Geometry and Trigonometry.pdf

GEOMETRY & TRIGONOMETRY Geometry Geometry Geo means earth and metre means measurement is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Branches of Geometry Algebraic...

GEOMETRY & TRIGONOMETRY Geometry Geometry Geo means earth and metre means measurement is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Branches of Geometry Algebraic geometry Discrete geometry Differential geometry Euclidean geometry & Non- Euclidean geometry Convex geometry Egyptian Geometry Vladimir Golenishchev known as Wladimir or WoldemarGolenischeff was one of the first and one of the most accomplished Russian Egyptologists. He was one of the founders of the Cairo School of Egyptology and one of the most recognized authorities of the schools of Assyriology and Egyptology in Russia. contribution Moscow Mathematical Papyrus also named the Golenishchev Mathematical Papyrus 14th problem of the Moscow Mathematical Papyrus Ahmes was an ancient Egyptian scribe who lived towards the end of the Fifteenth Dynasty (and of the Second Intermediate Period) and the beginning of the Eighteenth Dynasty (and of the New Kingdom). Ahmes He transcribed the Rhind Mathematical Papyrus, a work of ancient Egyptian mathematics that dates to approximately 1550 BC; he is the earliest contributor to mathematics whose name is known. Ahmes claimed not to be the writer of the work but rather just the scribe. He claimed the material came from an even older document from around 2000 B.C. Rhind Mathematical Papyrus is one of the best known examples of ancient Egyptian mathematics. Ø is the larger, but younger, of the two. consists of geometry problems A portion of the Rhind Papyrus  Greek Geometry Thales of Miletus 6th century BCE He began the process of using deduction from first principles and believed that he travelled to Egypt and Babylon, picking up geometric techniques from these cultures, and he certainly would have had access to their work. Thales strongly believed that reasoning should supersede experimentation and intuition, and began to look for solid principles upon which he could build theorems. This introduced the idea of proof into geometry and he proposed some axioms that he believed to be mathematical truths. Thales A circle is bisected by any of its diameters The base angles of an isosceles triangle are equal When two straight lines cross, the opposing angles are equal An angle drawn in a semi-circle is a right angle Two triangles with one equal side and two equal angles are congruent Pythagoras of Samos (570 –495 BC) Øwas an ancient Ionian Greek philosopher, polymath and the eponymous founder of Pythagoreanism. ØPythagoras was credited with many mathematical and scientific discoveries, including the Pythagorean theorem, Pythagorean tuning, the five regular solids, the Theory of Proportions, the sphericity of the Earth, and the identity of the morning and evening stars as the planet Venus. Pythagorean theorem is a cornerstone of math that helps us find the missing side length of a right triangle. In a right triangle with sides A, B, and hypotenuse C, the theorem states that A² + B² = C². Euclid of Alexandria was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", Øknown for the Elements treatise, which established the foundations of geometry that largely dominated the field until the early 19th century. Euclidean geometry studies solid figures and planes on the foundation of theorems and axioms. He describes this in his book of geometry, Euclid’s Elements. Euclid’s method consists in believing a small set of axioms and deducing many other theorems from these. Archimedes of Syracuse ( 287-212 BC ) was an Ancient Greek mathematician, physicist, engineer, astronomer, and inventor from the ancient city of Syracuse in Sicily. he is regarded as one of the leading scientists in classical antiquity. Considered the greatest mathematician of ancient history, and one of the greatest of all time Archimedes anticipated modern calculus and analysis by applying the concept of the infinitely small and the method of exhaustion to derive and rigorously prove a range of geometrical theorems. These include the area of a circle, the surface area and volume of a sphere, the area of an ellipse, the area under a parabola, the volume of a segment of a paraboloid of revolution, the volume of a segment of a hyperboloid of revolution, and the area of a spiral. Archimedes Archimedes made significant advances in geometry, particularly with his work on the area and volume of geometric shapes. His method for approximating the value of π (pi) by inscribing and circumscribing polygons around a circle was groundbreaking Apollonius of Perga 249 BC- 190 BC was an ancient Greek geometer and astronomer known for his work on conic sections. Apollonius is generally considered among the greatest mathematicians of antiquity. he brought them to the state prior to the invention of analytic geometry 'The Great Geometer'. Apollonius his famous book Conics introduced terms which are familiar to us today such as parabola, ellipse and hyperbola. Conics consisted of 8 books. Books one to four form an elementary introduction to the basic properties of conics. book one the relations satisfied by the diameters and tangents of conics are studied while in book two Apollonius investigates how hyperbolas are related to their asymptotes, and he also studies how to draw tangents to given conics.  