When I am studying the Law of Cosines, this theorem, which describes the relationship of the three sides of the triangles, reminds me of another special theorem using right triangles that I learned in primary school, which is the Pythagorean Theorem, as the Law of Cosines is the generalization of the Pythagorean Theorem. Surprisingly, I found myself not familiar with the proving of the Pythagorean Theorem. I knew what the Pythagorean Theorem was, but never paid attention to how to prove it. Also, the connections between these two theorems (the Law of Cosines and Pythagorean Theorem) interested me as well. I decided I would try to discover the proofs of both theorems while using the Pythagorean Theorem to prove the Law of Cosines.
The Pythagorean Theorem has been known since the days of the ancient Babylonians and Egyptians. It was also found by Greek mathematician Pythagoras (569-500) BC , the discovery most well known today, and also in a Chinese mathematical dissertation called Zhou Bei Suan Jing (1000 BC) . Another theorem, the Law of Cosines was found in Euclid’s Element predate the word “cosine”, which contains an early geometric theorem equivalent to the law of cosines.
Pythagorean Theorem
Pythagorean Theorem is a relationship of the length of three sides of a triangle containing a right angle and is often written as a² + b² = c². It states that “The area of the square built upon the hypotenuse of a right triangle is equal to the sum of the areas of the squares upon the remaining sides" , which can be shown as the picture below.
In the picture, the blue area, which is the area of square C, is equal to the red area, which is the sum of areas of square A and B.
So for the conclusion of the theorem, it states that the squar...
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...ythagorean Theorem and Distance Formula." Accessed April 12, 2014. http://www.academia.edu/1524756/Pythagorean_Theorem_and_Distance_Formula.
Cullen, Christopher. Astronomy and Mathematics in Ancient China The Zhou Bi Suan Jing. Cambridge: Cambridge Univ Pr , 2007.
Rossi, Richard. Theorems, Corollaries, Lemmas, and Methods of Proof. John Wiley & Sons : New York, 2011.
Posamentier, Alfred S., and Herbert A. Hauptman. The Pythagorean Theorem: The Story of Its Power and Beauty . Prometheus Books: New York, 2010.
Mathematica. (2012). Pythagorean Theorem. Retrieved March 3, 2012, from Mathematica: http://mathematica.ludibunda.ch/pythagoras6.html Morris, S. J. (1997, May 29). The University of Georgia. Retrieved 3 2, 2012, from Department of Mathematics Education:http://jwilson.coe.uga.edu/emt669/student.folders/morris.stephanie/emt.669/essay.1/pythagorean.htm
[IMAGE] ½ (a2 + b2) times it by the ratio of its real area to a
Through history, as said before, many philosophers have supported and developed what Pythagoras first exposed to the world. One of the most important philosophers to support Pythagoras’s ideas was Plato. In some of his writings he discusses the creation of the universe based on the musical proportions discovered by Pythagoras (Timaeus), and the explanation of the sound emitted by the planets, which is exposed in the “Myth of Er” in The Republic. It talks about a man who died and came back to life who narrates how he saw the space and how, in every “sphere,” there was a being singing constantly, each one in a different tone, so a perfect harmony was built. Nevertheless, not everyone agreed with this theory, being one of its most important critics Aristotle, who claimed that Plato’s arguments where false in his text On the Heavens. He acknowledges that it is a creative and innovative theory, but it is absurd to think that such music, which is imperceptible to us, exists in a harmonic way up in the heavens. I am not going to go deeper into that for it is not relevant for the text. As the years went on, many people continued developing this theory. Nevertheless, this philosophical theory, not truly explained until later on, was an inspiration for many artists and that is why not only philosophers but many other artists mention and base their works upon this theory.
After 3rd century BC, Eratosthenes calculation about Earth's circumference was used correctly in different locations such as Alexandria and syene (Aswan now) by simple geometry and the shadows cast. Eratosthenes's results undertaken in 1ST century by Posidonius, were corroborated in Alexandria and Rhodes by the comparison between remarks is excellent.
Geometry, a cornerstone in modern civilization, also had its beginnings in Ancient Greece. Euclid, a mathematician, formed many geometric proofs and theories [Document 5]. He also came to one of the most significant discoveries of math, Pi. This number showed the ratio between the diameter and circumference of a circle.
