific functions of angles and their application to calculations. There are six trigonometric functions - sine, cosine, tangent, cotangent, secant, and cosecant. Trigonometry originated in ancient civilizations for practical geometry applications and was further developed by Greek mathematicians like Hipparchus and Ptolemy. Indian and later Islamic mathematicians made important contributions, including the first tables of sines and tangents. Trigonometry was an important tool for astronomy and passed to Europe during the Middle Ages, with major works by Menelaus and Regiomontanus.

1. Trigonometry

2. Introduction
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).
TRIGONOMETRY

3. Introduction
• Six trigonometric functions in relation
to a right triangle.
TRIGONOMETRY

4. Introduction
• 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.
TRIGONOMETRY

5. Introduction
• 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.
TRIGONOMETRY

6. Introduction
• Problems involving angles and
distances in one plane are covered in
plane trigonometry.
• Applications to similar problems in
more than one plane of three-
dimensional space are considered in
spherical trigonometry.
TRIGONOMETRY

7. History of Trigonometry
Classical Trigonometry
Modern Trigonometry

8. 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.

9. Classical Trigonometry
oFor 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.
oSuch calculations distinguish
trigonometry from geometry, which
mainly investigates qualitative
relations.

10. 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.

11. 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.

12. Classical Trigonometry
 Ancient Egypt and the Mediterranean world
 India and the Islamic world
 Passage to Europe
HISTORY OF TRIGONOMETRY

13. 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.

14. • 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.

15. • 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.

16. 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 “proto-trigonometry.”

17. • 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.

18. • 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).

19. 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
side of an inscribed
triangle) . Triangle inscribed
in a circle

20. • 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.

21. 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.

22. 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

23. 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.

24. • 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.

25. • 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.

26. • 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.

27. • This is essentially a table of sines,
which can be seen by denoting the
radius r, the arc A, and the length of
the subtended chord c (see the figure),
to obtain c = 2r sin A/2.
• Constructing a table
of chords
• c = 2r sin (A/2).
• Hence, a table of
values for chords in a
circle of fixed radius
is also a table of
values for the sine of
angles (by doubling
the arc).

30. 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.
CLASSICAL TRIGONOMETRY

31. Indian and Islamic World
• Thus, rewriting Ptolemy's formula as
c/120 = sin B, where B = A/2, the relation
relation 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.

32. 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
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.

33. 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
Arabic means “fold” or “bay.”

34. Indian and Islamic World

35.  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.

36.  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.

37. Indian and Islamic World
oAnother 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.
oAl-Bāttāni's rule, s = h sin (90° − θ)/sin θ,
is equivalent to the formula s = h cot θ.

38. 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.

39. Passage to Europe
• The first definition of a spherical
triangle is contained in Book 1 of the
Sphaerica, a three-book treatise by
by Menelaus of Alexandria (c. AD 100)
in which Menelaus developed the
spherical equivalents of Euclid's
propositions for planar triangles.

40. 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

41. 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.

42. 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°.

43. Passage to Europe
Several Arab scholars, notably Naṣīr
al-Dīn 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.

44. 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).

45. Passage to Europe
On Triangles was greatly
admired by future generations of
scientists; the astronomer
Copernicus (1473–1543) studied
thoroughly, and his annotated
survives.
The final major development in
classical trigonometry was the
invention of logarithms by the
mathematician John Napier in

46. The end