hoeever this isnt really trigonometric, but rather graphical. Its just that you call a/c=sin(α) but at that point this is just a name. So it has nothing to do at all with trigonometry itselve.
@@neutronenstern. I think the point is that if you draw triangles with the same angles, the lengths of the sides may vary but the ration between the side lengths stays the same. This is the whole basis behind trigonometry, and is why we can say that there is such a thing as sin(α). It's a little like how Ohm's Law in physics actually says that in a conductor the current that flows is (all things being the same) proportional to the voltage. This means that you can define resistance as being this ratio, and can say that V = IR, which is what many people (including some electronics experts) think is Ohm's Law.
@@pauln7869 yes, but with sin(x) its special, i think. Cause in this proove no property of sin was used,without deriving it geometrically first. no actual value of sin was used. So its kind of a trick to say, this is trigonometry. This proove works exactly the same, without needing the help of any sine.
"This would have made a great undergraduate senior project..." Why do you need it to be college students for it to be "great"? It is high school students, it is a great project, they deserve all credit accorded mathematicians of any age or school level, for having developed this proof.
Já tem inúmeras aplicações. Pq o método delas se relaciona diretamente com limites e cálculo infinitesimal que é o que se faz em matemática na prática. Adaptando esse método já deve ter surgido vários métodos baseados em pós-graduação associados a matemática computacional e muito mais.
The most fascinating thing that no one is talking about is that we human beings uses shared knowledges across time periods and always figure out a way to make things better, and discoveries aren't always made by people that society deems genius, but people that are curious enough to put in the time and effort.
And then published it in a closed form so it’s (again) not commonly available until someone reverse engineers it - at least without solving a riddle or poem in this case
To be fair, trigonometry is based on the parallel postulate, which is equivalent to the Pythagoras theorem. The students theorem is based on the assumption the extended lines must intersect.
True, and that is dependent on whether alpha and beta are equal. If you take the limit as they approach the same angle, you would get the answer for when those two are equal.
@@frogmorely Nope. The fifth postulate presupposes Euclidian (flat) space. Euclidian space may be more formally described as a space in which distances between points in the coordinate system are calculated using pythagoras. (There are other possibilities.) Euclidian space has the feature where non-parallel lines intersect in exactly one point. This feature is required for the construction used. IOW, Pythagoras is actually smuggled into the construction when the lines are extended to the intersection. Nevertheless, still a clever way of tackling the problem.
I can see how all this mathematics is accessible to high schools (albeit advanced high schoolers), but it also took the creativity and grit of two young mathematicians to put it all together. Bravo!
@@michaelgum97so if someone figures something out you couldn't then that means they are geniuses? I am not saying they may not be very smart or their accomplishment shouldn't be acknowledged.
It depends a lot on what we mean by *only* trigonometry. In particular, it means that there is something used in the common proofs of the theorem that has to be excluded. If not one could just apply a thin varnish of trigonometry on another proof. An obvious candidate would be to exclude geometry (which would then exclude this new proof) but then how do we define the trigonometric functions? We should do it axiomatically. If we give differential equations as axioms, it might be possible to prove the theorem using analysis but would not really fit the notion of what trigonometry *is*. If we want to use some of the common trigonometric formulas as axioms, we would need to not include sin²x+cos²x=1 (as that is essentially what we want to prove) but even from other formulas (like the formulas for sum of angles), we can quite trivially 'prove' the sum of squares (and so decide that we kind of put the result in the axioms). Nevertheless it is an interesting proof with non-trivial trigonometry, geometry and limits in the long tradition of alternative proves of this most famous theorem. Congrats to them!
For some reason adding a character different from a space right before or right after your *bold text* breaks the formatting (my guess is that they didn't want people censoring words to have their 3 asterisks reduced to one bold one) At least on mobile it does
Jason Zimba, who provided a trigonometric proof of Pythagoras in 2009, prefaces the paper with this: "In a remarkable 1940 treatise entitled The Pythagorean Proposition, Elisha Scott Loomis (1852-1940) presented literally hundreds of distinct proofs of the Pythagorean theorem. Loomis provided both “algebraic proofs” that make use of similar trian- gles, as well as “geometric proofs” that make use of area reasoning." The proof presented in the video uses both similar triangles and area reasoning. We should consider this proof "algebraic" or "geometric" or both. I've little doubt that similar strategies are found elsewhere in the 367 proofs published by Elisha Scott Loomis.
I think the most beautiful proof of all is the following. You have a right triangle. Divide the triangle into two triangles by a line that is perpendicular to the hypotenuse and connects to the vertex of the right angle. Now you have three similar triangles: the original one and the two pieces. Because of the fact that the triangles are similar, each occupies the same fraction 'x' of the square on its hypothenuse. If the hypotenuses and surface areas of the three triangles are a, b, c and A, B, C, respectively, you therefore have A = x*a^2, B = x*b^2 and C = x*c^2. Because of the fact that A + B = C, then x*a^2 + x*b^2 = x*c^2 and therefore a^2 + b^2 = c^2.
"occupies the same fraction 'x' of the square on its hypotenuse" -- I think you mean "of the square _of_ its hypotenuse". But beautiful proof! Thanks for posting this, very educational and much appreciated.
@@yurenchu You are right, that might have been a better wording. I originally found out about this proof when I was watching a video by James Grime that is called "A Pythagorean Theorem for Pentagons + Einstein's Proof." Definitely one on my favorite math videos on RUclips so check that out if you're interested.
@@jimi02468 Thanks, I went to watch it. It's been quite some time since I've watched one of "Singing Banana" 's videos. But to be honest, I found the video not so surprising anymore, after having read the proof that you posted here. :-) And I was expecting to see some equivalent theorem that's valid for pentagons instead of for (right) triangles, but there wasn't such a thing. :-( But still interesting to learn how Einstein looked at it, so thanks anyway!
I'm an old PhD in mathematics & I never heard the assertion anywhere at any time that it was "impossible to prove the Pythagorean theorem using trigonometry alone". Lol! Perhaps you can send me a link to someone making that assertion? ;)
Elisha scott loomis did make that asseration in a book titled "The Pythagorean Proposition" however it was disproven by Jason Zimba way before the students did.and most likely someone probably disproved it before zimba
@@tank2256 It's not even that Elisha Scott was wrong, very often the chosen analytic _definitions_ of the trigonometric functions are based on the Pythagorean Theorem. If your trig functions are defined that way, you can't prove Pythagorean Theorem with trig. Elisha was probably referring to that.
Indeed that is crap. You have to tell what axioms you use before claiming such a thing. I don't know what "using trigonometry alone" means. The Pythagorean theorem is very elementary. Trigonometry is a very sophisticated tool. This amounts to boasting you can crack a nut with a hydrogen bomb.
1) It's not the first trigonometric proof of the Pythagorean Theorem. 2) It's not pure trigonometry. 3) It's only superficially trigonometric. 4) It's almost an exact copy of an existing proof. Look up John Arioni "Pythagorean Theorem via Geometric Progression"
I don't know... To me Pyth and Trig were always different formulations of the same qualities of triangles, expressed differently, but NOT from different principles. It always felt like they both started from the way that right triangles can be inscribed into circles. But I never devoted much thought into this. So it's interesting but ultimately I don't think a proof from Trig is independent at all.
I would assume this proof was overlooked until now because it relied on taking the sum of an inifinite series, which is a relatively modern concept. By the time that idea had been well established, the dogma that there were no trigonometric proofs was well established, so no one went looking until now. Bravo!
