Trigonometry
History of Trigonometry
Principles of Trigonometry
Classical Trigonometry
Modern Trigonometry
Trigonometry
History of Trigonometry
Principles of Trigonometry
Classical Trigonometry
Modern Trigonometry
Trigonometric Functions
Has everything needed for a CBSE Student to make up a project on Trigonometry. I hope this helps you all...
Topics:-
Intro, Information, Formulas, Summary and overview
This lesson is the second of the series I am working on. It really should have come first, though. This lesson introduces trigonometry, detailing what it is, what is uses and a few important topics and formulas you'll find yourself using quite frequently.
Has everything needed for a CBSE Student to make up a project on Trigonometry. I hope this helps you all...
Topics:-
Intro, Information, Formulas, Summary and overview
This lesson is the second of the series I am working on. It really should have come first, though. This lesson introduces trigonometry, detailing what it is, what is uses and a few important topics and formulas you'll find yourself using quite frequently.
It is a ppt on Trigonometry for th students of class 10 .
The basic concepts of trigonometry are provided here with examples Hope that that you like it .!! Thankyou ..!! :)
Lecture Presentation on Trigonometry, types of angle, angle measurement, pythagorean theorem, trigonometric function, trigonometric relationship, circle function, co function, reference angle, odd even function,graphing of trigonometric function, special angles and terminology and history of trigonometry
This is a school standard presentation for class 10 students .
It will be very helpful to you all.
Hope you all like this .
And pass your exams with flying colors
The Presentation explains 'The Father Of Geometry' - "Euclid" with his life history and some of his most influential and remarkable works which contribute to The Modern Mathematics.
The data is present below the pictures so as to edit it as per your needs. I wanted to use good fonts and this was the only way i could do it as the fonts would not be available on your computer.
It is a ppt on Trigonometry for th students of class 10 .
The basic concepts of trigonometry are provided here with examples Hope that that you like it .!! Thankyou ..!! :)
Lecture Presentation on Trigonometry, types of angle, angle measurement, pythagorean theorem, trigonometric function, trigonometric relationship, circle function, co function, reference angle, odd even function,graphing of trigonometric function, special angles and terminology and history of trigonometry
This is a school standard presentation for class 10 students .
It will be very helpful to you all.
Hope you all like this .
And pass your exams with flying colors
The Presentation explains 'The Father Of Geometry' - "Euclid" with his life history and some of his most influential and remarkable works which contribute to The Modern Mathematics.
The data is present below the pictures so as to edit it as per your needs. I wanted to use good fonts and this was the only way i could do it as the fonts would not be available on your computer.
Trigonometry
History of Trigonometry
Principles of Trigonometry
Classical Trigonometry
Modern Trigonometry
Trigonometry
History of Trigonometry
Principles of Trigonometry
Classical Trigonometry
Modern Trigonometry
Trigonometric Functions
Mathematics(History,Formula etc.) and brief description on S.Ramanujan.Mayank Devnani
A brief description on the history of math, many famous mathematicians and also women mathematicians..
And very huge description ( bio-data, formulas etc.) on famous mathematician S.Ramanujan.
"Application of 3D and 2D geometry" explains the importance of geometry in our lives. Geometry is found everywhere from nature to human made machines. I have tried to inculcate all
its applications.
I hope it helps in providing guidance to those who are aspiring to understand geometry. I have taken help from internet and some books to acquire knowledge.
thank you for clicking my slide.
June 3, 2024 Anti-Semitism Letter Sent to MIT President Kornbluth and MIT Cor...Levi Shapiro
Letter from the Congress of the United States regarding Anti-Semitism sent June 3rd to MIT President Sally Kornbluth, MIT Corp Chair, Mark Gorenberg
Dear Dr. Kornbluth and Mr. Gorenberg,
The US House of Representatives is deeply concerned by ongoing and pervasive acts of antisemitic
harassment and intimidation at the Massachusetts Institute of Technology (MIT). Failing to act decisively to ensure a safe learning environment for all students would be a grave dereliction of your responsibilities as President of MIT and Chair of the MIT Corporation.
This Congress will not stand idly by and allow an environment hostile to Jewish students to persist. The House believes that your institution is in violation of Title VI of the Civil Rights Act, and the inability or
unwillingness to rectify this violation through action requires accountability.
Postsecondary education is a unique opportunity for students to learn and have their ideas and beliefs challenged. However, universities receiving hundreds of millions of federal funds annually have denied
students that opportunity and have been hijacked to become venues for the promotion of terrorism, antisemitic harassment and intimidation, unlawful encampments, and in some cases, assaults and riots.
The House of Representatives will not countenance the use of federal funds to indoctrinate students into hateful, antisemitic, anti-American supporters of terrorism. Investigations into campus antisemitism by the Committee on Education and the Workforce and the Committee on Ways and Means have been expanded into a Congress-wide probe across all relevant jurisdictions to address this national crisis. The undersigned Committees will conduct oversight into the use of federal funds at MIT and its learning environment under authorities granted to each Committee.