Chinese Geometry Shiing-Shen Chern was widely regarded as a leader in geometry and one of the greatest mathematicians of the 20th century and was awarded the Wolf Prize for his contributions to mathematics. “More than any other mathematician, Shiing-Shen Chern defined the subject of global differential geometry, a central area in contemporary mathematics. In work that spanned almost seven decades, he helped to shape large areas of modern mathematics Shiing-Shen Chern His work extended over all the classic fields of differential geometry as well as more modern ones including general relativity, invariant theory, characteristic classes, cohomology theory, Morse theory, Fiber bundles, Sheaf theory, Cartan's theory of differential forms, etc. His work included areas currently-fashionable, perennial, foundational, and nascent Shing-Tung Yau is a Chinese-American mathematician. He is the director of the Yau Mathematical Sciences Center at Tsinghua University and Professor Emeritus at Harvard University. Yau is considered one of the major contributors to the development of modern differential geometry and geometric analysis. Shing-Tau Yau Yau's work are also seen in the mathematical and physical fields of convex geometry, algebraic geometry, enumerative geometry, mirror symmetry, general relativity, and string theory, while his work has also touched upon applied mathematics, engineering, and numerical analysis.  Indian Geometry Baudhayana (800 BC – 740 BC) is said to be the Original Mathematician behind the Pythagoras Theorem. was an Indian Mathematician who was born in 800 BC and dies in 740 BC. He was a Vedic brahmin priest. He is said to be the original founder of Pythagoras’s Theorem. He was the first-ever Indian Mathematician who came up with several concepts in Mathematics. Baudhayana sutras are a group of Vedic Sanskrit texts which cover dharma, daily ritual, mathematics and is one of the oldest Dharma-related texts of Hinduism that have survived into the modern age from the 1st-millennium BCE. Aryabhata (476–550 CE) was the first of the major mathematician- astronomers from the classical age of Indian mathematics and Indian astronomy. His works include the Āryabhaṭīya and the Arya-siddhanta. Aryabhatiya or Aryabhatiyam a compendium of mathematics and astronomy, was referred to in the Indian mathematical literature and has survived to modern times. The mathematical part of the Aryabhatiya covers arithmetic, algebra, plane trigonometry, and spherical trigonometry. It also contains continued fractions, quadratic equations, sums-of-power series, and a table of sines. Brahmagupta (598-668 CE) was an Indian mathematician and astronomer. in 628 CE, Brahmagupta first described gravity as an attractive force, and used the term "gurutvākarṣaṇam (गुरुत्वाकर्षणम्)" in Sanskrit to describe it also credited with the first clear description of the quadratic formula (the solution of the quadratic equation) his main work, the Brāhma-sphuṭa-siddhānta. Brahmagupta theorem In geometry, Brahmagupta's theorem states that if a cyclic quadrilateral is orthodiagonal (that is, has perpendicular diagonals), then the perpendicular to a side from the point of intersection of the diagonals always bisects the opposite side. It is named after the Indian mathematician Brahmagupta (598-668). Acharya Virasena (792-853 CE) was an Indian mathematician and Jain philosopher and scholar. He was also known as a famous orator and an accomplished poet. Virasena was a noted mathematician. He gave the derivation of the volume of a frustum by a sort of infinite procedure. He worked with the concept of ardhachheda: the number of times a number can be divided by 2. This coincides with the binary logarithm when applied to powers of two, but gives the 2-adic order rather than the logarithm for other integers. Acharya Virasena Virasena gave the approximate formula C = 3d + (16d+16)/113 to relate the circumference of a circle, C, to its diameter, d. For large values of d, this gives the approximation π ≈ 355/113 = 3.14159292..., which is more accurate than the approximation π ≈ 3.1416 given by Aryabhata in the Aryabhatiya.  French Geometry René Descartes 31 March 1596 – 11 February 1650) was a French philosopher, scientist, and mathematician, widely considered a seminal figure in the emergence of modern philosophy and science. Mathematics was paramount to his method of inquiry, and he connected the previously separate fields of geometry and algebra into analytic geometry. x for unknown; exponential notation Descartes "invented the convention of representing unknowns in equations by x, y, and z, and knowns by a, b, and c". He also "pioneered the standard notation" that uses superscripts to show the powers or exponents; for example, the 2 used in x2 to indicate x squared. Analytic geometry also known as coordinate geometry or Cartesian geometry, is the study of geometry using a coordinate system , is the study of geometry using a coordinate system. is used in physics and engineering, and also in aviation, rocketry, space science, and spaceflight. Blaise Pascal (19 June 1623 – 19 August 1662) was a French mathematician, physicist, inventor, philosopher, and Catholic writer. His earliest mathematical work was on projective geometry; he wrote a significant treatise on the subject of conic sections at the age of 16. projective geometry projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting, projective space, and a selective set of basic geometric concepts. conic section conic section, conic