"I would assume this proof was overlooked until now" It would be safer to assume it's been proposed by students many times and there are thousands of proofs like it that students have proposed that were never mentioned outside of their classrom. But since these chicks are black, it's national news.
the sine rule is based on right angle triangle drawn by taking one of the heights. so even though law of sine extends to any triangle the law inherently is based on basic definition of sin in relation to a right angled triangle. so I don't think the paragraph is wrong, these students found nothing new but just a longer proof of proving something which can be itself written as sin²a + cos²a=1
yes you're right, but all the idiots can't understand that. the greeks already discovered it 2000 years ago, and their whole point was to make it simple, not unnecessarily convoluted
1) It's not the first trigonometric proof of the Pythagorean Theorem. 2) It's not pure trigonometry. 3) There is an exception: It doesn't work for isosceles triangles.
Put a small square inside a larger square with their centers coinciding. Rotate the smaller square until its 4 corners touch the sides of the large square. The large square will be comprised of 4 identical right triangles and the small square. Write an equation stating that the area of the large square is equal to the area of the small square + the area of the 4 identical triangles. Rearrange the terms in the equation and voila!
Of course it is required that the area of the smaller square is at least half of the area of the larger square (otherwise the corners of the smaller square would never touch the sides of the larger square, and we wouldn't have any right triangles in the diagram). But yes, this is an excellent proof! If the right triangles have right sides a and b (with a ≤ b) and hypotenuse c, then the larger square has side (a+b) while the smaller square has side c. The area of each triangle equals ½ab . Since the area of the larger square equals the area of the smaller square + the area of the four triangles, the equation is (a+b)² = c² + 4*(½ab) . After expanding, we get a² + b² + 2ab = c² + 2ab , and hence a² + b² = c² . Additionally, to physically demonstrate the result, the four right triangles can also be paired to each other so that they form two separate, identical rectangles (with their diagonal equal to the triangle's hypotenuse), and these two rectangles can then be placed back inside the (emptied) large square: one rectangle horizontally in the bottom left corner, and the other rectangle vertically into the top right corner. The area of the large square that is not covered by the two rectangles consists of exactly one square in the top left corner with side b and area b² , and one square in the bottom right corner with side a and area a² ; and of course, it follows that these two squares together have the same area as the rotated smaller square (with side c and area c² ) that we had at the beginning.
@@yurenchu Yes, U got it. I never calc'd exactly the minimum size of the small square such that its corners eventually contact the sides of the big square when rotated but your 1st statement does precisely define it. The length of the diagonal from the small square's center to its corners must be greater than half the length of the big square's side, which works out to what you said.
My go at it; The angle addition formula can be prooven without pythagoras theorem cos(a+b) = cos(a)cos(b) - sin(a)sin(b) let b = -a cos(a-a) = cos(a)cos(-a) - sin(a)sin(-a) = cos(a)cos(a) + sin(a)sin(a) = cos²(a) + sin²(a) We also have cos(a-a) = cos(0) = 1 cos²(a) + sin²(a) = cos(a-a) = 1 q.e.d.
The identities cos(-x) = cos(x) and sin(-x) = -sin(x) can also be prooven with the angle addition formula. Assuming cos and sin are well defined on the interval 0≤x≤0.5π, I am going to extended their definition to all real numbers, in a way that preserves the angle addition formula. First, I evaluate sin and cos at π, 1.5π and 2π, by evaluating their angle addition formulas at 0.5π + 0.5π, π + 0.5π and π + π respectively. Then I can proove that they have a period of 2π, by evaluating their angle addition formulas at x + 2π cos(x + 2π) = cos(x)cos(2π) - sin(x)sin(2π) = cos(x)•1 - sin(x)•0 = cos(x) Similar proof for sin(x+2π) = sin(x) The identities cos(0.5π-x) = sin(x) and sin(0.5π-x) = cos(x) in the interval 0≤x≤0.5π can be directly observed in a right triangle. Final step, evaluate cos and sin at -x, by evaluating them at 1.5π + (0.5π-x). This is the same as evaluating them at -x, due to the period of 2π. cos(1.5π + (0.5π - x)) = cos(1.5π)cos(0.5π - x) - sin(1.5π)sin(0.5π - x) = 0•sin(x) - (-1)•cos(x) = cos(x) Similarly sin(-x)=sin(2π-x)=sin(1.5π+(0.5π-x)) evaluates to -sin(x) Insert pythagoras theorem proof here Things like cos(π-x)=-cos(x) can all be prooven in a similar manner, extending the definition of cos and sin to all real numbers
There are well over 371 Pythagorean Theorem proofs, originally collected and put into a book in 1927, which includes those by a 12-year-old Einstein (who uses the theorem two decades later for something about relatively), Leonardo da Vinci and President of the United States James A. Garfield.
@@hoochygucci9432 Not to mention that there are countless proofs that also claimed to be fully trigonometric, which is supposedly the thing that makes this proof interesting. The most recent example was a proof by Zimba that was made in the early 2000s. So clearly the video is bogus, and whatever these students are being rewarded for is clearly not what the video claims, because we've already had fully trigonometric proofs of Pythagoras for a looooong time. And when I say fully trigonometric I really mean it. Like, really really mean it.
@@tiranito2834why is this your perspective on life. There’s no need to be so negative. It’s impressive that high schoolers whose highest math is probably basic calculus were able to come up with a new and interesting proof for this theorem. Also the video isn’t the one making a big deal about it it’s just explaining the proof. Why can’t you just appreciate math, why does it have to be a contest that requires putting passionate people down.
@@joshuafrank1246 The problem is that they didn't make a new proof, look it up, they copied parts of multiple proofs, most of it straight out of Zimba's proof. It is precisely because I appreciate math that this situation infuriates me. Because the fact that this is being treated as news is nothing but proof of lost knowledge. Things that used to be common knowledge just about a decade ago are now cutting edge and people act as if they had never heard of fully trigonometric proofs for pythagoras before.
1) It's not the first trigonometric proof of the Pythagorean Theorem. 2) It's not pure trigonometry. 3) There is an exception: It doesn't work for isosceles triangles.
I like the history you are providing in recent days. is this related to your earlier poll and will you discuss more mathematical history? It's definitely interesting to know, especially considering how despite different cultures and different times, mathematics connects us all. I remember seeing depictions of 'Pascal's Triangle' from a Chinese mathematician and Ancient Indian Mathematician. Even without knowing the language, I can understand it as Pascal's triangle.
The "gameplay vs lore" Meme absolutely sums up maths perfectly While doing maths in general is.... "Fun", the history that maths itself has is insanely interesting and intricate. Throughout hundreds to thousands of years, humans have advanced in this language, the language of truth itself
Honestly, I still don't really get what is so special about this proof. Of course, it's a very unique proof and those teenagers are apparantly brilliant. But the 'never done before' part of it is not entirely clear to me. A statement in a book that some kind of proof is impossible is not the same as a real mathematical conjecture. So I can't really tell if the media is blowing things up, but I'm suspicious.
The media but also other RUclipsrs are clueless on this. The majority of this "proof" is showing the trigonometry identity: sin(2 alpha) = 2*tan(alpha) / ( 1 + tan(alpha)^2 ) But this is already a known fact and has nothing to do with proving the Pythagorean theorem. Replacing tan(alpha) with a/b gets you pretty close to the end result. Nevertheless, these students discovered this by themselves and that is quite an achievement.
its because its a bunch of kids. Why can't you just let them feel like they did something cool? Imagine if you were a highschooler who discovered something and online everyone is just saying how you are a fraud and your work isn't important. You all are obviously jealous because instead of being positive you are going out of your way to find something negative to say. Its so sad.
