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INTRODUCTION
Trigonometry (from Greek trigōnon "triangle" + metron "measure")[1] is a branch of
mathematics that studies triangles and the relationships between their sides and the
angles between the sides. Trigonometry defines the trigonometric functions, which
describe those relationships and have applicability to cyclical phenomena, such as
waves. The field evolved during the third century B.C. as a branch of geometry used
extensively for astronomical studies.[2]
Trigonometry is usually taught in middle and secondary schools either as a separate
course or as part of a precalculus curriculum. It has applications in both pure
mathematics and applied mathematics, where it is essential in many branches of
science and technology. A branch of trigonometry, called spherical trigonometry,
studies triangles on spheres, and is important in astronomy and navigation
A branch of trigonometry, called spherical trigonometry, studies triangles on spheres,
and is important in astronomy and navigation..
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1.1 History of trigonometry
Pre-Hellenic societies such as the ancient Egyptians and Babylonians lacked the
concept of an angle measure, but they studied the ratios of the sides of similar
triangles and discovered some properties of these ratios. Ancient Greek
mathematicians such as Euclid and Archimedes studied the properties of the chord
of an angle and proved theorems that are equivalent to modern trigonometric
formulae, although they presented them geometrically rather than algebraically. The
sine function in its modern form was first defined in the Surya Siddhanta and its
properties were further documented by the 5th century Indian mathematician and
astronomer Aryabhata. These Indian works were translated and expanded by
medieval Islamic scholars
By the 10th century Islamic mathematicians were using all six trigonometric
functions, had tabulated their values, and were applying them to problems in
spherical geometry. At about the same time, Chinese mathematicians developed
trignometry independently, although it was not a major field of study for them.
Knowledge of trigonometric functions and methods reached Europe via Latintranslations of the works of Persian and Arabic astronomers such as Al Battani and
Nasir al-Din al-Tusi
One of the earliest works on trigonometry by a European mathematician is De
Triangulis by the 15th century German mathematician Regiomontanus. Trigonometry
was still so little known in 16th century Europe that Nicolaus Copernicus devoted two
chapters of De revolutionibus orbium coelestium to explaining its basic concept.
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A painting of the famous Greek geometries’, and "father of measurement", Euclid. In
the times of the Greeks, trigonometry and geometry were important mathematical
principles used in building, agriculture and education.
It is easy to explain in word-terms what trigonometry means, but it is moreimportant to understand what mechanisms of thinking, which will help understand not
only trigonometry, but everything in life in a much more vivid way.
In mathematics, trigonometry is an important set of disciplines which relate to
two and three dimensional objects; practically anything that you can see around you
can be related to the principles of trigonometry and algebra -- in the real-world, it is
very useful in engineering and construction, where its principles are important inaccurately determining the lengths, sizes and areas of objects without having to
actually create them first. Imagine the need to build a structure with only the basic
land-area given to you: using the principles of trigonometry, you can easily calculate
the geometric properties of objects to an unerring degree of accuracy.
Trigonometry, however, isn't just about using formulae to find the correct
angle or size in school. It describes the relationships that occur naturally betweenobjects and their similarity in structure. When we compare them using a similar set of
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ideas, it gives us a lot of power to understand the basis of other things in life beyond
that of just their appearance. Even though we can look at a circle, an oval, square or
rectangle, we can know that there are principles we can apply to their shape which
can be expressed through one entity: the triangle.
The ancient Greeks transformed trigonometry into an ordered science. Astronomy
was the driving force behind advancements in trigonometry. Most of the early
advancements in trigonometry were in spherical trigonometry mostly because of its
application to astronomy. The three main figures that we know of in the development
of Greek trigonometry are Hipparchus, Menelaus, and Ptolemy. There were likely
other contributors but over time their works have been loss and their names have
been forgotten.
All of the trigonometric functions of an
angle θ can be constructed geometrically in
terms of a unit circle centered at O
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1.2 EARLY TRIGONOMETRY
The ancient Egyptians and Babylonians had known of theorems on the ratios of the
sides of similar squares for many centuries. But pre-Hellenic societies lacked the
concept of an angle measure and consequently, the sides of triangles were studied
instead, a field that would be better called "trilaterometry‖
The Babylonian astronomers kept detailed records on the rising and setting of stars,
the motion of the planets, and the solar and lunar eclipses, all of which required
familiarity with angular distances measured on the celestial sphere. Based on one
interpretation of the Plimpton 322 cuneiform tablet (circa 1900 BC), some have even
asserted that the ancient Babylonians had a table of secants. There is, however,
much debate as to whether it is a table of Pythagorean triples, a solution of quadratic
equations, or a trigonometric table.
The Egyptians, on the other hand, used a primitive form of trigonometry for building
pyramids in the 2nd millennium BC.[4] The Rhind Mathematical Papyrus, written by
the Egyptian scribe Ahmes (circa 1680-1620 BC), contains the following problem
related to trigonometry:
"If a pyramid is 250 cubits high and the side of its base 360 cubits long, what is its
seked ?‖
Ahmes' solution to the problem is the ratio of half the side of the base of the pyramid
to its height, or the run-to-rise ratio of its face. In other words, the quantity he found
for the seked is the cotangent of the angle to the base of the pyramid and its face.
1.2.1Greeks and Trigonometry
In the second century B.C., Hipparchus derived a trigonometric table measuring
chord lengths of a circle having a fixed radius. Hipparchus built the values in
increasing degrees, beginning with 71 and ending with 180, incrementing in units of
71 degrees. In the second century A.D., Ptolemy defined Hipparchus' value for the
radius as 60 and created a table of chords incrementing one degree, from 0 degrees
to 180 degrees. This table of chords also showed how to find unknown parts of
triangles from given parts.
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Trigonometry was originally created by the Greeks to aid in the study of astronomy.
Hipparchus of Bithynia (190-120 B.C.) tabulated trigonometric ratios, to enable the
calculation of a planet's position as formulated by Apollonius. Angles were also
defined, taking the Babylonian measure of 360 degrees. The chord was defined, and
the cosine and sine loosely defined. The results sin2x + cos2x = 1 and the half-angle
formulae were also derived, geometrically.
