...1. Hipparchus is known as the inventor of trigonometry. He is the first person who we have documentary evidence of the use of trigonometry. He had the first computation of the table of chords. Menelaus is another person known as an inventor of trigonometry. He expanded the work of Hipparchus, with focus on traversals. The third person known well for his contributions to trigonometry is Ptolemy. He expanded the table of chords, which was founded by Hipparchus. 2. The first recorded step in the history of trigonometry is the creation of the first trigonometric tables. These were created by Hipparchus. These tables were compiled of trigonometric values of arc and chord for a series of angle measurements. 3. There are many important facts in the history of trigonometry. In my opinion, the most important are the findings of Hipparchus. He developed the first table of chords. Without these, trigonometry might not be as developed as it is now. 4....
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...Project - Article Research Hamid Naderi Yeganeh uses a series of cosine formulas to create art masterpieces into shapes that look like everyday surroundings, such as animals. I found it interesting that Yegeneh, being a math whiz, takes trigonometry and makes beautiful art with it. He started out with just making symmetrical shapes, which eventually led him into creating art with the trigonometric functions by creating endpoints and segments. Some of his work includes a fox, a bird in flight, and a multitude of pieces using thousands of varied circles. I am quite intrigued by how the author of the article, Stephy Chung, compares Yeganeh to one of the most famous artists of all time in Leonardo Da Vinci. Da Vinci is responsible for painting the...
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...UNIT CIRCLE TRIGONOMETRY The Unit Circle is the circle centered at the origin with radius 1 unit (hence, the “unit” circle). The equation of this circle is x 2 + y 2 = 1 . A diagram of the unit circle is shown below: y 1 x2 + y2 = 1 x -2 -1 -1 1 2 -2 We have previously applied trigonometry to triangles that were drawn with no reference to any coordinate system. Because the radius of the unit circle is 1, we will see that it provides a convenient framework within which we can apply trigonometry to the coordinate plane. Drawing Angles in Standard Position We will first learn how angles are drawn within the coordinate plane. An angle is said to be in standard position if the vertex of the angle is at (0, 0) and the initial side of the angle lies along the positive x-axis. If the angle measure is positive, then the angle has been created by a counterclockwise rotation from the initial to the terminal side. If the angle measure is negative, then the angle has been created by a clockwise rotation from the initial to the terminal side. θ in standard position, where y Terminal side θ is positive: θ in standard position, where y θ is negative: θ Initial side Initial side x θ x Terminal side Unit Circle Trigonometry Drawing Angles in Standard Position Examples The following angles are drawn in standard position: 1. θ = 40 y 2. θ = 160 θ y x θ x y 3. θ = −320 θ x Notice that...
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...Difference Between Euclidean and Spherical Trigonometry 1 Non-Euclidean geometry is geometry that is not based on the postulates of Euclidean geometry. The five postulates of Euclidean geometry are: 1. Two points determine one line segment. 2. A line segment can be extended infinitely. 3. A center and radius determine a circle. 4. All right angles are congruent. 5. Given a line and a point not on the line, there exists exactly one line containing the given point parallel to the given line. The fifth postulate is sometimes called the parallel postulate. It determines the curvature of the geometry’s space. If there is one line parallel to the given line (like in Euclidean geometry), it has no curvature. If there are at least two lines parallel to the given line, it has a negative curvature. If there are no lines parallel to the given line, it has a positive curvature. The most important non-Euclidean geometries are hyperbolic geometry and spherical geometry. Hyperbolic geometry is the geometry on a hyperbolic surface. A hyperbolic surface has a negative curvature. Thus, the fifth postulate of hyperbolic geometry is that there are at least two lines parallel to the given line through the given point. 2 Spherical geometry is the geometry on the surface of a sphere. The five postulates of spherical geometry are: 1. Two points determine one line segment, unless the points are antipodal (the endpoints of a diameter of the sphere), in which case ...