• The Committee on Education and the Workforce has been investigating your institution since December 7, 2023. The Committee has broad jurisdiction over postsecondary education, including its compliance with Title VI of the Civil Rights Act, campus safety concerns over disruptions to the learning environment, and the awarding of federal student aid under the Higher Education Act.
• The Committee on Oversight and Accountability is investigating the sources of funding and other support flowing to groups espousing pro-Hamas propaganda and engaged in antisemitic harassment and intimidation of students. The Committee on Oversight and Accountability is the principal oversight committee of the US House of Representatives and has broad authority to investigate “any matter” at “any time” under House Rule X.
• The Committee on Ways and Means has been investigating several universities since November 15, 2023, when the Committee held a hearing entitled From Ivory Towers to Dark Corners: Investigating the Nexus Between Antisemitism, Tax-Exempt Universities, and Terror Financing. The Committee followed the hearing with letters to those institutions on January 10, 202
Introduction to AI for Nonprofits with Tapp NetworkTechSoup
Dive into the world of AI! Experts Jon Hill and Tareq Monaur will guide you through AI's role in enhancing nonprofit websites and basic marketing strategies, making it easy to understand and apply.
2024.06.01 Introducing a competency framework for languag learning materials ...Sandy Millin
http://sandymillin.wordpress.com/iateflwebinar2024
Published classroom materials form the basis of syllabuses, drive teacher professional development, and have a potentially huge influence on learners, teachers and education systems. All teachers also create their own materials, whether a few sentences on a blackboard, a highly-structured fully-realised online course, or anything in between. Despite this, the knowledge and skills needed to create effective language learning materials are rarely part of teacher training, and are mostly learnt by trial and error.
Knowledge and skills frameworks, generally called competency frameworks, for ELT teachers, trainers and managers have existed for a few years now. However, until I created one for my MA dissertation, there wasn’t one drawing together what we need to know and do to be able to effectively produce language learning materials.
This webinar will introduce you to my framework, highlighting the key competencies I identified from my research. It will also show how anybody involved in language teaching (any language, not just English!), teacher training, managing schools or developing language learning materials can benefit from using the framework.
Operation “Blue Star” is the only event in the history of Independent India where the state went into war with its own people. Even after about 40 years it is not clear if it was culmination of states anger over people of the region, a political game of power or start of dictatorial chapter in the democratic setup.
The people of Punjab felt alienated from main stream due to denial of their just demands during a long democratic struggle since independence. As it happen all over the word, it led to militant struggle with great loss of lives of military, police and civilian personnel. Killing of Indira Gandhi and massacre of innocent Sikhs in Delhi and other India cities was also associated with this movement.
Honest Reviews of Tim Han LMA Course Program.pptxtimhan337
Personal development courses are widely available today, with each one promising life-changing outcomes. Tim Han’s Life Mastery Achievers (LMA) Course has drawn a lot of interest. In addition to offering my frank assessment of Success Insider’s LMA Course, this piece examines the course’s effects via a variety of Tim Han LMA course reviews and Success Insider comments.
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Unit 8 - Information and Communication Technology (Paper I).pdfThiyagu K
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Palestine last event orientationfvgnh .pptxRaedMohamed3
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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
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
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.
HISTORY OF TRIGONOMETRY
9. 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
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.
HISTORY OF TRIGONOMETRY
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.
HISTORY OF TRIGONOMETRY
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.
HISTORY OF TRIGONOMETRY CLASSICAL TRIGONOMETRY
14. 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
15. 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
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.‖
HISTORY OF TRIGONOMETRY CLASSICAL TRIGONOMETRY
17. 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
18. 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
20. 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
21. 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
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. 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
24. 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
25. 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
26. 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
27. Ancient Egypt and the
Mediterranean world
• 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.
HISTORY OF TRIGONOMETRY CLASSICAL TRIGONOMETRY
•
•
•
28. 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
29. 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
30. Classical Trigonometry
• Ancient Egypt and the
Mediterranean world
• India and the Islamic world
• Passage to Europe
HISTORY OF TRIGONOMETRY
31. 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
32. Indian and Islamic World
• Thus, rewriting Ptolemy's formula as
c/120 = sin B, where B = A/2, the 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.
HISTORY OF TRIGONOMETRY CLASSICAL TRIGONOMETRY
33. 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
34. 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
35. 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
36. 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
37. Indian and Islamic World
HISTORY OF TRIGONOMETRY
• 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.
CLASSICAL TRIGONOMETRY
38. 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
39. 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
40. Classical Trigonometry
• Ancient Egypt and the
Mediterranean world
• India and the Islamic world
• Passage to Europe
HISTORY OF TRIGONOMETRY
41. 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
42. 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
43. 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
44. 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
45. 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
46. 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.
HISTORY OF TRIGONOMETRY CLASSICAL TRIGONOMETRY
47. 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
48. 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
49. 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
50. 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.
HISTORY OF TRIGONOMETRY CLASSICAL TRIGONOMETRY