Claudius Ptolemy worked further on Hipparchus' chord table and came up with a
more complete one. He used Euclid's propositions to aid in his work and developed a
method of calculating square roots, though he never explained how. Using his
theorem (for a quadrilateral inscribed in a circle, the product of the diagonals equals
the sum of the products of the opposite sides) and the half-angle formula, he derived
the sum and difference (addition) formulae.
Ptolemy then proceeded to work on plane triangles. In this process, he developed
the idea of inverse trigonometric functions. He also derived, in modern terms, the
Sine and Cosine Rules.
1.2.2 India and Trigonometry
In the sixth century, India based its trigonometry on the sine function, which was the
length of the side opposite the angle in a right triangle of a specific hypotenuse
instead of a ratio. The Indians used various values for the hypotenuse. They built
sine tables from these functions and later introduced a cosine function and tables.
The Indians were the next to advance the study of trigonometry. They developed
their own sine tables, using the Greek half-angle formula. Later, the cosine table was
also constructed. Techniques of approximation to a relatively high accuracy were
also introduced.
1.2.3 Muslims and Trigonometry
The Indian works were translated and read by the Islamic mathematicians, who alsoworked on trigonometry. Similar to the Greeks and Indians, they related trigonometry
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and astronomy. The Indian sine was used, as well as the chord. The cosine was also
formally introduced, by Abu Abdallah Muhammad ibn Jabir al-Battani.Abu Abdallah
Muhammad ibn Jabir al-Battani, who lived from 858 to 929, formally introduced the
cosine function, as he built on the work of the Indians and Greeks.
Muslim mathematicians also introduced the polar triangle for spherical triangles, sine
and tangent tables created in 1/60th-of-a-degree steps. Nasir ad-Din at-Tusi, who
lived from 1201 to 1274, wrote a book separating plane and spherical trigonometry
into its own field of study, called the Book of the Transversal Figure. Muslim
mathematicians revived the long-dead tangent function, invented by the Chinese but
lost, and added the co-tangent, co-secant, and secant functions.
The tangent function resurfaced; and the cotangent, cosecant and scant functions
were introduced. Although their definitions were initially geometric, it was soon
realized that they were the reciprocal functions of tangent, sine and cosine
respectively. Highly accurate tables were developed for the trigonometric functions.
The triple-angle formulae, already derived, were used for this.
1.2.4 China and Trigonometry
Early forms of trigonometry appeared in Chinese mathematics in the sixth century,
but major advances in trigonometry did not happen until the 12th and 13th centuries,
even though astronometrical calculations and calendar science demanded it. Shen
Kuo, who lived from 1031 to 1095, used trigonometry to solve problems of chords
and arcs. Guo Shoujing worked on arcs and circles, which formed the foundation of
spherical trigonometry during the 12th and 13th centuries. Most of China's
mathematics was lost after the Yuan Dynasty took root in 1271 until the 19th century.
The Chinese in the medieval times studied astronomy and hence trigonometry. They
introduced the tangent function. However, most of their works are in the field of
astronomy, and many of their trigonometric advancements were not continued.
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1.2.5 European Developments
Georges Joachim defined trigonometric functions as ratios instead of lengths of lines
during the 13th century. French mathematician François Viète, who lived from 1540
to 1653, introduced the polar triangle into spherical trigonometry and published two
books, Canon Mathematicus, and Universalium Inspectionum Liber Singularis, in
1579. These two books were mathematical tables in which the values for sine are
computed to 10 to the negative eighth power. In the 17th century, John Napier, a
Scottish mathematician, invented logarithms, memory tricks to remember the 10
laws of how to solve spherical triangles; he also came up with what are now called
Napier's analogies to help mathematicians solve oblique spherical triangles. In the
18th century, Leonhard Euler defined trigonometric functions in terms of complex
numbers showing how basic laws of trigonometry were the consequences of
arithmetic of complex numbers.
Trigonometry reached Europe in the medieval times. Richard of Wallingford wrote a
text on trigonometry, Quadripartium. He related the Indian sine to the ancient chords.
He used Euclid's Elements as a basis for his arguments in plane trigonometry. Levi
ben Gerson worked on plane trigonometry, particularly the laws of sines and
cosines.
In the 16th century, trigonometry was incorporated into geography and navigation.
Knowledge of trigonometry was used to construct maps, determining the position of
a land mass in relation to the longitudes and latitudes.
Johannes Muller, or more popularly known as Regiomontanus, wrote a text On
Triangles. He studied plane trigonometry, including results for solving triangles. He
expanded on Levi ben Gerson's work. He proved the Sine Rule, and also considered
the ambiguous case in using the rule.
Later works improved the tables of sines, which has been worked on extensively; as
well as included tables for the other functions. Thomas Finck was the first to use the
modern terms "tangent" and "secant".
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The worked so far applied trigonometric concepts in astronomy. It was only until
Bartholomew Pitiscus when there was a text considering the solving of a plane
triangle on earth. He invented the word "trigonometry", in his title Trigonometriae
sive, de dimensione triangulis, Liber (Book of Trigonometry, or the Measurement of
Triangles). He developed his own sine and tangent tables. However, like all the
tables that had been calculated before, the values are actually the lengths of certain
lines in a fixed circle.
Later developments in trigonometry are mainly the use of trigonometric ratios in
calculus; analysis, differential equations and integration, just to name a few.
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2.1 Euclid
Euclid
Artist's depiction of Euclid
Born fl. 300 BC
Residence Alexandria, Egypt
Ethnicity Greek
Fields Mathematics
Known for Euclidean geometry
Euclid's Elements
Euclid (Greek: Εὐκλείδης — Eukleídēs), fl. 300 BC, also known as Euclid of
Alexandria, was a Greek mathematician and is often referred to as the "Father of
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Geometry." He was active in Hellenistic Alexandria during the reign of Ptolemy I
(323 –283 BC). His Elements is the most successful textbook and one of the most
influential works in the history of mathematics, serving as the main textbook for
teaching mathematics (especially geometry) from the time of its publication until the
late 19th or early 20th century. In it, the principles of what is now called Euclidean
geometry were deduced from a small set of axioms. Euclid also wrote works on
perspective, conic sections, spherical geometry, number theory and rigor.
2.1.1 Elements
One of the oldest surviving fragments of Euclid's Elements, found at Oxyrhynchus
and dated to circa AD 100. The diagram accompanies Book II, Proposition 5.