The math concept of Geometry or shapes will be taught to a second-grade classroom during and after the reading of The Greedy Triangle (1994) by Marilyn Burns. We will discuss the different shapes, their attributes, how they are used and how many sides and angles each shape has.
Here Pythagoras, better known as a mathematician for the famous theorem named for him, applied theoretical mathematics and the theory of numbers to the natural sciences (Nordqvist, 1). Pythagoras equated the duration of the lunar cycle to the female menstrual cycle and related the biblical equation of infinity as the product of the number seventy and forty to the normal length of pregnancy at 280 days (Nordqvist, 1). More practical, Pythagoras also contributed the idea of medical quarantine to the practice of medicine setting a forty-day period standard quarantine to avoid the spread of disease. While Pythagoras chose the number forty for its perceived divine nature his practical application of a quarantine must have been based on the observation that in some instances disease spreads through contact. The concept of Quarantine is still in use to this day and is an example of how Pythagoras contributed to modern medicine even while his methods were based on “mystical aspects of the number system” Pythagoras and his followers did “attempt to use mathematics to quantify nature” and as a result, medical practice (Ede, Cormack,
The following three propositions are contained in Archimedes' book. The book also showed Archimedes had given the closest approximation of what we now call pi to date. i) The area of a circle is equal to that of a right-angled triangle where the sides, including the right angle, are equal to the radius and circumference of the circle.
and 'c' (a being the shortest side, c the hypotenuse): a2 + b2 = c2
This source provided a lot of background information on Euclid and his discoveries. This source gave details about the many geometrical theories of Euclid, as well as his practical geometrical uses. This source also explained how geometry helped Greece a long time ago, and how it is used by many people everyday.
In 1665, the Binomial Theorem was born by the highly appraised Isaac Newton, who at the time was just a graduate from Cambridge University. He came up with the proof and extensions of the Binomial Theorem, which he included it into what he called “method of fluxions”. However, Newton was not the first one to formulate the expression (a + b)n, in Euclid II, 4, the first traces of the Binomial Theorem is found. “If a straight line be cut at random, the square on the whole is equal to the squares on the segments and twice the rectangle of the segments” (Euclid II, 4), thus in algebraic terms if taken into account that the segments are a and b:
Euclid also came up with a number of axioms and proofs, which he called “postulates.” Some of these postulates relate to all sciences, while other postulates relate only to geometry. An example of a Euclidean postulate that relates to all sciences is “The whole is greater than the part.” An example of a Euclidean postulate relating only to geometry is “You can draw a straight line between any two points.” Although these postulates seem extremely simple and obvious to us, Euclid was the first person to state them, as well as prove them to be true without question. These simple postulates really help with more complicated math and sciences, such as advanced geometry. For example, when doing advanced geometry involving a lot of lines and shapes, it is extremely helpful to know for sure that any single line can never contain more than one parallel line.
Trigonometry is the branch of mathematics that is based on the study of triangles. This study helps defining the relations between the different angle measures of a triangle with the lengths of their sides. Trigonometry functions such as sine, cosine, and tangent, and their reciprocals are used to find the unknown parts of a triangle. Laws of sines and cosines are the most common applications of trigonometry that we have used in our pre-calculus class. Historically. Trigonometry was developed for astronomy and geography as it helped early explorers plot the stars and navigate the seas, but scientists have been using it for centuries for other purposes, too. Besides other fields of mathematics, it is used in physics,
Burton, D. (2011). The History of Mathematics: An Introduction. (Seventh Ed.) New York, NY. McGraw-Hill Companies, Inc.
There is a triangle called the Heronian triangle. It has area and side lengths that are all integers. The Heronian triangle is named after the great hero of Alexandria. The term is sometimes applied more widely to triangles whose sides and area are all rational numbers. An Isosceles triangle is a triangle that has two sides of equal length. Sometimes is specified as having two and only two sides of equal length. Triangles are polygons with the least possible number of sides, which is
The 17th Century saw Napier, Briggs and others greatly extend the power of mathematics as a calculator science with his discovery of logarithms. Cavalieri made progress towards the calculus with his infinitesimal methods and Descartes added the power of algebraic methods to geometry. Euclid, who lived around 300 BC in Alexandria, first stated his five postulates in his book The Elements that forms the base for all of his later Abu Abd-Allah ibn Musa al’Khwarizmi, was born abo...