@@Brad-qw1telol don't waste your energy on toxic people they are always there in our lives . I mean if the scientific community acknowledges that this is something to celebrate then some nonentities comes and dispute that what can you say? You should just let them swallow their toxicity alone
The funny thing is that neither of them wanted to pursue mathematics in college but instead, if my memory is correct, environmental engineering and biochemistry. Of course both are math heavy but only math. Also, I believe it originated out of school competition and not just a paragraph in a textbook.
Well... Actually this proof uses not only trigonometry. It uses also limits and similarity. And you can easily prove the theorem using similarity only. :-)
@@PHlophe Hi PHlophe, please consider a right triangle ABC, with the right angle at C. Let AB be the hypotenuse, so AC and BC are the legs. Now draw the altitude from C to AB, which will intersect AB at point D. Then AB + BD = AB. This will create two smaller triangles, triangle ADC and triangle BDC, both of which are SIMILAR to triangle ABC by angle-angle criterion (angle ADC = angle ABC and angle DAC = angle BAC) Triangle BDC is similar to ABC for the same reasons Since ADC and ABC are similar, we can form the ratio of sides AD / AC = AC / AB (1) and since BDC and ABC are also similar, we have the ratio BD / BC = BC / AB (2) Now ADC and BDC have the same altitude CD. So that it looks like the pythagorean theorem, let AC = a, BC = b, and AB = c, which gives AD = a^2 / c and BD = b^2 / c after cross multiplying and substituting ratios (1) and (2) Then AD + BD = a^2 + b^2 / c Since AD + BD = c, we have a^2 + b^2 / c = c, or a^2 + b^2 = c^2 I'm not sure how I was able to do that through all the jealousy and rage eating me up inside, but it looks like I deserve a Fields medal :)
7:46 If alpha + beta is 90 degrees then you're assuming the euclidian parallel postulate, which is equivalent to the pythagorean theorem. Seems to me that the students are implicitly assuming that pythagoras is true in this proof, but I might be wrong.
@@rohangeorge712 The parallel postulate is that a line can be drawn parallel to anyother line, which if you remember 7th grade math, is used to prove the angle sum property.
1) It's not the first trigonometric proof of the Pythagorean Theorem. 2) It's not pure trigonometry. 3) The same proof could be done without any trigonometric equations at all. It's only superficially trigonometric.
There is a separate proof for the case of isosceles triangles, it's in the first few minutes of the video. Seems like lots of commenters literally didn't even watch it...
At 12:18 you have expressions for all three sides of the big triangle, which is a right triangle. You can show that the sum of the squares of the two shorter sides is equal to the square of the hypotenuse. That proves the pythagorean theorem without needing to go back to the original triangle.
he needs to prove that a^2 + b^2 = c^2, just because the sum of the square of the two shorter sides is equal to the hypotenuse, that doesn't prove a^2 + b^2 = c^2. that would just prove the case for the super large triangle, which could just be a coincidence.
@@rohangeorge712no it wouldn’t because all the triangles are similar are form a right triangle cause of corresponding angles. It wouldn’t be a “coincidence”
@@prod_EYESTechnically yeah, it would prove the Pythagoras Theorem but I guess they did that because the main objective was to prove specifically 'a²+b²=c²'.
I don't know if I missed it in the video; but I was a tad unconvinced, at first, that the similarity factor of all of the waffle-triangles would be the same. They ARE definitely similar, but I think it just needed a short explanation as to how you can tell the similarity factor for all of the triangles would be the same. Your old side "a" becomes your new side "b" (ie, your short side length from the larger triangle is now your shorter side length of the smaller triangle). Based on the similar triangles, the ratio of the short side to the long side is a/b. That's basically it. Every time you want to calculate the next unknown perpendicular length, you would use this ratio (its also the tangent ratio!), so your similarity scaling factor is a/b. After typing this out, I think this was touched on, but I hope this extra explanation is helpful for anyone who might have been initially unconvinced.
Congrats to these high students. I would not say they used trigonometry to prove the theorem. Instead, they used calculus plus a clever geometry construction to prove it, which is a great effort. BTW, I never heard such comments from mathematicians that a particular tool is IMPOSSIBLE to solve a specific problem.
1) It's not the first trigonometric proof of the Pythagorean Theorem. 2) It's not pure trigonometry. 3) There is an exception: It doesn't work for isosceles triangles.
Perhaps even more amazing than these students coming up with this never-before-discovered proof, is that the proof they came up with is relatively understandable by lay people. Great job!!
Wow! When I first heard about this proof from my mom, who hardly knows any math, by the way, and she told me that the students used trig to prove PT, naturally I was quite skeptical, but after watching this video, I now see that this is indeed a remarkable proof! Congrats to Ne'Kiya Jackson and Calcea Johnson! I wish I could have done something half this great when I was in high school!
It's possible to prove the law of cosines without using the Pythagorean theorem. The proof can be found on Wikipedia for example. Now using the law of cosines on a right triangle just gives the Pythagorean theorem as a special case.
1:27 I wonder, why they came up with this rather strange statement. The sine theorems in (right) triangles can easily be deducted purely from the definition of the sine, just like it's done in the video. Also it seems quite obvious that sin alpha = a/c will hold if and only if gamma=90° . So anyway I just googled this book... it's from 1940. Not that people couldn't have known better back then, but it is quite imaginable that nowadays such books would be checked much more thouroughly by many more people. So that might explain it in conjection with someone being a bit too excited about the topic.
It's called a geometric series and scaling. It's infinitely repetitive. It says a^2 + b^2 = C^2 because a^2 + b^2 = c^2 over and over and over again for infinity. It's was also already done in a previous proof: John Arioni "Pythagorean Theorem via Geometric Progression"
There are literally hundreds of trigonometric proofs of the Pythagorean Theorem. Every trigonometric function is only a ratio of 2 geometric knowns. EVERY geometric proof to the pythagorean theorem can be expressed in terms of trigonometric functions.
This strikes me as more of a geometric proof than a trigonometric one. Most of the steps in Case 2 are using trig functions to add extra steps to simply taking ratios of similar triangles. Nonetheless, it is a very nice and creative proof.
I love the new proof using infinite series … I think i recall learning the first alternate/other proof in high school, - a new proof - genius! these students are legends in high school. Thanks everyone for shining light on this!
There is another trigonometric proof I have been taught at school. The only thing you need is to come up with a shape of an outer square with the side a+b, inner square with side c and the rest of space filled with 4 equal right triangles rotated 4 ways (0, 90, 180, 270 degrees). Essentially, you put those in the corners of a big square. Then you write the area of the big square two ways and voila - very easy;)
The thing that is still missing is that it is BY DEFINITION what sin and cos mean, and they are based on right triangles. There is nothing new in the “proof” here, except to take as defined one thing vs another, neither of which are independent (of course not, otherwise Pyth theorem would not be true).
1) It's not the first trigonometric proof of the Pythagorean Theorem. 2) It's not pure trigonometry. 3) There is an exception: It doesn't work for isosceles triangles.