Euclid's most celebrated work is the Elements , which is primarily a treatise on
geometry contained in 13 books. Although many of the results in Elements originated
with earlier mathematicians, one of Euclid's accomplishments was to present them in
a single, logically coherent framework, making it easy to use and easy to reference,
including a system of rigorous mathematical proofs that remains the basis of
mathematics 23 centuries later.
Euclid's Elements owed its enormously high status to a number of reasons.
The most influential single feature was Euclid's use of the axiomatic method whereby
all the theorems were laid out as deductions from certain self-evident basic
propositions or axioms in such a way that in each successive proof only propositions
already proved or axioms were used. This became accepted as the paradigmatically
rigorous way of setting out any body of knowledge, and attempts were made to apply
it not just to mathematics, but to natural science, theology, and even philosophy and
ethics.
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The Elements consists of 13 books. Within each book is a sequence of propositions
or theorems, varying from about 10 to 100, preceded by definitions.
In Book I, 23 definitions are followed by five postulates. After the postulates,
five common notions or axioms are listed. The first is, "Things which are equal
to the same thing are also equal to each other." The usual elementary course
in Euclidean geometry is based on Book I.
Book II is a continuation of Book I, proving geometrically what today would be
called algebraic identities, such as (a + b )2 = a 2 + b 2 + 2ab, and generalizing
some propositions of Book I.
Book III is on circles, intersections of circles, and properties of tangents to
circles.
Book IV continues with circles, emphasizing inscribed and circumscribed
rectilinear figures.
Book V of the Elements is one of the finest works in Greek mathematics. The
theory of proportions discovered by Eudoxus is here expounded masterfully
by Euclid. The theory of proportions is concerned with the ratios of
magnitudes (rational or irrational numbers) and their integral multiples.
Book VI applies the propositions of Book V to the figures of plane geometry. A
basic proposition in this book is that a line parallel to one side of a triangle will
divide the other two sides in the same ratio.
As in Book V, Books VII, VIII, and IX are concerned with properties of
(positive integral) numbers. In
Book VII a prime number is defined as that which is measured by a unit alone
(a prime number is divisible only by itself and 1).
In Book IX proposition 20 asserts that there are infinitely many prime
numbers, and Euclid's proof is essentially the one usually given in modern
algebra textbooks.
Book X is an impressively well-finished treatment of irrational numbers or,
more precisely, straight lines whose lengths cannot be measured exactly by a
given line assumed as rational.
Books XI-XIII are principally concerned with three-dimensional figures. In
Book XII the method of exhaustion is used extensively. The final book shows
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how to construct and circumscribe by a sphere the five Platonic, or regular,
solids: the regular pyramid or tetrahedron, octahedron, cube, icosahedron,
and dodecahedron.
2.1.2 Other works
In addition to the Elements , at least five works of Euclid have survived to the present
day. They follow the same logical structure as Elements , with definitions and proved
propositions.
Data deals with the nature and implications of "given" information in
geometrical problems; the subject matter is closely related to the first four
books of the Elements .
On Divisions of Figures , which survives only partially in Arabic translation,
concerns the division of geometrical figures into two or more equal parts or
into parts in given ratios. It is similar to a third century AD work by Heron of
Alexandria.
Phenomena , a treatise on spherical astronomy, survives in Greek; it is quite
similar to On the Moving Sphere by Autolycus of Pitane, who flourished
around 310 BC.
Optics is the earliest surviving Greek treatise on perspective. In its definitions
Euclid follows the Platonic tradition that vision is caused by discrete rays
which emanate from the eye. One important definition is the fourth: "Things
seen under a greater angle appear greater, and those under a lesser angle
less, while those under equal angles appear equal." In the 36 propositions
that follow, Euclid relates the apparent size of an object to its distance from
the eye and investigates the apparent shapes of cylinders and cones when
viewed from different angles. Proposition 45 is interesting, proving that for any
two unequal magnitudes, there is a point from which the two appear equal.
Pappus believed these results to be important in astronomy and included
Euclid's Optics , along with his Phaenomena , in the Little Astronomy , a
compendium of smaller works to be studied before the Syntaxis (Almagest ) of
Claudius Ptolemy.
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Statue of Euclid in the Oxford University
Museum of Natural History
Other works are credibly attributed to Euclid, but have been lost.
Conics was a work on conic sections that was later extended by Apollonius of
Perga into his famous work on the subject. It is likely that the first four books
of Apollonius's work come directly from Euclid. According to Pappus,
"Apollonius, having completed Euclid's four books of conics and added four
others, handed down eight volumes of conics." The Conics of Apollonius
quickly supplanted the former work, and by the time of Pappus, Euclid's work
was already lost.
Surface Loci concerned either loci (sets of points) on surfaces or loci which
were themselves surfaces; under the latter interpretation, it has been
hypothesized that the work might have dealt with quadric surfaces.
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2.2 Ptolemy
Ptolemy
An early Baroque artist's rendition of Claudius Ptolemaeus.
Born c. AD 90
Egypt
Died c. AD 168
Alexandria, Egypt
Occupation mathematician, geographer, astronomer, astrologer
A medieval artist's rendition of Claudius Ptolemy
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Although it is not known when the systematic use of the 360° circle came into
mathematics, it is known that the systematic introduction of the 360° circle came a
little after Aristarchus of Samos composed On the Sizes and Distances of the Sun
and Moon (ca. 260 B.C.), since he measured an angle in terms of a fraction of a
quadrant.[8] It seems that the systematic use of the 360° circle is largely due to
Hipparchus and his table of chords. Hipparchus may have taken the idea of this
division from Hypsicles who had earlier divided the day into 360 parts, a division of
the day that may have been suggested by Babylonian astronomy. In ancient
astronomy, the zodiac had been divided into twelve "signs" or thirty-six "decans". A
seasonal cycle of roughly 360 days could have corresponded to the signs and
decans of the zodiac by dividing each sign into thirty parts and each decan into ten
parts. It is due to the Babylonian sexagesimal number system that each degree is
divided into sixty minutes and each minute is divided into sixty seconds.