or a quadratic curve is a curve obtained from a cone's surface intersecting a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a special case of the ellipse, though it was sometimes called as a fourth type. Trigonometry Introduction What is trigonometry Trigonometry is the branch of mathematics that explores the relationship between the length of the triangles sides and angles 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. THE SIX TRIGONOMETRIC FUNCTIONS There are six functions of an angle commonly used in trigonometry. The names and abbreviations are sine (sin), cosine (cos), tangent (tan), cotangent (cot), secant (sec), and cosecant (csc). six trigonometric functions in relation to a right riangle 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. Trigonometric function are used 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.  Plane trigonometry Problems involving angles and distances in one plane. Spherical trigonometry Types of Applications to similar problems in more than trigonometry one plane of three-dimensional space. Core trigonometry deals with the ratio between the sides of a right triangle and its angles.  Its applications are in various fields like Applications oceanography, seismology, meteorology, physical sciences, astronomy, acoustics, navigation, electronics, etc. of trigonometry It is also helpful to measure the height of the mountain, find the distance of long rivers, etc. HISTORY OF TRIGONOMETRY Classical trigonometry ( Ancient- Medieval Period ) Modern trigonometry ( 16th century onward ) Classical trigonemtry Several ancient civilizations-in particular, the egyptian, babylonian, hindus, and Islam possessed a considerable knowledge of practical geometry, including some concepts that were a prelude to trigonometry  Egyptian and Babylonian 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 (c. 1900 BC), some have even asserted that the ancient Babylonians had a table of secants but does not work in this context as without using circles and angles in the situation modern trigonometric notations will not apply. 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 and Babylonians possessed knowledge of the ratios of sides in similar triangles, but they lacked a concept of angle measure. Their focus was on practical applications like building pyramids and astronomical observations. - Rhind Mathematical Papyrus (c. 1680- 1620 BC): This Egyptian document contains problems related to the "seked," a concept similar to the cotangent of an angle, indicating early use of trigonometric ideas.  Egyptian and Babylonian For example: '' if a pyramid is 250 cubits long, what is its seked'' the solutuion is given as 5 1/25 palm per cubit; and since one cubitequals 7 palms, this fraction is equivalent to the pure ratio 18/25  GREEKS-Hellinistic period ( 3rd century onwards ) -Hipparchus of Nicaea (c. 190-120 BC): Often considered the "father of trigonometry," Hipparchus developed the first trigonometric tables, relating angles to chord lengths in a circle. These tables were crucial for astronomy and surveying. - Astronomical Applications: Hipparchus's work in trigonometry was driven by his astronomical research. He used trigonometry to calculate the distances and sizes of celestial objects, leading to more accurate predictions of planetary movements. (HIPPARCHUS MEASUREMENT OF THE DISTANCE TO THE MOON)  GREEKS-Hellinistic period ( 3rd century onwards ) - 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 form of plane trigonometry -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). -This figure illustrates the relationship between a central angle ( an angle formed by two radii in a circle) and its chord AB ( equal to one side of an inscribed triangle ) 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.  GREEKS-Hellinistic period ( 3rd century onwards ) -Claudius Ptolemy was a 2nd century Greek mathematician, astronomer and geographer famous for his controversial geocentric theory of the universe, which would form the basis of our understanding of the motions of stars and planets for over than a thousand years. Ptolemy's Almagest (circa 150 AD) built upon Hipparchus's work and established trigonometry as a formalized branch of mathematics. he used chord lengths to define trigonometric functions and created detailed tables of chords. GREEKS-Hellinistic period ( 3rd century onwards ) The Almagest is divided into 13 books. Book 1 gives arguments for a geocentric spherical cosmos and introduces the necessary trigonometry, along with a trigonometry table, that allowed Ptolemy in subsequent books to explain and predict the motions of the Sun, Moon, planets, and stars. Book 2 uses spherical trigonometry to explain cartography and astronomical phenomena (such as the length of the longest day) characteristic of various localities. Book 3 deals with the motion of the Sun and how to predict its position in the zodiac at any given time, and Books 4 and 5 treat the more difficult problem of the Moon’s motion. Book 5 also describes the construction of instruments to aid in these investigations. The theory developed to this point is applied to solar and lunar eclipses in Book 6. The next Greek mathematician to produce a table of chords was Menelaus in about 100 AD. Menelaus worked in Rome producing six books of tables of chords which have been lost but his work on spherics has survived and is the earliest known work on spherical trigonometry. Pythagorean