@@needtasteingames This Math PhD explains it better than me: Is a Trigonometric Proof Possible for the Theorem of Pythagoras? Michael de Villiers RUMEUS, University of Stellenbosch CONCLUDING COMMENTS To get back to the original question of whether a trigonometric proof for the theorem of Pythagoras is possible, the answer is unfortunately twofold: yes and no. 1) Yes, if we restrict the domain to positive acute angles, any valid similarity proof can be translated into a corresponding trigonometric one, or alternatively, we could use an approach like that of Zimba (2009) or Luzia (2015). 2) No, if we strictly adhere to the unit circle definitions of the trigonometric ratios as analytic functions, since that would lead to a circularity.
Can someone explain why exactly the other proofs told at the end of the video are not considered trigonometric proofs for the Pythagoras theorem, and why does this new proof special?
1) It's not the first trigonometric proof of the Pythagorean Theorem. 2) It's not pure trigonometry. 3) It comes straight out of a calculus book. It's a geometric series and the math is a Taylor series. 4) Homework questions phrased like "Find the length of the zigzag line in the triangle" are solved using this "proof".
By the way, the last proof that you showed is also one of the fundamental equalities in basic trigonometry, taken nearly straight from the definition of cosinus.
Did you know that the Pythagorean Theorem or the Equation to the Unit Circle is embedded within the simplest of all arithmetic equations: 1+1=2? And with that, so are all the definitions, identities and properties of Trigonometry. If you can construct a Unit Circle you already have the Pythagorean Theorem and from that Unit Circle you can construct a Right Triangle and this is why the Trigonometric Functions have a Pythagorean Identity. The equation 1+1 = 2 is the unit circle located with its center (h,k) located at (1,0). Each unit value of 1 is a radii and their sum of 2 is the unit circle's diameter. In order to understand this proof, you can not think of 1 and 2 as being a scalar value. You have to think of them as being unit vectors with the operation of addition (+) as being a linear transformation of those unit vectors. When you look at them within this context, then it becomes clear as to why the cosine also has a direct relationship to the dot product. I could even go one step deeper than this. All of these properties are actually embedded not just within this arithmetic or algebraic expression but they are also embedded within the identity properties of both addition and multiplication a+0 = a, a*1 = a, as well as in the equality or assignment operator as in a = a. If you don't think so, then consider the following equation: y = x. This is a specific case of the slope-intercept form of y = mx+b where b is the y-intercept and m is the slope of the line defined as rise over run which can be solved by m = (y2-y1)/(x2-x1) where (x1,y1) and (x2,y2) are any two points on that line. This can be simplified to dy/dx. With the case of y = x, the intercept is 0 as this line crosses through the origin (0,0) and the slope here is 1. The line y = x bisects the first and third quadrants at 45 and 225 degrees or PI/4 and 5*PI/4 radians. At 45 degrees or PI/4 radians, both the sine and cosine of those functions are equal, sqrt(2)/2 . This is also tan(45) or tan(PI/4). Within the linear equation in the slope-intercept form we can substitute m = dy/dx with sin(t)/cos(t) = tan(t) where t, theta is the angle between the line y=mx+b and the +x-axis. And this can be derived or constructed simply from the line y = x which is equivalent to x = x which is equivalent to x+0 = x and/or x*1 = x. x | y ==== .. | ... -3 | -3 -2 | -2 -1 | -1 0 | 0 1 | 1 2 | 2 3 | 3 ... | ... And this is just an simple expression or equation which doesn't even involve function composition. The reason this works is because the Pythagorean Theorem A^2 + B^2 = C^2 for all tense and purposes is equivalent to or a simplified form the equation to a circle (x-h)^2 + (y-k)^2 = r^2 where (x,y) is any point on the circumference (h,k) is the center of the circle and r is it's radius. So going back to the expression 1+1 = 2 (full circle) the tail of the first vector of (1,0) starts at (0,0) and it's head ends at (1,0). The addition operator (+) translates this vector in the same direction (+) by convention to the right by its magnitude of (1) since the second operand of the equation is also a unit vector. The result of this linear transformation (translation) now has the 2nd vector starting at (1,0) and its head is at (2,0). The center of the unit circle is (1,0) and both points (0,0) and (2,0) are on the circumference of the circle. We can substitute these in to the equation of the circle (x-h)^2 + (y-k)^2 = r^2 == (2-1)^2 + (0-0)^2 = 1^2 which == 1^2 = 1^2 = 1 which is the radius of the unit circle and is also the unit vector. The diameter of the unit circle is 2r which is 2*1. Therefore when you see any term in any mathematics of 2x or 2n, you actually have a solution to a given circle where x or n is its radius.This is also why the distance formula between any two points is directly related to the equations of a line as well as the Pythagorean Theorem. You don't need a "Triangle" to have a "Pythagorean Theorem". You only need a Line Segment. With that, everything within mathematics is related from basic arithmetics to algebra to geometry to trigonometry to linear algebra to calculus, and so on... These are all possible simply just because we can Count or Enumerate. This is why I appreciate numbers and mathematics.
I can dig it. You put a wonderful deal of effort into this reply and the relations are mutual. Excellent work! But. Trigonometry: tri. You need more than one line segment. In the circle, the perimeter is the so-called "continuous line", drawing not a boundary, but a measurable distance away from some point, for all points swept out by the so-called "full rotation" about the point... another line, here, delineates the distance r to some point p.
I think you are implicitly assuming the parallel postulate here. The sum angles in a triangle don’t have to be 180 degrees, but if they are, then you basically assuming thing that make the Pythagorean theorem true
What about it The parallel postule and the Pythagorean theorem are equivalent So all proofs must assume or imply something equivalent to the postule, else you have a proof that's independant of to it
Here's another way to prove it with trigonometry: d/dx (cosx) = -sinx and d/dx (sinx) = cosx is proven without pythagoras. We say that: (sinx)^2 + (cosx)^2 = f(x), for some function f DIFFERENTIATE BY X TO GET: 2sinx(cosx) + 2cosx(-sinx) = f'(x) 0 = f'(x) --> f(x) = C, where C is a constant (sin0)^2 + (cos0)^2 = 1 --> C = 1 --> (sinx)^2 + (cosx)^2 = 1
@MindYourDecisions I appreciate that you try to be historically sensitive. I really do. But there is a lot of value in naming conventions. If a large portion of people know the thing under the same name, the Pythagorean theorem, then that's an achievement all on its own. And it's not worth giving that up for the chase of "Who did it first".
The simplest is to take the limit as spherical trigonometry approaches plane trignometry. Use the formula cos A = cos B cos C + sin B sin C cos(a - b). A B and C are the sides and a, b, c the vertex angles of a spherical triangle - all dimensionless. In the limit A, B, and C are very small and cos A -> 1 - 1/2 A^2 and sin A -> A to highest order. Let the angle a-b be pi/2, then we get (1 - 1/2 A^2) = (1 - 1/2 B^2) (1 - 1/2 C^2) - working out to highest order, A^2 = B^2 + C^2. This proof does not presuppose the theorem in any way, just the power series expansions for sin and cos and the angular geometry of a sphere. It works in any number of dimensions as well.
Excellent work by these two young scholars!!! Now lets get the proof added to Algebra, Geometry, Trigonometry, & Calculus texts. This is good for young people to see they too can add to our math base. Most classes at the HS level are boring, uninspired, and cover nothing new or interesting. Students need to see they can connect to advanced mathematics and that math when connected to actual practical problems is useful, exciting and very much connected to their lives.
Using trig on a triangle with side a, b, c and angle A. a = c*cosA. b = c*sinA, a^2 + b^2 = c^2*(cosA)^2 + c^2*(sinA)^2 = c^2*((cosA)^2 + (sinA)^2) = c^2. I used only trignometry.