2.2.1 Menelaus of Alexandria (ca. 100 A.D.) wrote in three books his Sphaerica .
In Book I, he established a basis for spherical triangles analogous to the
Euclidean basis for plane triangles. He establishes a theorem that is without
Euclidean analogue, that two spherical triangles are congruent if
corresponding angles are equal, but he did not distinguish between congruent
and symmetric spherical triangles. Another theorem that he establishes is that
the sum of the angles of a spherical triangle is greater than 180°.
Book II of Sphaerica applies spherical geometry to astronomy.
Book III contains the "theorem of Menelaus". He further gave his famous "rule
of six quantities".
Later, Claudius Ptolemy (ca. 90 - ca. 168 A.D.) expanded upon Hipparchus' Chords
in a Circle in his Almagest , or the Mathematical Syntaxis . The thirteen books of the
Almagest are the most influential and significant trigonometric work of all antiquity. A
theorem that was central to Ptolemy's calculation of chords was what is still known
today as Ptolemy's theorem, that the sum of the products of the opposite sides of a
cyclic quadrilateral is equal to the product of the diagonals. A special case ofPtolemy's theorem appeared as proposition 93 in Euclid's Data . Ptolemy's theorem
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leads to the equivalent of the four sum-and-difference formulas for sine and cosine
that are today known as Ptolemy's formulas, although Ptolemy himself used chords
instead of sine and cosine. Ptolemy further derived the equivalent of the half-angle
formula . Ptolemy used these results to create his
trigonometric tables, but whether these tables were derived from Hipparchus' work
cannot be determined.
Neither the tables of Hipparchus nor those of Ptolemy have survived to the present
day, although descriptions by other ancient authors leave little doubt that they once
existed.
2.2.2 CONTRIBUTIONS
Ptolemy's theorem implies the theorem of Pythagoras. The latter serves as a
foundation of Trigonometry, the branch of mathematics that deals with
relationships between the sides and angles of a triangle. In the language of
Trigonometry, Pythagorean Theorem reads
sin²(A) + cos²(A) = 1,
where A is one of the internal angles of a right triangle. If the
hypotenuse of the triangle is of length 1, then sin (A) is the length of the side
opposite to the angle A, cos(A) is the length of the adjacent side.
Ptolemy's theorem also provides an elegant way to prove other
trigonometric identities. In a little while, I'll prove the addition and subtraction
formulas for sine:
(1) sin(A + B) = sin(A)cos(B) + cos(A)sin(B)
(2) sin(A - B) = sin(A)cos(B) - cos(A)sin(B)
But first let's have a simple proof for the Law of Sines .
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Proposition III.20 from Euclid's Elements says:
In a circle the angle at the center is double of the angle at the
circumference, when angles have the same circumference as
base.
The more common formulation asserts that an angle circumscribed in a
circle is equal to half the central angle that subtends the same chord. (As a
corollary, from here it follows that all circumscribed angles subtending the
same arc are equal irrespective of their position on the circle. This is
Proposition III.21) On the diagram, ∠BOC = 2∠BAC (= 2A.)
Drop a perpendicular from O on the side BC. Assuming the radius of
the circle is R, OB = OC = R. Also, ∠BOP = ∠POC. In ΔBOP, sin (∠BOP) =
BP/OB = BC/2R. Therefore, BC/sin (∠BOP) = 2R. When angle A is obtuse,
the center O is located outside ΔABC and the diagram looks differently. The
resulting identity is, however, the same. Repeating these steps with the other
two angles B and C of ΔABC we get the Law of Sines .
In the case, where the diameter of the circumscribed circle is 1, we
have a = sin(A), b = sin(B), and c = sin(C). This is all we need to apply
Ptolemy's theorem.
Consider a quadrilateral ABDC inscribed into a circle of diameter 1 so
that the diagonal BC serves as a diameter.
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From the definition of sine and cosine we determine the sides of the
quadrilateral. The Law of Sines supplies the length of the remaining diagonal.
The addition formula for sine is just a reformulation of Ptolemy's theorem.
To prove the subtraction formula, let the side BC serve as a diameter.
As a consequence, we obtain formulas for sine (in one step) and for cosine
(in two steps) of complementary angles:
sin(π/2 - α) = cos α,
cos(π/2 - α) = sin α.
From these and the addition formulas for sine it is not difficult to derive the
addition formulas for cosine:
cos(α + β) = cos(α) cos(β) - sin(α) sin(β),
cos(α - β) = cos(α) cos(β) + sin(α) sin(β).
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2.3 Hipparchus
Hipparchus or Hipparch
The first whose quantitative and accurate models for the motion of the Sun and
Moon survive
Born Nicaea (now Iznik, Turkey
Residence Alexandria, Egypt
Fields Mathematics
Occupation Greek astronomer, geographer, and mathematician of the
Hellenistic period
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Hipparchus or Hipparch (Greek: Ἵππαρχος, Hipparkhos ; c. 190 BC – c. 120 BC)
was a Greek astronomer, geographer, and mathematician of the Hellenistic period.
Hipparchus was born in Nicaea (now Iznik, Turkey), and probably died on the
island of Rhodes. He is known to have been a working astronomer at least from 147
BC to 127 BC. Hipparchus is considered the greatest ancient astronomical observer
and, by some, the greatest overall astronomer of antiquity.
He was the first whose quantitative and accurate models for the motion of the
Sun and Moon survive. For this he certainly made use of the observations and
perhaps the mathematical techniques accumulated over centuries by the Chaldeans
from Babylonia. He developed trigonometry and constructed trigonometric tables,and he has solved several problems of spherical trigonometry. With his solar and
lunar theories and his trigonometry, he may have been the first to develop a reliable
method to predict solar eclipses. His other reputed achievements include the
discovery of precession, the compilation of the first comprehensive star catalog of
the western world, and possibly the invention of the astrolabe, also of the armillary
sphere which first appeared during his century and was used by him during the
creation of much of the star catalogue. It would be three centuries before Claudius
Ptolemaeus' synthesis of astronomy would supersede the work of Hipparchus; it is
heavily dependent on it in many areas.
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2.4 Aryabhata
Hipparchus or Hipparch
The first whose quantitative and accurate models for the motion of the Sun and
Moon survive
Born India
Known as
Book The Siddhantas and the Aryabhatiya
Occupation Indian mathematician and astronomer
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Statue of Aryabhata. As there is no known information regarding his appearance,
any image of Aryabhata originates from an artist's conception.