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...The History of Geometry Geometry, from the ancient Greek “geo” meaning Earth and “metron” meaning measurement, arose as the field of knowledge dealing with spatial relationships. Geometry was revolutionized by Euclid, who introduced mathematical rigor still in use today. In modern times, geometric concepts have been generalized to a high level of abstraction and complexity, and have been subjected to the methods of calculus and abstract algebra, so that many modern branches of the field are barely recognizable as the descendants of early geometry. The earliest recorded beginnings of geometry can be traced to early peoples, who discovered obtuse triangles in the ancient Indus Valley, and ancient Babylonia from around 3000 BC. Early geometry was a collection of empirically discovered principles concerning lengths, angles, areas, and volumes, which were developed to meet some practical need in surveying, construction, astronomy, and various crafts. Among these were some surprisingly sophisticated principles, and a modern mathematician might be hard put to derive some of them without the use of calculus. Greek Geometry The early history of Greek geometry is unclear, because no original sources of information remain and all of our knowledge is from secondary sources written many years after the early period. For the ancient Greek mathematicians, geometry was the crown jewel of their sciences, reaching a completeness and perfection of methodology that no other branch of their...
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...Bottom of Form * Home Page » * Other Topics History of Indian Mathematics In: Other Topics History of Indian Mathematics MATHEMATICS IN INDIA The history of maths in india is very great & eventful.Indians gave the system of numerals, zero, geometry & equations to the world. The great Indian mathematician Aryabhata (476-529) wrote the Aryabhatiya ─ a volume of 121 verses. Apart from discussing astronomy, he laid down procedures of arithmetic, geometry, algebra and trigonometry. He calculated the value of Pi at 3.1416 and covered subjects like numerical squares and cube roots. Aryabhata is credited with the emergence of trigonometry through sine functions. Around the beginning of the fifteenth century Madhava (1350-1425) developed his own system of calculus based on his knowledge of trigonometry. He was an untutored mathematician from Kerala, and preceded Newton and Liebnitz by a century. The twentieth-century genius Srinivas Ramanujan (1887-1920) developed a formula for partitioning any natural number, expressing an integer as the sum of squares, cubes, or higher power of a few integers. Origin of Zero and the Decimal System The zero was known to the ancient Indians and most probably the knowledge of it spread from India to other cultures. Brahmagupta (598-668),who had worked on mathematics and astronomy, was the head of the astronomy observatory in Ujjain, which was at that point of time, the foremost mathematical centre in India; he and Bhaskar...
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...College Trigonometry Version π Corrected Edition by Carl Stitz, Ph.D. Lakeland Community College Jeff Zeager, Ph.D. Lorain County Community College July 4, 2013 ii Acknowledgements While the cover of this textbook lists only two names, the book as it stands today would simply not exist if not for the tireless work and dedication of several people. First and foremost, we wish to thank our families for their patience and support during the creative process. We would also like to thank our students - the sole inspiration for the work. Among our colleagues, we wish to thank Rich Basich, Bill Previts, and Irina Lomonosov, who not only were early adopters of the textbook, but also contributed materials to the project. Special thanks go to Katie Cimperman, Terry Dykstra, Frank LeMay, and Rich Hagen who provided valuable feedback from the classroom. Thanks also to David Stumpf, Ivana Gorgievska, Jorge Gerszonowicz, Kathryn Arocho, Heather Bubnick, and Florin Muscutariu for their unwaivering support (and sometimes defense) of the book. From outside the classroom, we wish to thank Don Anthan and Ken White, who designed the electric circuit applications used in the text, as well as Drs. Wendy Marley and Marcia Ballinger for the Lorain CCC enrollment data used in the text. The authors are also indebted to the good folks at our schools’ bookstores, Gwen Sevtis (Lakeland CC) and Chris Callahan (Lorain CCC), for working with us to get printed copies to the students...
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...Willy Ngin AMAT 452: History of Mathematics Mathematical History of China and India Since the beginning of time mathematics has been a part of history. Throughout time without mathematics we wouldn’t have been able to make fundamental advances in science, engineering, technology and much more. Although every country has different histories, cultures and lifestyles; one thing that remains the same is the universal language of Mathematics. If you go to any country in the world, mathematics will always be the same. Addition will always be addition and subtraction will always be subtraction anywhere. Some of the countries who have been able to help further our discoveries and advances in mathematics were China and India. China’s history included many different wars which led to a lot of different dynasties taking over the country. Still, ”the demands of the empire for administrative services, including surveying, taxation, and calendar making, required that many civil servants be competent in certain areas of mathematics” (Katz, 2009, p. 197). It wasn’t until 1984 when they opened the tombs that they found some of the mathematic history. “Among the books was discovered a mathematics text written on 200 bamboo strips. This work, called the Suan shu shu (Book of Numbers and Computation), is the earliest extant text of Chinese mathematics.” (Katz, 2009, p. 196). This work was created during the Han Dynasty. It consisted of different problems and their solution. Alongside...