I was taught in grad school the Greeks did not have algebra. The Greeks had a straight edge and a compass, so using trigonometry with algebra seems slightly anachronistic. I would be really blown away to see the proof using just a straight edge and a compass! On the other hand, my hat goes off to the two high school students. Great work.
Uncle Tony, we should praise the ladies for doing what 99% truly can't . but it is sad that older dudes in the comments are salty because 2 girls not only came up with something new and they did so with 2 separate solutions of their own.
This is a clever construction, but I would not be surprised at all if some previous mathematics student (undergrad or grad) made a similar observation that such an infinite series could be constructed to provide a demonstration of this famous theorem. I'm happy to see that a HS student had the patience and determination to explore this approach because that is how mathematics is done in the professional level, just with higher complexity of concepts/techniques and typically a much longer period of time.
There is already an infinite series proof by John Arioni. Johnson's "waffle cone" can be found in many calculus textbooks. Jackson's proof was a copy of another that can be found at cut-the-knot (proof 60) by B. F. Yanney and J. A. Calderhead.
The infinite geometric series part is a pre-existing proof. They just slapped a couple of triangles on top of it and it just happened to work. Makes me wonder if you could do the same with other known proofs.
I am surprised no one has thought of this before for 2000 years. It was so satisfying when I realized how they were going to attack the proof. Good job!
Well, the thing is that there are other proofs that use the same law of cosine, are much simpler and do not rely on limits. Kudos to the students for their creativity, but they have just reinvented a crooked wheel.
@@kyriakosphilitas6950 Jason Zimba did 14 years ago though, and you don't even know who he is. Why did the the news of the first to discover it not make it to you but this news did? Most likely because they are black, female, high school students. If they read this paper he wrote beforehand it is less impressive. forumgeom.fau.edu/FG2009volume9/FG200925.pdf
@@kyriakosphilitas6950 He most likely did considering how the first officially known and approved modern FULLY TRIGONOMETRIC proof of Pythagoras was published in the early 2000s by Jason Zimba so... it is clear that these student's didn't really come up with anything new. As a matter of fact, Zimba's proof was taught in highschool to us and is written in my book, which I still have in my posession to this day.
You're quite right. The use of trigonometry in this proof is quite unnecessary. The use of sines is simply a different way of expressing the similar triangles and adds nothing useful to the proof.
I'm an engineering student and I don't know who said no proof existed before. I've proven it with a basic method that takes 2 minutes to explain with 1 square
Even with everything it's really really hard to follow but hell I had a hell of a time with plain geometry so this was practically impossible for me to follow
My math teacher literally proved it to us using trigo like 10 years ago and nobody cared cuz we assumed it was common knowledge and a waste of time 💀💀💀
99% of trigonometry is dependent on a²+b²=c². If your teacher used something like sin cos tan, that's not REALLY considered proof. For the same reason you can't have two essays that cite eachother as a source.
@@josephwodarczyk977 Well, the thing is that there are other proofs that use the same law of cosine, are much simpler and do not rely on limits. Kudos to the students for their creativity, but they have just reinvented a crooked wheel.
Just draw in any right angled triangle the vertical to the hypotenuse c going through the opposite corner. Name the 2 segments a and b. With similarity follows a^2 + b^2 = c^2
No mathematician ever said it was 'impossible'. Nor has any professional mathematician been trying to do a trig proof in the past 2000 years. There is so much nonsense in this story that it makes my brain hurt.
I love when a proof that seems to be spiraling away into unintelligible complexity suddenly is resolved into simplicity itself, seemingly out of the blue. Bravo!
É incrível como na matematica existem coisas que ainda não descobrimos, e ainda não provamos A matemática sempre me fascina pois ela é um alfabeto completo que podemos descobrir coisas com puro raciocínio e apenas lendo de forma alfabética, é legal resolver problemas nessa elegância de ler matematicamente, e por isso apoio totalmente olimpíadas de matemática, são incríveis A matemática é cheia de incompletudes e apesar de parecer estressante, só me deixa mais fascinado com essa idéia A matemática tem muita coisa ainda a ser tratada. Mesmo que demore 300 anos pra provar algum problema proposto por algum matematico maldoso. Kkkkk quem sabe mexer de verdade com a matemática, pode usufruir de tudo dela e dos problemas, mas também pode ser um cara chato que aterroriza todos do ensino médio com sua fórmula que ele cria para provar alguma coisa bem doida. Amo
While proofs like the one at 15:10 use trigonometry, it doesn’t really require it - you could do almost the same proof just using similar triangles and not mentioning sine. The student’s proof is arguably “more trigonometric”, since it uses the law of sines, while still not requiring anything beyond trigonometry.
proof of Pythagorean theorem involving only trigonometry: assuming cos(b) != 0 (for b != Pi/2 ) cos(b) = cos(a-(a-b)) = cos(a)cos(a-b) + sin(a)sin(a-b) =cos(a)cos(a)cos(b) + cos(a)sin(a)sin(b) +sin(a)(sin(a)cos(b)-cos(a)sin(b)) =cos(b)[ cos^2(a) +sin^2(b) ] cos^2(a) +sin^2(b) = 1 the teenagers haven't proven anything that hasn't been known for thousands of years the general formula for triangles is c^2 = a^2 + b^2 - 2ab cos theta, where theta is the angle between sides a and b trigonometry (i.e., sin, cos, tan, etc.) depends on an understanding of Pi and of radians -- that the circumference of a circle is proportional to its diameter... once you have that then the trig functions are straightforward to understand: the hypotenuse of an inscribed triangle grows with the radius of the circle and so does the "sine" and the "co-sine" of the arc generated by the chord (which are the lengths of an imaginary bow, with the bow itself being the circles arc, etc.)
The true master equation of planar geometry is Norman J. Wildberger's Cross Law, which is the cosine law squared (=1-sin²), out of which Pythagora's theorem is a limit case when the spread ("sine squared") between two sides of a triangle is 1. The other limit case is when a spread of a (degenerate) triangle is 0, hence about 3 collinear points, as strong and as fundamental as Pythagora's theorem. This video is much wind for nothing, NJWildberger cracked geometry properly, without the sine qua none need for infinite processes 😊 Nevertheless, I enjoy the channel and content very much!! Keep up the good work 😁🙏🙋
Pythagoras' theorem is not part of trigonometry as it does not involve angles, but only sides...it is wrongly lumped with trigonometry because it's a triangle...
I really wish math instructors would focus more on the details. I get that this is youtube, and there is a certain standard for the length of videos, but I would have loved to learn all the reasoning for example visually of all the connecting ideas within the equations. (and can literally be a second-long, you can throw a picture in with text that explains how you were able to move forward). I loved that I remembered how to simplify complex formulae, but even for youth, I can see it being extremely useful for reviewing material. Also, how much do I bet that length and rate of 'pause' in a video is the single most defining factor in how interested people are in your videos.
This proof is implicitly using the flatness of euclidean geometry in that current form of the law of sines (there is a modified form in the other geometries), and is using this flatness in saying sin(alpha)=a/c etc. since this is directly derived from the law of sines. In general, this just simply is not the case. Therefor, this proof is using implicitly the assumption of a flat euclidean geometry, which is the geometry which assumes the parallel postulate, and the parallel postulate is equivalent to the pythagorean theorem. Therefor, the proof is starting with the assumption that the pythagorean theorem is true, albeit subtly and implicitly. It is a smart proof, but it is circular.