The next significant developments of trigonometry were in India. Influential
works from the 4th –5th century, known as the Siddhantas (of which there were five,
the most complete survivor of which is the Surya Siddhanta) first defined the sine as
the modern relationship between half an angle and half a chord, while also defining
the cosine, versine, and inverse sine. Soon afterwards, another Indian
mathematician and astronomer Aryabhata (476 –550 AD), collected and expanded
upon the developments of the Siddhantas in an important work called the
Aryabhatiya . The Siddhantas and the Aryabhatiya contain the earliest surviving
tables of sine values and versine (1 − cosine) values, in 3.75° intervals from 0° to
90°, to an accuracy of 4 decimal places. They used the words jya for sine, kojya for
cosine, ukramajya for versine, and otkram jya for inverse sine. The words jya and
kojya eventually became sine and cosine respectively after a mistranslation
described above.
Other Indian mathematicians later expanded on these works of trigonometry. In the
6th century, Varahamihira used the formulas
(equivalent to formulas known by Thales and
Pythagoras[18])
(equivalent to a formula known to Ptolomy; see above)
In the 7th century, Bhaskara I produced a formula for calculating the sine of
an acute angle without the use of a table. He also gave the following approximation
formula for sin(x), which had a relative error of less than 1.9%:
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Later in the 7th century, Brahmagupta redeveloped the formula
(also derived earlier, as mentioned
above) as well as the Brahmagupta interpolation formula for computing sine values.
Another later Indian author on trigonometry was Bhaskara II in the 12th century.
Madhava (c. 1400) made early strides in the analysis of trigonometric functions and
their infinite series expansions. He developed the concepts of the power series and
Taylor series, and produced the trigonometric series expansions of sine, cosine,
tangent, and arctangent. Using the Taylor series approximations of sine and cosine,
he produced a sine table to 12 decimal places of accuracy and a cosine table to 9
decimal places of accuracy. He also gave the power series of π and the θ, radius,
diameter, and circumference of a circle in terms of trigonometric functions. His works
were expanded by his followers at the Kerala School up to the 16th century.
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2.5 Al-Khwārizmī
Muhammad ibn Mūsā al-Khwārizmī
The medieval Islamic world by Muslim mathematicians of mostly Persian descent
Born c. 780
Died c. 850
Book Al-Harrānī al-Battānī (Albatenius)
Occupation Greek astronomer, geographer, and mathematician of the
Hellenistic period
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Al-Khwārizmī depicted on a Soviet stamp
The Indian works were later translated and expanded in the medieval Islamic world
by Muslim mathematicians of mostly Persian descent. They enunciated a large
number of theorems which freed the subject of trigonometry from dependence upon
the complete quadrilateral, as was the case in Hellenistic mathematics due to the
application of Menelaus' theorem. According to E. S. Kennedy, it was after this
development in Islamic mathematics that "the first real trigonometry emerged, in the
sense that only then did the object of study become the spherical or plane triangle,
its sides and angles."
A page from al-Khwārizmī's Algebra
Al- Kitāb al -mukhta ṣar fī ḥisāb al -jabr wa-l- muqābala (Arabic: اكتب امختصر في حب اجر
―The Compendious Book on Calculation by Completion and Balancing‖) is aوامقة
mathematical book written approximately 830 CE. The book was written with the
encouragement of the Caliph Al-Ma'mun as a popular work on calculation and is
replete with examples and applications to a wide range of problems in trade,
surveying and legal inheritance. The term algebra is derived from the name of one of
the basic operations with equations (al-jabr ) described in this book. The book was
translated in Latin as Liber algebrae et almucabala by Robert of Chester (Segovia,
1145) hence "algebra", and also by Gerard of Cremona. A unique Arabic copy is
kept at Oxford and was translated in 1831 by F. Rosen. A Latin translation is kept in
Cambridge.
The al-jabr is considered the foundational text of modern algebra. It provided
an exhaustive account of solving polynomial equations up to the second degree, andintroduced the fundamental methods of "reduction" and "balancing", referring to the
transposition of subtracted terms to the other side of an equation, that is, the
cancellation of like terms on opposite sides of the equation.
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Al-Khwārizmī's method of solving linear and quadratic equations worked by first
reducing the equation to one of six standard forms (where b and c are positive
integers)
squares equal roots (ax 2 = bx )
squares equal number (ax 2 = c )
roots equal number (bx = c )
squares and roots equal number (ax 2 + bx = c )
squares and number equal roots (ax 2 + c = bx )
roots and number equal squares (bx + c = ax 2)
by dividing out the coefficient of the square and using the two operations al- ǧabr (Arabic: ―restoring‖ or ―completion‖) andاجر al- muqābala ("balancing"). Al-ǧabr is
the process of removing negative units, roots and squares from the equation by
adding the same quantity to each side. For example, x 2 = 40x − 4x 2 is reduced to 5x 2
= 40x . Al-muqābala is the process of bringing quantities of the same type to the
same side of the equation. For example, x 2 + 14 = x + 5 is reduced to x 2 + 9 = x .
In the early 9th century, Muhammad ibn Mūsā al-Khwārizmī produced
accurate sine and cosine tables, and the first table of tangents. He was also a
pioneer in spherical trigonometry. In 830, Habash al-Hasib al-Marwazi produced the
first table of cotangents. Muhammad ibn Jābir al-Harrānī al-Battānī (Albatenius)
(853-929) discovered the reciprocal functions of secant and cosecant, and produced
the first table of cosecants for each degree from 1° to 90°. He was also responsible
for establishing a number of important trigometrical relationships, such as:
By the 10th century, in the work of Abū al-Wafā' al-Būzjānī , Muslim
mathematicians were using all six trigonometric functions. Abu al-Wafa had sine
tables in 0.25° increments, to 8 decimal places of accuracy, and accurate tables of
tangent values. He also developed the following trigonometric formula:
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Dīn al-Tūsī was the first to treat trigonometry as a mathematical discipline
independent from astronomy, and he developed spherical trigonometry into its
present form. He listed the six distinct cases of a right-angled triangle in spherical
trigonometry, and in his On the Sector Figure , he stated the law of sines for plane
and spherical triangles, discovered the law of tangents for spherical triangles, and
provided proofs for both these laws.
