What is pythagoras theorem

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...

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...

Jeanne Rast Jeanne Rast has taught Mathematics in grades 7-12 and college for over 30 years. She has a Ph.D. in Math Education and a M.Ed. in Math both from Georgia State University, as well as a B.A. in Math from The University of the South. Dr. Rast is a certified teacher for the State of Georgia for Mathematics grades 7-12 • Instructor The Pythagorean Theorem rule is that the length of one leg squared plus the length of the other leg squared is equal to the hypotenuse squared. Step by step this means 1) Square one leg 2) Square the other leg 3) Add these together 4) Take the square root of the sum 5) The answer is equal to the hypotenuse The Pythagorean Theorem is a rule for right triangles that is used to find the length of one side when two sides are given. The rule is that the sum of the squares of the two shorter sides is equal to the square of the longest side. The shorter sides form the right angles and are called the legs. The longest side is opposite the right angle and is the hypotenuse. The Pythagorean Theorem is a rule for right triangles that is used to find the length of one side when two sides are given. To solve a problem using the Pythagorean first square the lengths of each leg. Add these results together. Take the square root of the answer. If the hypotenuse is given, then square the length of the hypotenuse and subtract the side squared. The square root of this result is the length of the other leg. The The Pythagorean Theorem is a specific case of th...

The Pythagorean theorem consists of a formula a^2+b^2=c^2 which is used to figure out the value of (mostly) the hypotenuse in a right triangle. The a and b are the 2 "non-hypotenuse" sides of the triangle (Opposite and Adjacent). Once you progress, you will be given the hypotenuse and would be needed to find the opposite or the adjacent side (a or b). The Pythagorean theorem is a simple formula which uses the squared value of a and b; for example "a=3 and b=4, what is the value of c?" you square a (3^2=9=a) and b (4^2=16=b) and add the 2 values (9+16=25) to get to c. To complete the question, you have to square root c's value (square root of 25=5) because the formula says c^2 and not just c. Once you have done that, you can check your answer by squaring a,b and c to see if you have added and divided (Square-rooted) correctly. Hope this helps! 5:27 he said that in order to complete the equation you have to take the positive square root of both sides, which for 25 would equal 5. But what does that mean? How did he get 5 from 25? What did he do, what did he divide 25 by and why did he divide that and not another number? I will be waiting for a response thank you to those that reply, I will be very thankful because I know I would be taking time away from you just so you can answer my question. Thanks! A square root is a number that produces a specified quantity when multiplied by itself. It goes hand in hand with exponents and squares. 2 squared is 4, and the square root of 4 ...

Pythagorean theorem, Rule relating the lengths of the sides of a right triangle. It says that the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse (the side opposite the right angle). That is, a 2 + b 2 = c 2, where c is the length of the hypotenuse. Triads of whole numbers that satisfy it (e.g., 3, 4, and 5) are called Pythagorean triples. See also law of cosines; law of sines. Related Article Summaries

The legend tells that Pythagoras was looking at the square tiles of Samos' palace, waiting to be received by Polycrates, when he noticed that if one divides diagonally one of those squares, it turns out that the two halves are right triangles (whose area is half the area of the tile). The area of a square whose side is the hypotenuse of one of those right triangles, is double the area of a square tile. Here's an image that shows this (the dashed square is made of 4 half tiles, i.e. 2 tiles in total) Now, since the two legs of the right triangle are sides of a tile, the squares built on them are two tiles. Hence, the sum of their area equals the area of the square built on the hypotenuse (the dashed one). So Pythagoras' theorem holds for isosceles right triangles. The legend tells that, after proving this result, Pythagoras generalized his theorem, to make it hold for right triangles whose sides are not equal. He concentrated his attention on the 4 tiles on which the square built on the hypotenuse was constructed, and he noticed that the drawing could be modified in the following way (on the left the former drawing, on the right the modified version): In the right image, the two squares built on the legs of the red-filled right triangle are equivalent to the two squares delimited by the blue dots. Each of the two remaining rectangles, delimited by the blue dots, is made by two copies of the red-filled triangle. Together, the two squares and the two rectangles build 4 tiles ...