Identities Pythagorean identities, as the name suggests, are derived from the Pythagoras theorem. According to this theorem, in any right-angled triangle, the square of the hypotenuse (longest side) is equal to the sum of the squares of the other two sides (legs). This theorem can be applied to trigonometric ratios (as they are defined for a right-angled triangle) that results in Pythagorean identities. n Pythagoras was a Greek Pythagorean identities are important identities in trigonometry that are philosopher, mathematician, and derived from the Pythagoras theorem. These identities are used in solving many trigonometric problems where one trigonometric ratio is given and the founder of the Pythagorean other ratios are to be found. The fundamental Pythagorean identity gives the brotherhood. He was born in relation between sin and cos and it is the most commonly used Pythagorean Samos, Greece in around 570 B.C. identity which says: sin2θ + cos2θ = 1 (which gives the relation between sin and cos) There are other two Pythagorean identities that are as follows: Pythagorean Identities The fundamental Pythagorean identity gives the relation between sin and cos and it is the most commonly used Pythagorean identity which says: sin2θ + cos2θ = 1 (which gives the relation between sin and cos) There are other two Pythagorean identities that are as follows: sec2θ - tan2θ = 1 (which gives the relation between sec and tan) csc2θ - cot2θ = 1 (which gives the relation between csc and cot)  Chinese period The development of trigonometry in China during the classical period, roughly spanning from the ancient dynasties to the Song dynasty (960-1279 AD), was distinct from the trajectory seen in other civilizations. While the Chinese excelled in various mathematical fields, including algebra, geometry, and astronomy, they did not develop trigonometry as an independent subject in the same way as the Greeks or later the Islamic mathematicians.  Indian Contributions (5th-6th Centuries AD) the next signifant development of trigonometry were in india. Influential works from the 4th-5th century, known as the siddhantas, first defined the sine as the modern relationship between half angle and a half chord, while also defining the cosine, versine (1-cosine), and inverse sine Aryabhata (c. 476-550 AD): This Indian mathematician introduced the sine function (originally called "jya") and developed tables of sine values. His work significantly advanced trigonometry. Aryabhata collected and expanded upon the developments of the siddhantas in an important work called the Aryabhatiya  Indian Contributions (5th-6th Centuries AD) In the 6th century, Varahamihira used the formula sin x = cos(p/2 - x), sin2x + cos2x = 1, and (1 - cos 2x)/2 = sin2x. In the 7th century, Bhaskara 1 produced a formula for calculating the sine of an acute angle without the used of a table. Later in the 7th century, Brahmagupta redeveloped the formula 1 sin^2 (x) = cos^2(x)=sin^2 /2 - x  Indian Contributions (5th-6th Centuries AD) 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 power series expansions of sine, cosine, tangent, and arctangent.  Medieval islamic world (5th-6th Centuries AD) Al-Khwarizmi (c. 780-850 AD): While primarily known for his work in algebra, Al- Khwarizmi also contributed to trigonometry, further developing the study of trigonometric functions. Abu al-Wafa' (c. 940-998 AD): This Persian mathematician introduced all six trigonometric functions and developed more accurate tables of sine and tangent values. Nasir al-Din al-Tusi (c. 1201-1274 AD): He established trigonometry as an independent mathematical discipline, developing spherical trigonometry into its present form. Modern trigonometry ( 16th Century onwards )  European Renaissance and Beyond Regiomontanus (1436-1476): This German mathematician wrote one of the earliest European works on trigonometry, De Triangulis, which further developed the field. Bartholomaeus Pitiscus (1561-1613): He coined the term “trigonometry” and published a significant work on the subject. 17th Century: The development of analytical geometry and symbolic algebra led to a shift towards a more analytical approach to trigonometry. Leonhard Euler (1707-1783): Euler’s work fully incorporated complex numbers into trigonometry, leading to a deeper understanding of trigonometric functions.  Key Developments in Modern Trigonometry Development of Trigonometric Identities: These identities are crucial for simplifying expressions and solving trigonometric equations. Trigonometric Series: The development of infinite series representations of trigonometric functions opened new avenues for analysis and applications. Applications in Various Fields: Modern trigonometry is essential in fields like physics, engineering, navigation, and computer graphics.  Conclusion Geometry & trigonometry has evolved from its early roots in practical applications to a sophisticated branch of mathematics with numerous applications in modern science and technology. The classical period laid the foundation with the development of solving practical problems, trigonometric tables and functions, while the modern period witnessed a shift towards a more analytical and abstract approach, leading to a deeper understanding of the subject and its vast applications. Questions and Clariffication Thank You!  MEMBERS Bastida, Keith Jersey Geslaga,Joan Cañeda,Maxine Pandoro, Rhycel Dandan,John Vincent Sabellano, EJ Dwight Genilza, Kimberly Rose