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...MAPÚA INSTITUTE OF TECHNOLOGY Department of Mathematics COURSE SYLLABUS 1. Course Code: Math 10-3 2. Course Title: Algebra 3. Pre-requisite: none 4. Co-requisite: none 5. Credit: 3 units 6. Course Description: This course covers discussions on a wide range of topics necessary to meet the demands of college mathematics. The course discussion starts with an introductory set theories then progresses to cover the following topics: the real number system, algebraic expressions, rational expressions, rational exponents and radicals, linear and quadratic equations and their applications, inequalities, and ratio, proportion and variations. 7. Student Outcomes and Relationship to Program Educational Objectives Student Outcomes Program Educational Objectives 1 2 (a) an ability to apply knowledge of mathematics, science, and engineering √ (b) an ability to design and conduct experiments, as well as to analyze and interpret from data √ (c) an ability to design a system, component, or process to meet desired needs √ (d) an ability to function on multidisciplinary teams √ √ (e) an ability to identify, formulate, and solve engineering problems √ (f) an understanding of professional and ethical responsibility √ (g) an ability to communicate effectively √ √ (h) the broad education necessary to understand the impact of engineering solutions in the global and societal context √ √ (i) a recognition of the need for, and an ability to engage...
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...MATHEMATICS IN INDIA The history of maths in india is very great & eventful.Indians gave the system of numerals, zero, geometry & equations to the world. The great Indian mathematician Aryabhata (476-529) wrote the Aryabhatiya ─ a volume of 121 verses. Apart from discussing astronomy, he laid down procedures of arithmetic, geometry, algebra and trigonometry. He calculated the value of Pi at 3.1416 and covered subjects like numerical squares and cube roots. Aryabhata is credited with the emergence of trigonometry through sine functions. Around the beginning of the fifteenth century Madhava (1350-1425) developed his own system of calculus based on his knowledge of trigonometry. He was an untutored mathematician from Kerala, and preceded Newton and Liebnitz by a century. The twentieth-century genius Srinivas Ramanujan (1887-1920) developed a formula for partitioning any natural number, expressing an integer as the sum of squares, cubes, or higher power of a few integers. Origin of Zero and the Decimal System The zero was known to the ancient Indians and most probably the knowledge of it spread from India to other cultures. Brahmagupta (598-668),who had worked on mathematics and astronomy, was the head of the astronomy observatory in Ujjain, which was at that point of time, the foremost mathematical centre in India; he and Bhaskar the second (1114-1185), who reached understanding on the number systems and solving equations, have together provided many rules for arithmetical...
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...th Trying trig Everything you need to Know By: Noah Gregory subject Page Radians & Degree Measure 3 Unite Circle 4 Right Triangle Trig 5-7 trig functions of any angle 8-10 graphs 11-15 using fundamental trig identities 16-17 verifying trig identities 18-20 solving trig equations 21-23 sum & difference formulas 24 law of sines 25-27 laws of cosines 28-29 vectors 30-31 Definitions 32-33 Radians & Degree Measure Converting radians to degrees: To convert radians to degrees, we make use of the fact that p radians equals one half circle, or 180º. [pic] This means that if we divide radians by p, the answer is the number of half circles. Multiplying this by 180º will tell us the answer in degrees. So, to convert radians to degrees, multiply by 180/p, like this: [pic] To convert degrees to radians, first find the number of half circles in the answer by dividing by 180º. But each half circle equals p radians, so multiply the number of half circles by p. Example 1 (p= Pie) 10º in radians would be 18 Radians. First put your degree over 1 R= 10°/1 (p/180°) Next multiply & divide & you will get 18p ------------------------------ Example 2 1.4 Radians would be 80.2° put your radian over 1 D= 1.4/1 (180°/p) Next multiply & divide & you will get 80.2 ° Unite Circle ...