In the 15th century, Jamshīd al-Kāshī provided the first explicit statement of
the law of cosines in a form suitable for triangulation. In France, the law of cosines is
still referred to as the theorem of Al-Kashi . He also gave trigonometric tables of
values of the sine function to four sexagesimal digits (equivalent to 8 decimal places)
for each 1° of argument with differences to be added for each 1/60 of 1°. Ulugh Beg
also gives accurate tables of sines and tangents correct to 8 decimal places around
the same time.
In the 16th century, Taqi al-Din contributed to trigonometry in his Sidrat al-
Muntaha , in which he was the first mathematician to extract the precise value of Sin
1°. He discusses the values given by his predecessors, explaining how Ptolemy
used an approximate method to obtain his value of Sin 1° and how Abū al -Wafā, Ibn
Yunus, al-Kashi, Qāḍī Zāda al-Rūmī , Ulugh Beg and Mirim Chelebi improved on the
value. Taqi al-Din then solves the problem to obtain the precise value of Sin 1°:
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2.6 ABŪ AL-WAFĀ' BŪZJĀNĪ
Abū al-Wafāʾ Muḥammad ibn Muḥammad ibn Yaḥyā ibn Ismāʿīl ibn al-ʿAbbās al-
Būzjānī
The medieval Islamic world by Muslim mathematicians of mostly Persian descent
Born 10 June 940
Died 1 July 998
Book Abul Wáfa on the Moon
Occupation principally in the field of trigonometry
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BACKGROUND
Abul Wafa Buzjani (10 June 940 – 1 July 998) (Persian: ,(اواوف وزجی
extended name: Abū al-Wafāʾ Muḥammad ibn Muḥammad ibn Yaḥyā ibn
Ismāʿīl ibn al-ʿAbbās al-Būzjānī (Persian: اواوف محمد ن محمد ن حی ن امعل ن
) was aاس اوزجی Persian mathematician and astronomer. He was born
in Buzhgan, (now Torbat-e Jam) in Iran.
In 959 AD, he moved to Iraq. He studied mathematics and worked
principally in the field of trigonometry. He wrote a number of books, most of
which no longer exist. He also studied the movements of the moon. Thecrater Abul Wáfa on the Moon is named after him.
Buzjani, the Persian mathematician and astronomer.
CONTRIBUTIONS
He devised a wall quadrant for the accurate astronomy measurement of the
declination of stars. He also introduced the tangent function and improved
methods of calculating trigonometry tables and developed novel ways of
solving some problems of spherical triangles.
He established the trigonometric identities:
sin(a + b ) = sin(a )cos(b ) + cos(a )sin(b )
cos(2a ) = 1 − 2sin2(a )
sin(2a ) = 2sin(a )cos(a )
and discovered the law of sines for spherical triangles:[2]
List of trigonometric identities
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In mathematics, trigonometric identities are equalities that involve
trigonometric functions that are true for every single value of the occurring
variables (see Identity (mathematics)). Geometrically, these are identities
involving certain functions of one or more angles. These are distinct
from triangle identities, which are identities involving both angles and side
lengths of a triangle. Only the former are covered in this article.
These identities are useful whenever expressions involving
trigonometric functions need to be simplified. An important application is
the integration of non-trigonometric functions: a common technique involves
first using the substitution rule with a trigonometric function, and then
simplifying the resulting integral with a trigonometric identity.
Notation
Angles
This article uses Greek letters such as alpha (α ), beta ( β), gamma (γ ),
and theta (θ ) to represent angles. Several different units of angle measure are
widely used, including degrees,radians, and grads:
1 full circle = 360 degrees = 2π radians = 400 grads.
The following table shows the conversions for some common angles:
Degrees 30° 60° 120° 150° 210° 240° 300° 330°
Radians
Grads33⅓
grad
66⅔
grad
133⅓
grad
166⅔
grad
233⅓
grad
266⅔
grad
333⅓
grad
366⅔
grad
Degrees 45° 90° 135° 180° 225° 270° 315° 360°
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Radians
Grads50
grad
100
grad
150
grad
200
grad
250
grad
300
grad
350
grad
400
grad
Unless otherwise specified, all angles in this article are assumed to be
in radians, though angles ending in a degree symbol (°) are in degrees.
Trigonometric functionsThe primary trigonometric functions are the sine and cosine of an angle.
These are usually abbreviated sin(θ ) and cos(θ ), respectively, where θ is the
angle. In addition, the parentheses around the angle are sometimes omitted,
e.g. sin θ and cos θ .
The tangent (tan) of an angle is the ratio of the sine to the cosine:
Finally, the reciprocal functions secant (sec), cosecant (csc), and
cotangent (cot) are the reciprocals of the cosine, sine, and tangent:
These definitions are sometimes referred to as ratio identities.
Inverse functions
The inverse trigonometric functions are partial inverse functions for the
trigonometric functions. For example, the inverse function for the sine, known
as the inverse sine (sin−1) or arcsine(arcsin or asin), satisfies
and
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This article uses the following notation for inverse trigonometric functions:
Function Sin cos tan sec csc cot
Inverse Arcsin arccos arctan arcsec arccsc arccot
The Pythagorean identity
The basic relationship between the sine and the cosine is the Pythagorean
trigonometric identity:
This can be viewed as a version of the Pythagorean theorem, and
follows from the equation x 2 + y 2 = 1 for the unit circle. This equation can be
solved for either the sine or the cosine:
Related identities
Dividing the Pythagorean identity through by either cos2 θ or sin2 θ yields two
other identities:
Using these identities together with the ratio identities, it is possible to express
any trigonometric function in terms of any other (up to a plus or minus sign):
Each trigonometric function in terms of the other five.
sinθ
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cosθ
tanθ
cscθ
secθ
cotθ
Historic shorthands
The versine, coversine, haversine, and exsecant were used in navigation. For
example the haversine formula was used to calculate the distance between
two points on a sphere. They are rarely used today.
Name(s) Abbreviation(s) Value
versed sine, versine
versed cosine, vercosine,
coversed sine, coversine
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half versed sine, haversine
half versed cosine,
havercosine,
hacoversed sine, half
coversine,
cohaversed sine, cohaversine
exterior secant, exsecant
exterior cosecant, excosecant
Symmetry, shifts, and periodicity
By examining the unit circle, the following properties of the trigonometric
functions can be established.