It is not an impossible discovery. All we have is another proof of the Theorem, which is nice for the students, but other than that, not a remarkable discovery that goes into history books. The reason why this is "hyped up" is because the Pythagorean Theorem relates to a lot of people through their high school math. So easier accessible for general audience. The proof of Fermats theorem by Wiles was far more ground breaking than anything else but then...who knows Fermat anyway?
Since when infinite series is "trigonometry alone"? There is another branch of mathematics that gives as the sum of it, it's called calculus. That was NOT a proof using "trigonometry alone".
They used trigonometry to prove that x = 2ac/b (see ruclips.net/video/juFdo2bijic/видео.html) , but they could have proven it by the proportionality of sides in similar triangles (in fact, that's what Pythagoras used). In other words, it is really just one more proof of the theorem. Anyway, it is great for high school students.
It is, first, a circular reasoning test... pun subtended. Using the definition of sin to prove sin is literally circular reasoning. This is a big gut check: drop nuts and call a spade a spade.
I'm confused... you say the students found the impossible: A trigonometric proof of the Pythagorean Theorem. Then you proceed to show it, and it uses infinite iterations of smaller triangles. That's fine. But then in the "More Proofs" section of this video you show a different, much simpler trigonometric proof. Wasn't that simpler proof already known?
you are right. People like to think about these things as if they're mystical so they fall into the trap of fruitless erroneous imagination (and also fall under the illusion that they are smart and very special)
The part that I don’t understand is how does the geometric series count under trigonometry? I would’ve considered that to be a more algebra or calculus concept
In this "proof" sin(alpha) is just the ratio a/c where a is the side opposite alpha and c is the hypotenuse. So we don't need to mention the sin function and in that sense, it's not "trigonometric". The "proof" assumes the ratio properties of similar triangles, which themselves are proved from the Euclidean axioms. Euclid gives a far simpler ratio proof. He also gives an area proof, but this depends on further axioms to do with areas.
I'll give you that this proof technically doesn't use trigonometry, though I don't know why you put proof in quotes. But in your same line of reasoning, nobody uses trigonometry because it's all ratios anyway.
@@mike1024. OK the quotes were unnecessary. The point with the ratios is yes, they provide definitions of sin, cos, tan etc, but they don't tell you how to calculate them. In fact, the functions were tabulated from mediaeval times or before, in various places and by various methods, culminating in the modern series definitions. Anyway Euclid + trig functions is different from Euclid by itself, and necessary for astronomy, surveying and the like.
@@pwmiles56 I see what you're saying! I'm aware that the trig ratios were originally used for astronomy hundreds of years ago, but I've never actually studied their history. It still might be worth doing! I have no idea how they calculated the values previously.
@@mike1024. There's an article on Wiki, History of Trigonometry. They used various formulae, in the process discovering some of them, e.g. sin(a+b) = sin a cos b + cos a sin b With this and the corresponding formula for cos(a+b) you could make a a general angle and b a very small one and kind of inch forward that way.
they are only tricking us into thinking, its a trigonometric proove. However its a graphical one, cause it only gives a name to a/c and call it sin(α). But at that point its just a name,and has nothing to do with trigonometry itselve.
It's not actually their proof. They took a known proof and just added a little bit of trig so it's not actually a trig proof. It's only superficially trig.
I would guess because there wasn’t really any knowledge about infinite geomteric series back then, or infinite series in general. Those were concepts that were only really developed much later. That would me my guess.
Technically you can prove the theorem with sin²x + cos²x = 1 without circuluar reasoning. You can prove sin²x + cos²x = 1 wothout the pythagoras theorem. Pf: Since sin x and cos x are continuous functions, so are the sum of their squares. Then, take the derivative of sin²x + cos²x, which is 0. Since sin²x + cos²x is continuous and has a derivative equal to 0, sin²x + cos²x is just a constant. Substitute in a good value for x: sin²0 + cos²0 = 1. Since sin²x + cos²x is constant, it is equal to 1.
@@gealbert5737 We wanna prove lim x->a sin x = sin a. We use epsilon delta: Using the difference of sines formula, sin x - sin a = 2 cos ((x + a)/2) sin ((x - a)/2). Thus abs(sin x - sin a) = abs(2cos((x + a)/2)sin((x - a)/2))
I once sat a test and I wrote "therefore by Baudhayan a^2 + b^2 = c^2" and I got 0/10 The teacher commented, "It's the Pythagoras theorem, what is wrong with you?"
I've always thought that the idea of a "trigonometric proof" was silly, as trigonometry and geometry are the same. Consider the law of sines, it may seem that its application here is somehow "escaping geometry by using trigonometry" when in reality its all the same. Consider a triangle with sides A,B,C and angles a,b,c (angle a opposite of side A and so on). Then let h_a be the height of the triangle when A is the base (also called the altitude off angle a) and h_b and h_c defined similarly. Then we can show that h_a/BC = h_b/AC = h_c/AB. We can do this using an area argument or a similar triangles argument. This relationship is equivalent to the law of sines, notice that if we were using trig sin(a) = h_c/B and so on for the other angles. It is less intuitive sure, but we could have used the geometric relationship and similar triangles to arrive at the same result, and I image that while it may be convoluted to do so we could "triggify" many geometric proofs. This does not take away from these girls unique approach involving the geometric series, which was something I'd never thought I would see in a geometry proof.
This is not groundbreaking in math. It is just a nice geometry problem plus a little bit of calculus. Groundbreaking works in mathematics generally shed light on a subject in a way that has never been considered before. This does not really come close to any revelation about Pythagoras' theorem, it is just another proof using limits, in fact you never have to mention sin and the proof stays the same. Groundbreaking work is stuff like the introduction of perfectoid spaces, or the development of the Lebesgue measure, central models, etc etc. These are brand new ideas that fundamentally change their subject.
![Pythagoras Would Be Proud: High School Students' New Proof of the Pythagorean Theorem [TRIGONOMETRY]](/img/1.gif)
It took these 18 year olds 2000 years to solve this math problem. That’s what I call dedication
wut? Why 2000 years?
@@shadesmarerik4112It is quite an obvious joke, and like the first thing said in the video
@@sovietmogus3263 and no, its not nearly what was said in the video
@@shadesmarerik4112 copium just admit you missed the joke
@@shadowsavage3693 ow its kinda revealing, that u project onto me that this "joke" appeared intellectually challenging to u. i feel sorry for u
Very clever high school students. This would have made a great undergraduate senior project in mathematics.
I think this is already known
hoeever this isnt really trigonometric, but rather graphical. Its just that you call a/c=sin(α) but at that point this is just a name. So it has nothing to do at all with trigonometry itselve.
@@neutronenstern. I think the point is that if you draw triangles with the same angles, the lengths of the sides may vary but the ration between the side lengths stays the same. This is the whole basis behind trigonometry, and is why we can say that there is such a thing as sin(α).
It's a little like how Ohm's Law in physics actually says that in a conductor the current that flows is (all things being the same) proportional to the voltage. This means that you can define resistance as being this ratio, and can say that V = IR, which is what many people (including some electronics experts) think is Ohm's Law.
@@pauln7869 yes, but with sin(x) its special, i think. Cause in this proove no property of sin was used,without deriving it geometrically first. no actual value of sin was used.
So its kind of a trick to say, this is trigonometry. This proove works exactly the same, without needing the help of any sine.
"This would have made a great undergraduate senior project..." Why do you need it to be college students for it to be "great"? It is high school students, it is a great project, they deserve all credit accorded mathematicians of any age or school level, for having developed this proof.