Jeanne Rast Jeanne Rast has taught Mathematics in grades 7-12 and college for over 30 years. She has a Ph.D. in Math Education and a M.Ed. in Math both from Georgia State University, as well as a B.A. in Math from The University of the South. Dr. Rast is a certified teacher for the State of Georgia for Mathematics grades 7-12 • Instructor The Pythagorean Theorem rule is that the length of one leg squared plus the length of the other leg squared is equal to the hypotenuse squared. Step by step this means 1) Square one leg 2) Square the other leg 3) Add these together 4) Take the square root of the sum 5) The answer is equal to the hypotenuse The Pythagorean Theorem is a rule for right triangles that is used to find the length of one side when two sides are given. The rule is that the sum of the squares of the two shorter sides is equal to the square of the longest side. The shorter sides form the right angles and are called the legs. The longest side is opposite the right angle and is the hypotenuse. The Pythagorean Theorem is a rule for right triangles that is used to find the length of one side when two sides are given. To solve a problem using the Pythagorean first square the lengths of each leg. Add these results together. Take the square root of the answer. If the hypotenuse is given, then square the length of the hypotenuse and subtract the side squared. The square root of this result is the length of the other leg. The The Pythagorean Theorem is a specific case of th...

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...

The Pythagorean theorem consists of a formula a^2+b^2=c^2 which is used to figure out the value of (mostly) the hypotenuse in a right triangle. The a and b are the 2 "non-hypotenuse" sides of the triangle (Opposite and Adjacent). Once you progress, you will be given the hypotenuse and would be needed to find the opposite or the adjacent side (a or b). The Pythagorean theorem is a simple formula which uses the squared value of a and b; for example "a=3 and b=4, what is the value of c?" you square a (3^2=9=a) and b (4^2=16=b) and add the 2 values (9+16=25) to get to c. To complete the question, you have to square root c's value (square root of 25=5) because the formula says c^2 and not just c. Once you have done that, you can check your answer by squaring a,b and c to see if you have added and divided (Square-rooted) correctly. Hope this helps! 5:27 he said that in order to complete the equation you have to take the positive square root of both sides, which for 25 would equal 5. But what does that mean? How did he get 5 from 25? What did he do, what did he divide 25 by and why did he divide that and not another number? I will be waiting for a response thank you to those that reply, I will be very thankful because I know I would be taking time away from you just so you can answer my question. Thanks! A square root is a number that produces a specified quantity when multiplied by itself. It goes hand in hand with exponents and squares. 2 squared is 4, and the square root of 4 ...

The legend tells that Pythagoras was looking at the square tiles of Samos' palace, waiting to be received by Polycrates, when he noticed that if one divides diagonally one of those squares, it turns out that the two halves are right triangles (whose area is half the area of the tile). The area of a square whose side is the hypotenuse of one of those right triangles, is double the area of a square tile. Here's an image that shows this (the dashed square is made of 4 half tiles, i.e. 2 tiles in total) Now, since the two legs of the right triangle are sides of a tile, the squares built on them are two tiles. Hence, the sum of their area equals the area of the square built on the hypotenuse (the dashed one). So Pythagoras' theorem holds for isosceles right triangles. The legend tells that, after proving this result, Pythagoras generalized his theorem, to make it hold for right triangles whose sides are not equal. He concentrated his attention on the 4 tiles on which the square built on the hypotenuse was constructed, and he noticed that the drawing could be modified in the following way (on the left the former drawing, on the right the modified version): In the right image, the two squares built on the legs of the red-filled right triangle are equivalent to the two squares delimited by the blue dots. Each of the two remaining rectangles, delimited by the blue dots, is made by two copies of the red-filled triangle. Together, the two squares and the two rectangles build 4 tiles ...