Pythagoras theorem

Two high school students have proved the Calcea Johnson and Ne’Kiya Jackson, both at St. Mary’s Academy in New Orleans, announced their achievement last month at an American Mathematical Society meeting. “It’s an unparalleled feeling, honestly, because there’s just nothing like it, being able to do something that ... people don’t think that young people can do,” If verified, Johnson and Jackson’s proof would contradict mathematician and educator Elisha Loomis, who stated in his 1927 book The Pythagorean Proposition that no trigonometric proof of the Pythagorean theorem could be correct. Their work joins a handful of other trigonometric proofs that were added to the mathematical archives over the years. Each sidestepped “circular logic” to prove the pivotal theorem. So what exactly is a trigonometric proof of the Pythagorean theorem, and why was Loomis so closed off to the idea? The Pythagorean theorem provides an equation to calculate the longer side of a right triangle by summing the squares of the other two sides. It is often phrased as a 2 + b 2 = c 2. In this equation, a, b and c represent the lengths of the three sides of a right triangle, a triangle with a 90-degree angle between two of its sides. The quantity c is the length of the longest side, called the hypotenuse. Though the theorem is named for the ancient Greek philosopher Pythagoras, some historians believe The theorem “connects algebra and geometry,” says Stuart Anderson, a professor emeritus of mathematics ...

Little of what is known about Pythagoras comes from contemporary accounts, and the first fragmentary accounts of his life came in the fourth century bce, about 150 years after his death. Pythagoras was born in Samos and likely went to bce, apparently to escape bce for Metapontum (now Metaponto, Italy) where he died.

Although he is most famous for his mathematical theorem, Pythagoras also made extraordinary developments in astronomy and geometry. He also developed a theory of music while and founded a philosophical and religious school in Croton, Italy. It was here he taught that "the whole cosmos is a scale and a number", according to the University of St Andrews . – Julius Caesar biography: Facts & history – Who were the ancient Persians? – The Renaissance: The 'Rebirth' of science & culture – Who was Herodotus? – Ancient Rome: From city to empire in 600 years While playing on his lyre, which was an ancient Greek stringed instrument, Pythagoras discovered that the vibrating strings created a beautiful sound when the ratios of the lengths of the wires were whole numbers, and that this was also true of other instruments. He combined this discovery with his understanding of the planets, conceiving the theory that when the planets were in harmony, it created beautiful music that man was incapable of hearing. It was claimed amongst other things that he had taken part in the Olympics and was awarded laurels for pugilism, or boxing, when he was a young man. It was also said that he had fought in the Trojan Wars during a previous life. This last myth reflects Pythagoras' genuine belief in metempsychosis, which argues that all souls are everlasting and, when the physical body dies, it simply floats away and finds a new body to live in, according to Stanford University. Later reports stated th...

Pythagoras's Proof Given any right triangle with legs \( a \) and \(b \) and hypotenuse \( c\) like the above, use four of them to make a square with sides \( a+b\) as shown below: This forms a square in the center with side length \( c \) and thus an area of \( c^2. \) However, if we rearrange the four triangles as follows, we can see two squares inside the larger square, one that is \( a^2 \) in area and one that is \( b^2 \) in area: Since the larger square has the same area in both cases, i.e. \( (a+b)^2 \), and since the four triangles are also the same in both cases, we must conclude that the two squares \( a^2 \) and \( b^2 \) are in fact equal in area to the larger square \( c^2 \). Thus, \( a^2 + b^2 = c^2 \). \( _\square \) Euclid's Proof In outline, here is how the proof in Let \(A, B, C\) be the vertices of a right triangle with the right angle at \(A.\) Drop a perpendicular from \(A\) to the square's side opposite the triangle's hypotenuse (as shown below). That line divides the square on the hypotenuse into two rectangles, each having the same area as one of the two squares on the legs. For the formal proof, we require four elementary lemmata: • If two triangles have two sides of the one equal to two sides of the other, each to each, and the angles included by those sides equal, then the triangles are congruent (side-angle-side). • The area of a triangle is half the area of any parallelogram on the same base and having the same altitude. • The area of a recta...