Good for them. Even if this proof may not be commonly used in everyday-life solving the Pythagoras theorem, it is still a spectacular discovery.
Applications for their discovery may be discovered in the future.
This proof has many indirect applications and it reminds me of Archimedes' Method of Exhaustion.
Já tem inúmeras aplicações. Pq o método delas se relaciona diretamente com limites e cálculo infinitesimal que é o que se faz em matemática na prática. Adaptando esse método já deve ter surgido vários métodos baseados em pós-graduação associados a matemática computacional e muito mais.
has it been verified that no one else came up with this proof first?
These proofs will take years for them to be put into action.
Thanks so much for the shout out! You didn't have to be so forthright with your citation but you were and i greatly appreciate it.
You are the high school student ??
@@kiloperson5680 no, this channel was presh talwalker’s inspiration for the video, as seen at 2:05
Lol what do you mean he didn't have to be forthright with his citation?
The most fascinating thing that no one is talking about is that we human beings uses shared knowledges across time periods and always figure out a way to make things better, and discoveries aren't always made by people that society deems genius, but people that are curious enough to put in the time and effort.
But every person who society deems a genius has discovered something or made something new, because otherwise they wouldn't be considered a genius.
yeah bro. no one has ever talked about scaffolding.
And then published it in a closed form so it’s (again) not commonly available until someone reverse engineers it - at least without solving a riddle or poem in this case
@@-AxisA-That's just completely untrue.
@@jasonnelson9141 Name 1 or a few generally recognized geniuses then who didn't invent or discover something.
To be fair, trigonometry is based on the parallel postulate, which is equivalent to the Pythagoras theorem. The students theorem is based on the assumption the extended lines must intersect.
1:19
Elegantly stated. :)
True, and that is dependent on whether alpha and beta are equal. If you take the limit as they approach the same angle, you would get the answer for when those two are equal.
I might have missed something, but doesn’t using an infinite series avoid having to rely on Euclid’s fifth postulate?
@@frogmorely Nope.
The fifth postulate presupposes Euclidian (flat) space. Euclidian space may be more formally described as a space in which distances between points in the coordinate system are calculated using pythagoras. (There are other possibilities.)
Euclidian space has the feature where non-parallel lines intersect in exactly one point. This feature is required for the construction used.
IOW, Pythagoras is actually smuggled into the construction when the lines are extended to the intersection.
Nevertheless, still a clever way of tackling the problem.
I can see how all this mathematics is accessible to high schools (albeit advanced high schoolers), but it also took the creativity and grit of two young mathematicians to put it all together. Bravo!
These two are geniuses for figuring this out.
One thing that I was interested in hearing is your opinions on Animation vs. Math.
yesss
absolutely
why do you say they are geniuses? don't you think you couldve done the same thing??
@@leif1075 It's called a compliment
I called them geniuses because I would've never been able to guess that.
@@michaelgum97so if someone figures something out you couldn't then that means they are geniuses? I am not saying they may not be very smart or their accomplishment shouldn't be acknowledged.
It depends a lot on what we mean by *only* trigonometry. In particular, it means that there is something used in the common proofs of the theorem that has to be excluded. If not one could just apply a thin varnish of trigonometry on another proof.
An obvious candidate would be to exclude geometry (which would then exclude this new proof) but then how do we define the trigonometric functions?
We should do it axiomatically. If we give differential equations as axioms, it might be possible to prove the theorem using analysis but would not really fit the notion of what trigonometry *is*. If we want to use some of the common trigonometric formulas as axioms, we would need to not include sin²x+cos²x=1 (as that is essentially what we want to prove) but even from other formulas (like the formulas for sum of angles), we can quite trivially 'prove' the sum of squares (and so decide that we kind of put the result in the axioms).
Nevertheless it is an interesting proof with non-trivial trigonometry, geometry and limits in the long tradition of alternative proves of this most famous theorem. Congrats to them!
does einstein's proof on Wikipedia of pythagoras theorem not constitute as using trigonometry without circular logic?
the Taylor series was published in 1715, over 300 years ago
For some reason adding a character different from a space right before or right after your *bold text* breaks the formatting (my guess is that they didn't want people censoring words to have their 3 asterisks reduced to one bold one)
At least on mobile it does
Jason Zimba, who provided a trigonometric proof of Pythagoras in 2009, prefaces the paper with this:
"In a remarkable 1940 treatise entitled The Pythagorean Proposition, Elisha Scott
Loomis (1852-1940) presented literally hundreds of distinct proofs of the Pythagorean
theorem. Loomis provided both “algebraic proofs” that make use of similar trian-
gles, as well as “geometric proofs” that make use of area reasoning."
The proof presented in the video uses both similar triangles and area reasoning. We should consider this proof "algebraic" or "geometric" or both. I've little doubt that similar strategies are found elsewhere in the 367 proofs published by Elisha Scott Loomis.
Man, his voice by the end of the second proof sounds so happy and excited, it shows genuine love to mathematics and the topic in hand.
it's a computer voice.
@@sb3nder proof? It's better to voice the video than do an AI voice
I think the most beautiful proof of all is the following. You have a right triangle. Divide the triangle into two triangles by a line that is perpendicular to the hypotenuse and connects to the vertex of the right angle. Now you have three similar triangles: the original one and the two pieces. Because of the fact that the triangles are similar, each occupies the same fraction 'x' of the square on its hypothenuse. If the hypotenuses and surface areas of the three triangles are a, b, c and A, B, C, respectively, you therefore have A = x*a^2, B = x*b^2 and C = x*c^2. Because of the fact that A + B = C, then x*a^2 + x*b^2 = x*c^2 and therefore a^2 + b^2 = c^2.
just a fractal simple yet full of beauty i think about this when in traffic
"occupies the same fraction 'x' of the square on its hypotenuse" -- I think you mean "of the square _of_ its hypotenuse".
But beautiful proof! Thanks for posting this, very educational and much appreciated.
@@yurenchu You are right, that might have been a better wording. I originally found out about this proof when I was watching a video by James Grime that is called "A Pythagorean Theorem for Pentagons + Einstein's Proof." Definitely one on my favorite math videos on RUclips so check that out if you're interested.
@@jimi02468 Thanks, I went to watch it. It's been quite some time since I've watched one of "Singing Banana" 's videos.
But to be honest, I found the video not so surprising anymore, after having read the proof that you posted here. :-)
And I was expecting to see some equivalent theorem that's valid for pentagons instead of for (right) triangles, but there wasn't such a thing.
:-(
But still interesting to learn how Einstein looked at it, so thanks anyway!
The most beautiful proofs of this are picture proofs with no words needed.
I'm an old PhD in mathematics & I never heard the assertion anywhere at any time that it was "impossible to prove the Pythagorean theorem using trigonometry alone". Lol! Perhaps you can send me a link to someone making that assertion? ;)
Elisha scott loomis did make that asseration in a book titled "The Pythagorean Proposition" however it was disproven by Jason Zimba way before the students did.and most likely someone probably disproved it before zimba
@@tank2256 It's not even that Elisha Scott was wrong, very often the chosen analytic _definitions_ of the trigonometric functions are based on the Pythagorean Theorem. If your trig functions are defined that way, you can't prove Pythagorean Theorem with trig. Elisha was probably referring to that.
Indeed that is crap. You have to tell what axioms you use before claiming such a thing. I don't know what "using trigonometry alone" means. The Pythagorean theorem is very elementary. Trigonometry is a very sophisticated tool. This amounts to boasting you can crack a nut with a hydrogen bomb.