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...occurs at 4:00 a.m. with a depth of 6 meters. Low tide occurs at 10:00 a.m. with a depth of 2 meters. To find , A, B, C, and D have to be solved. Below is the solution in terms of t, hours after midnight: The vertical shift, D, is found by using the maximum, high tide, and minimum, low tide, and finding the mean, average. The amplitude is the difference between high tide and low tide. Once that is done, divide the solution by 2. A period is one full cycle of high tide to high tide. Since the first high tide occurs 4 hours after midnight and the second high tide occurs 16 hours after midnight, 16-4=12. This means that a period lasts for 12 hours. Use 12 hours as the period, and then solve for B. To find the phase shift we know the formula is Use the information in the row above for B. The shift occurs from 0 to 4 hours after midnight. Therefore the phase shift is 4. Use this information to find C. The information above shows the work, which leads to , where hours after midnight. Figure 1, to the left, (Desmos, 2011), shows the graph from 0≤ t ≥24. High tide, or the maximum, at 4 am is 6 meters, and low tide, or the minimum, at 10 am is 2 meters. FIGURE 1: over 24 hour period Part 2: At noon, the water depth is 3 meters. This is shown in two different ways. The first way is to look at the graph in Figure 1, which shows you when t = 12, y (water depth) = 3 meters. The other way is to use the equation that was found to solve the problem. This way is demonstrated...
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...historically been called "the calculus of infinitesimals", or "infinitesimal calculus". The word "calculus" comes from Latin (calculus) and refers to a small stone used for counting. More generally, calculus (plural calculi) refers to any method or system of calculation guided by the symbolic manipulation of expressions. Some examples of other well-known calculi are propositional calculus, calculus of variations, lambda calculus, and process calculus. he ancient period introduced some of the ideas that led to integral calculus, but does not seem to have developed these ideas in a rigorous and systematic way. Calculations of volume and area, one goal of integral calculus, can be found in the Egyptian Moscow papyrus (c. 1820 BC), but the formulas are simple instructions, with no indication as to...
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...UNNECESSARY MATH Hengki Agus Rifa’i Math, as many say, seems to have been the most difficult subject to cope in schools. There are a number of reasons of saying so, ranging from the complexity of formula to its logical intricacies. Despite its terrible assumption in today’s status quo, math, in most curriculums across the world, is still included as one of compulsory subjects in almost all level of education as it is considered as the subject determining the students’ competency in other subjects. However, concerning the fact that many students remain fail, there are always reasons to claim that math should not be a compulsory subject in schools. First and foremost, it is important to think that math is not engaging for the students. Compared to other subjects, math is one of the least engaging subjects taught at schools. Subjects like chemistry are full of experiments which help them see what they are being taught in front of them. History, similarly, starts with telling stories, and even though that is not what the subject is really about, it offers a simple view into it. By contrast, math has almost nothing similar. It does not make sense for the students to use the formula of trigonometry to find the height of a tree or a building. In short, math does really make no significant understanding of what is being taught that the students get nothing from spending hours learning it in class. Secondly, it is also a fundamental principle of education that different people think...
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...Describe the work of Gauss, Bolyai and Lobachevsky on non-Euclidean geometry, including mathematical details of some of their results. What impact, if any, did the rise of non-Euclidean geometry have on subsequent developments in mathematics? Word Count: 1912 Euclidean geometry is the everyday “flat” or parabolic geometry which uses the axioms from Euclid’s book The Elements. Non-Euclidean geometry includes both hyperbolic and elliptical geometry [W5] and is a construction of shapes using a curved surface rather than an n-dimensional Euclidean space. The main difference between Euclidean and non-Euclidean geometry is the nature of parallel lines. There has been much investigation into the first five of Euclid’s postulates; mainly into proving the formulation of the fifth one, the parallel postulate, is totally independent of the previous four. The parallel postulate states “that, if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.” [W1] Many mathematicians have carried out extensive work into proving the parallel postulate and into the development of non-Euclidean geometry and the first to do so were the mathematicians Saccheri and Lambert. Lambert based most of his developments on previous results and conclusions by Saccheri. Saccheri looked at the three possibilities of the sum of the...
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