Symmetry
When the trigonometric functions are reflected from certain angles, the result
is often one of the other trigonometric functions. This leads to the following
identities
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Reflected in θ = 0 Reflected in θ = π / 2
(co-function identities)Reflected in θ = π
Shifts and periodicity
By shifting the function round by certain angles, it is often possible to find
different trigonometric functions that express the result more simply. Some
examples of this are shown by shifting functions round by π/2, π and 2π
radians. Because the periods of these functions are either π or 2π, there are
cases where the new function is exactly the same as the old function without
the shift.
Shift by π/2 Shift by π
Period for tan and cot
Shift by 2π
Period for sin, cos, csc
and sec
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2.7 Guo Shoujing
Guo Shoujing
The first whose quantitative and accurate models for the motion of the Sun and
Moon survive
Born (1231 –1316)
Famous Book Kaiyuan Zhanjing,
Fields Mathematics
Occupation Chinese trigonometry
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Guo Shoujing (1231 –1316)
In China, Aryabhata's table of sines were translated into the Chinese mathematical
book of the Kaiyuan Zhanjing , compiled in 718 AD during the Tang Dynasty.
Although the Chinese excelled in other fields of mathematics such as solid geometry,
binomial theorem, and complex algebraic formulas, early forms of trigonometry were
not as widely appreciated as in the earlier Greek and then Indian and Islamic worlds.
Instead, the early Chinese used an empirical substitute known as chong cha , while
practical use of plane trigonometry in using the sine, the tangent, and the secant
were known. However, this embryonic state of trigonometry in China slowly began to
change and advance during the Song Dynasty (960 –1279), where Chinese
mathematicians began to express greater emphasis for the need of spherical
trigonometry in calendrical science and astronomical calculations. The polymath
Chinese scientist, mathematician and official Shen Kuo (1031 –1095) used
trigonometric functions to solve mathematical problems of chords and arcs. Victor J.
Katz writes that in Shen's formula "technique of intersecting circles", he created an
approximation of the arc of a circle s given the diameter d , sagita v , and length of the
chord c subtending the arc, the length of which he approximated as .
Sal Restivo writes that Shen's work in the lengths of arcs of circles provided the
basis for spherical trigonometry developed in the 13th century by the mathematician
and astronomer Guo Shoujing (1231 –1316). As the historians L. Gauchet and
Joseph Needham state, Guo Shoujing used spherical trigonometry in his calculations
to improve the calendar system and Chinese astronomy. Along with a later 17th
century Chinese illustration of Guo's mathematical proofs, Needham states that:
Guo used a quadrangular spherical pyramid, the basal quadrilateral of which
consisted of one equatorial and one ecliptic arc, together with two meridian arcs, one
of which passed through the summer solstice point...By such methods he was able
to obtain the du lü (degrees of equator corresponding to degrees of ecliptic), the ji
cha (values of chords for given ecliptic arcs), and the cha lü (difference between
chords of arcs differing by 1 degree).
Despite the achievements of Shen and Guo's work in trigonometry, another
substantial work in Chinese trigonometry would not be published again until 1607,
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with the dual publication of Euclid's Elements by Chinese official and astronomer Xu
Guangqi (1562 –1633) and the Italian Jesuit Matteo Ricci (1552 –1610).
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2.9 Isaac Newton
Isaac Newton
first mathematician in Europe to treat trigonometry
Born Nicaea (now Iznik, Turkey
Famous Book De triangulis omnimodus
Fields Mathematics
Occupation Greek astronomer, geographer, and mathematician of the
Hellenistic period
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Isaac Newton in a 1702 portrait by Godfrey Kneller.
Regiomontanus was perhaps the first mathematician in Europe to treat trigonometryas a distinct mathematical discipline, in his De triangulis omnimodus written in 1464,
as well as his later Tabulae directionum which included the tangent function,
unnamed.
The Opus palatinum de triangulis of Georg Joachim Rheticus, a student of
Copernicus, was probably the first to define trigonometric functions directly in terms
of right triangles instead of circles, with tables for all six trigonometric functions; this
work was finished by Rheticus' student Valentin Otho in 1596.
In the 17th century, Isaac Newton and James Stirling developed the general
Newton-Stirling interpolation formula for trigonometric functions.
In the 18th century, Leonhard Euler's Introductio in analysin infinitorum (1748)
was mostly responsible for establishing the analytic treatment of trigonometric
functions in Europe, defining them as infinite series and presenting "Euler's formula"
e ix = cosx + i sinx . Euler used the near-modern abbreviations sin., cos., tang., cot.,
sec., and cosec.
Also in the 18th century, Brook Taylor defined the general Taylor series and
gave the series expansions and approximations for all six trigonometric functions.
The works of James Gregory in the 17th century and Colin Maclaurin in the 18th
century were also very influential in the development of trigonometric series.
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2.9 Leonhard Euler
Leonhard Euler
Born 15/April/1707
Basel, Switzerland
Died 18/September/1783(aged 76)
Residence Prussia,Russia
Switzerland
Nationality Swiss
Fields Mathematician and Physicist
Alma mater
University of Basel
Doctoral advisor Johann Bernoulli
Known for See full list
Religious stance Calvinist
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Contributions to mathematics
Euler worked in almost all areas of mathematics: geometry, calculus, trigonometry,
algebra, and number theory, as well as continuum physics, lunar theory and other
areas of physics. He is a seminal figure in the history of mathematics; if printed, his
works, many of which are of fundamental interest, would occupy between 60 and 80
quarto volumes. Euler's name is associated with a large number of topics.
Mathematical notation
Euler introduced and popularized several notational conventions through his
numerous and widely circulated textbooks. Most notably, he introduced the concept
of a function and was the first to write f (x ) to denote the function f applied to the
argument x . He also introduced the modern notation for the trigonometric functions,
the letter e for the base of the natural logarithm (now also known as Euler's number),
the Greek letter Σ for summations and the letter i to denote the imaginary unit. The
use of the Greek letter π to denote the ratio of a circle's circumference to its diameter
was also popularized by Euler, although it did not originate with him.