" " Pythagoras, an ancient Greek thinker — equal parts philosopher, mathematician and mystical cult leader — lived from 570 to 490 B.C.E and is credited with devising one of the most famous theorems of all time. Wikimedia Commons (CC By-SA 4.0)( OK, time for a pop quiz. You've got a right-angled triangle — that is, one where two of the sides come together to form a 90-degree angle. You know the length of those two sides. How do you figure out the length of the remaining side? That's easy, provided that you took geometry in high school and know the The Pythagorean theorem states that with a right-angled triangle, the sum of the squares of the two sides that form the right angle is equal to the square of the third, longer side, which is called the a 2 + b 2 = c 2, in which a and b represent the two sides of the right angle and c is the long side. Pythagoras With a hyperbolic reputation like that, it's little wonder that Pythagoras was credited with devising one of the most famous theorems of all time, even though he wasn't actually the first to come up with the concept. Chinese and Babylonian mathematicians beat him to it by a millennium. "What we have is evidence they knew the Pythagorean relationship through specific examples," writes a 2 + b 2 = c 2." The Pythagorean theorem isn't just an intriguing mathematical exercise. It's utilized in a wide range of fields, from construction and manufacturing to navigation. As Allen explains, one of the classic uses of the Pythagorea...

The historian Robinson writes, “The statement that `Pythagoras worked very hard at the arithmetical side of geometry' is further borne out by the tradition that he investigated the arithmetical problem of finding triangles having the square on one side equal to the sum of the squares on the other two” and did so, early on, by using stones in rows to understand the truths he was trying to convey (1968). The Pythagorean Theorem states that a² + b² = c². This is used when we are given a triangle in which we only know the length of two of the three sides. C is the longest side of the angle known as the hypotenuse. If a is the adjacent angle then b is the opposite side. If b is the adjacent angle then a is the opposite side. If a = 3, and b = 4, we could then solve for c. 32 + 42 = c². 9 + 16 = c². 25 = c². c = 5. This is one of the prime uses of the Pythagorean Theorem. YouTube Follow us on YouTube! Proposition: In right-angled triangles the square on the hypotenuse is equal to the sum of the squares on the legs. Euclid started with a Pythagorean configuration and then drew a line through a diagram illustrating the equalities of the areas. He concluded that AB/AC = AC/HA, therefore (AC)² = (HA)(AB). Since AB=AJ, the area of the rectangle HAJG corresponds to the area of the square on side AC. Similarly, AB/BC = BC/BH also written as (BC)² = (BH)(AB) = (BH)(BD) and since AB=BD. Thus we see that the sum of the areas of the rectangles is the area of the square on the hypotenuse. I...

Relation between sine and cosine The Pythagorean trigonometric identity, also called simply the Pythagorean identity, is an The identity is sin 2 ⁡ θ + cos 2 ⁡ θ = 1. . Proofs and their relationships to the Pythagorean theorem [ ] θ Proof based on right-angle triangles [ ] Any θ. The elementary definitions of the sine and cosine functions in terms of the sides of a right triangle are: sin ⁡ θ = o p p o s i t e h y p o t e n u s e = b c Related identities [ ] 1 + cot 2 θ = csc 2 θ, and applied to the red triangle shows that 1 + tan 2 θ = sec 2 θ. The identities 1 + tan 2 ⁡ θ = sec 2 ⁡ θ Proof using the unit circle [ ] The unit circle centered at the origin in the Euclidean plane is defined by the equation: x 2 + y 2 = 1. [ how?] See also [ ] • • • • • Notes [ ] • Lawrence S. Leff (2005). PreCalculus the Easy Way (7thed.). Barron's Educational Series. p. 0-7641-2892-2. • This result can be found using the distance formula d = x 2 + y 2 . See Cynthia Y. Young (2009). Algebra and Trigonometry (2nded.). Wiley. p.210. 978-0-470-22273-7. This approach assumes Pythagoras' theorem. Alternatively, one could simply substitute values and determine that the graph is a circle. • Contemporary Precalculus: A Graphing Approach (5thed.). Cengage Learning. p.442. 978-0-495-10833-7. • James Douglas Hamilton (1994). Time series analysis. Princeton University Press. p.714. 0-691-04289-6. • Steven George Krantz (2005). Real analysis and foundations (2nded.). CRC Press. pp.269–270. 1-58488-4...