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Trigonometry
Trigonometry
IntroductionTrigonometry 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
Introduction• Six trigonometric functions in relation
to a right triangle.
TRIGONOMETRY
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
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
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
History of Trigonometry
Classical Trigonometry
Modern Trigonometry
Classical TrigonometryThe 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.
Classical TrigonometryoFor 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.
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.
Classical TrigonometryStill, 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.
Classical Trigonometry
Ancient Egypt and the Mediterranean world
India and the Islamic world
Passage to Europe
HISTORY OF 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.
• The Rhind papyrus, an Egyptiancollection of 84 problems in arithmetic, algebra, and geometry dating from about 1800 BC, contains five problems dealing with the seked.
• 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.
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.”
• 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.
• 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 ABCin the figure).
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
• 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.
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.
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
The first major ancient work on trigonometry to reach Europe intact after the Dark Ages was the Almagestby Ptolemy (c. AD 100–170).He lived in Alexandria, the intellectual
centre of the Hellenistic world, but little else is known about him.
• 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.
• 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.
• 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.
• 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).
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
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.
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.
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.”
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.
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.
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.
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.
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.
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.
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°.
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.
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).
Passage to EuropeOn 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
The end