Analysis
The development of calculus was at the forefront of 18th century mathematical
research, and the Bernoullis—family friends of Euler—were responsible for much of
the early progress in the field. Thanks to their influence, studying calculus became
the major focus of Euler's work. While some of Euler's proofs are not acceptable by
modern standards of mathematical rigour, his ideas led to many great advances.
Euler is well-known in analysis for his frequent use and development of power
series, the expression of functions as sums of infinitely many terms, such as
Notably, Euler discovered the power series expansions for e and the inverse tangent
function. His daring (and, by modern standards, technically incorrect) use of power
series enabled him to solve the famous Basel problem in 1735:
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Euler introduced the use of the exponential function and logarithms in analytic
proofs. He discovered ways to express various logarithmic functions using power
series, and he successfully defined logarithms for negative and complex numbers,
thus greatly expanding the scope of mathematical applications of logarithms.[26] He
also defined the exponential function for complex numbers, and discovered its
relation to the trigonometric functions. For any real number φ, Euler's formula states
that the complex exponential function satisfies
A special case of the above formula is known as Euler's identity,
Called "the most remarkable formula in mathematics" by Richard Feynman,
for its single uses of the notions of addition, multiplication, exponentiation, and
equality, and the single uses of the important constants 0, 1, e , i and π. In 1988,readers of the Mathematical Intelligencer voted it "the Most Beautiful Mathematical
Formula Ever". In total, Euler was responsible for three of the top five formulae in
that poll.
De Moivre's formula is a direct consequence of Euler's formula.
In addition, Euler elaborated the theory of higher transcendental functions by
introducing the gamma function and introduced a new method for solving quartic
equations. He also found a way to calculate integrals with complex limits,
foreshadowing the development of modern complex analysis, and invented the
calculus of variations including its best-known result, the Euler –Lagrange equation.
Euler also pioneered the use of analytic methods to solve number theory
problems. In doing so, he united two disparate branches of mathematics and
introduced a new field of study, analytic number theory. In breaking ground for this
new field, Euler created the theory of hypergeometric series, q-series, hyperbolic
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trigonometric functions and the analytic theory of continued fractions. For example,
he proved the infinitude of primes using the divergence of the harmonic series, and
he used analytic methods to gain some understanding of the way prime numbers are
distributed. Euler's work in this area led to the development of the prime number
theorem.
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X cm
AIM : To measure the height of a tree by using trigonometry ratios and rules.
TOOLS NEEDED:
MEASURING TAPE PROTACTOR BOOK TABLE
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WAYS TO MEASURE THE HEIGHT OF THE TREE.
1. A distance at 400cm length was measured between the lowest point of the
tree and the place where a person need to stand.
PICTURE 1
2. A table was placed at the end of 400cm length-distance. This is the place
where we want to measure the angle of elevation. A student is required tomeasure the angle of elevation from the table.
PICTURE 2
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3. A book is used to measure the angle of elevation. The angle is measured
using a protactor.
PICTURE 3
PICTURE 4
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4. Then, we measured the height of the table that we used to discover the angle.
PICTURE 5
5. Since, we got the angle and the length of the base, now we can calculate the
height of the tree.
400 cm
50⁰
74 cm
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Using trigonometry ratios,
Then, we add the height of the table,
Thus, the height of the tree is 550.7 cm.
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AIM: To measure the height of the tree using shadow.
PICTURE 1
Tools Needed.
Measuring tape rope
X cm
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WAYS TO MEASURE THE HEIGHT OF THE TREE USING SHADOW.
ACTIVITY A
1. This activity was carried out under a sunny day so that we will get the
shadow.
2. The height of the student and the length of the shadow of the student is
measured using measuring tape.
PICTURE 2
3. After we get the height and the length, we use trigonometry formula to find out
the angle between the student and his shadow.
172 cm
x
208 cm
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ACTIVITY B
1. You need to do this activity under a sunny day so that we will easily get
the shadow of the tree.
PICTURE 3
2. By using the rope, we mark the highest point of the tree on tha shadow.
PICTURE 4
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3. Then, we measure the length of the shadow by using measuring tape.
PICTURE 5
4. After we get the length of the shadow, we apply it in trigonometry rules.
We use the angle that we get based on activity A.
Thus, the height of the tree is 135.62 cm.
39.59
164 cm
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THIRD METHOD
AIM OF MEASUREMENT:
To calculate the height of a pillar using the concept of trigonometry.
TOOLS USED FOR MEASUREMENT:
Measuring Tape Rope
Protactor Pillar
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STEPS OF MEASUREMENT:
1. Firstly, put one end of a rope at the highest peak of the pillar and hold it
permanently as shown in Picture 1. Then, put another end of the same rope onto the
flat brick floor as shown in Picture 2.
Picture 1
Picture 2
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2. Secondly, measure the angle of elavation using a protactor.
Picture 3
3. Thirdly, remove the end of rope which is placed at the peak of the pillar to the
bottom of the pillar.
Picture 4
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4. After that, measure the straight line of the rope using a measuring tape.
Picture 5
Picture 6
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CALCULATION OF THE MEASUREMENT:
?
28°
328.5 cm
tan 28° =
Length of the pillar = tan 28° x 328.5 cm
= 174.7 cm
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FOURTH METHOD
AIM OF MEASUREMENT:
To calculate the height of a pillar using the concept of trigonometry.
TOOLS USED FOR MEASUREMENT:
Measuring Tape Rope
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STEPS OF MEASUREMENT:
1. Firstly, search for the most suitable distance to see the highest peak of the tree.
Then, measure the distance between the bottom of the tree and the most suitablepoint using a measuring tape. Take the measurement as shown in Picture 2.
Picture 1
Picture 2
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2. Secondly, the observant must bend his body until 180° to see the highest peak of
the tree. This action is done to get an angle of 45° from the eyes of the observant to
the higest peak of the tree. A picture of the highest peak of the tree has been taken
by the observant while he is bend. The picture is shown in Picture 4.
Picture 3
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Picture 4
3. Thirdly, measure the distance once again to ensure that the measurement is
accurate.
Picture 5
Picture 6 Picture 7
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CALCULATION OF THE MEASUREMENT:
128 cm
tan 45° =
Length of the tree = tan 45° x 128 cm
= 128 cm
45°
?