@@tank22561914 J. Versluys as cited at the bottom of Zimba’s paper.
1) It's not the first trigonometric proof of the Pythagorean Theorem.
2) It's not pure trigonometry.
3) It's only superficially trigonometric.
4) It's almost an exact copy of an existing proof. Look up John Arioni "Pythagorean Theorem via Geometric Progression"
I don't know... To me Pyth and Trig were always different formulations of the same qualities of triangles, expressed differently, but NOT from different principles. It always felt like they both started from the way that right triangles can be inscribed into circles. But I never devoted much thought into this. So it's interesting but ultimately I don't think a proof from Trig is independent at all.
Absolutely, no way are Pythagoras, Trigonometry separate. I don't remember ever being taught one without using the other.
@@quantisedspace7047Hence the point that it is now under peer review. so many haters in this comment section
I would assume this proof was overlooked until now because it relied on taking the sum of an inifinite series, which is a relatively modern concept. By the time that idea had been well established, the dogma that there were no trigonometric proofs was well established, so no one went looking until now. Bravo!
This.
Infinite series were around since the time of Euler and earlier, before the 19th century
How to reduce someone’s accomplishment on one hand while complimenting on the other. Well done
A trigonometric proof was published in 2009 by Jason Zima.. still cool cuz they're new proofs, good on them for sure
"I would assume this proof was overlooked until now"
It would be safer to assume it's been proposed by students many times and there are thousands of proofs like it that students have proposed that were never mentioned outside of their classrom. But since these chicks are black, it's national news.
the sine rule is based on right angle triangle drawn by taking one of the heights. so even though law of sine extends to any triangle the law inherently is based on basic definition of sin in relation to a right angled triangle. so I don't think the paragraph is wrong, these students found nothing new but just a longer proof of proving something which can be itself written as sin²a + cos²a=1
yes you're right, but all the idiots can't understand that. the greeks already discovered it 2000 years ago, and their whole point was to make it simple, not unnecessarily convoluted
1) It's not the first trigonometric proof of the Pythagorean Theorem.
2) It's not pure trigonometry.
3) There is an exception: It doesn't work for isosceles triangles.
In that last proof, after drawing in the rest of the chord that made up sin(theta), you can use the intersecting chord theorem to get sin^2+cos^2=1
Put a small square inside a larger square with their centers coinciding. Rotate the smaller square until its 4 corners touch the sides of the large square. The large square will be comprised of 4 identical right triangles and the small square. Write an equation stating that the area of the large square is equal to the area of the small square + the area of the 4 identical triangles. Rearrange the terms in the equation and voila!
Of course it is required that the area of the smaller square is at least half of the area of the larger square (otherwise the corners of the smaller square would never touch the sides of the larger square, and we wouldn't have any right triangles in the diagram). But yes, this is an excellent proof!
If the right triangles have right sides a and b (with a ≤ b) and hypotenuse c, then the larger square has side (a+b) while the smaller square has side c. The area of each triangle equals ½ab . Since the area of the larger square equals the area of the smaller square + the area of the four triangles, the equation is (a+b)² = c² + 4*(½ab) . After expanding, we get a² + b² + 2ab = c² + 2ab , and hence a² + b² = c² .
Additionally, to physically demonstrate the result, the four right triangles can also be paired to each other so that they form two separate, identical rectangles (with their diagonal equal to the triangle's hypotenuse), and these two rectangles can then be placed back inside the (emptied) large square: one rectangle horizontally in the bottom left corner, and the other rectangle vertically into the top right corner. The area of the large square that is not covered by the two rectangles consists of exactly one square in the top left corner with side b and area b² , and one square in the bottom right corner with side a and area a² ; and of course, it follows that these two squares together have the same area as the rotated smaller square (with side c and area c² ) that we had at the beginning.
@@yurenchu Yes, U got it. I never calc'd exactly the minimum size of the small square such that its corners eventually contact the sides of the big square when rotated but your 1st statement does precisely define it. The length of the diagonal from the small square's center to its corners must be greater than half the length of the big square's side, which works out to what you said.
My go at it;
The angle addition formula can be prooven without pythagoras theorem
cos(a+b) = cos(a)cos(b) - sin(a)sin(b)
let b = -a
cos(a-a) = cos(a)cos(-a) - sin(a)sin(-a)
= cos(a)cos(a) + sin(a)sin(a)
= cos²(a) + sin²(a)
We also have
cos(a-a) = cos(0) = 1
cos²(a) + sin²(a) = cos(a-a) = 1
q.e.d.
The identities cos(-x) = cos(x) and sin(-x) = -sin(x) can also be prooven with the angle addition formula.
Assuming cos and sin are well defined on the interval 0≤x≤0.5π, I am going to extended their definition to all real numbers, in a way that preserves the angle addition formula.
First, I evaluate sin and cos at π, 1.5π and 2π, by evaluating their angle addition formulas at 0.5π + 0.5π, π + 0.5π and π + π respectively.
Then I can proove that they have a period of 2π, by evaluating their angle addition formulas at x + 2π
cos(x + 2π) = cos(x)cos(2π) - sin(x)sin(2π) = cos(x)•1 - sin(x)•0 = cos(x)
Similar proof for sin(x+2π) = sin(x)
The identities cos(0.5π-x) = sin(x) and sin(0.5π-x) = cos(x) in the interval 0≤x≤0.5π can be directly observed in a right triangle.
Final step, evaluate cos and sin at -x, by evaluating them at 1.5π + (0.5π-x). This is the same as evaluating them at -x, due to the period of 2π.
cos(1.5π + (0.5π - x)) = cos(1.5π)cos(0.5π - x) - sin(1.5π)sin(0.5π - x) = 0•sin(x) - (-1)•cos(x) = cos(x)
Similarly sin(-x)=sin(2π-x)=sin(1.5π+(0.5π-x)) evaluates to -sin(x)
Insert pythagoras theorem proof here
Things like cos(π-x)=-cos(x) can all be prooven in a similar manner, extending the definition of cos and sin to all real numbers
There are well over 371 Pythagorean Theorem proofs, originally collected and put into a book in 1927, which includes those by a 12-year-old Einstein (who uses the theorem two decades later for something about relatively), Leonardo da Vinci and President of the United States James A. Garfield.
Exactly. The premise of the video is bogus.
@@hoochygucci9432 Not to mention that there are countless proofs that also claimed to be fully trigonometric, which is supposedly the thing that makes this proof interesting. The most recent example was a proof by Zimba that was made in the early 2000s. So clearly the video is bogus, and whatever these students are being rewarded for is clearly not what the video claims, because we've already had fully trigonometric proofs of Pythagoras for a looooong time. And when I say fully trigonometric I really mean it. Like, really really mean it.
@@tiranito2834why is this your perspective on life. There’s no need to be so negative. It’s impressive that high schoolers whose highest math is probably basic calculus were able to come up with a new and interesting proof for this theorem. Also the video isn’t the one making a big deal about it it’s just explaining the proof. Why can’t you just appreciate math, why does it have to be a contest that requires putting passionate people down.
@@joshuafrank1246 The problem is that they didn't make a new proof, look it up, they copied parts of multiple proofs, most of it straight out of Zimba's proof.
It is precisely because I appreciate math that this situation infuriates me. Because the fact that this is being treated as news is nothing but proof of lost knowledge. Things that used to be common knowledge just about a decade ago are now cutting edge and people act as if they had never heard of fully trigonometric proofs for pythagoras before.
1) It's not the first trigonometric proof of the Pythagorean Theorem.
2) It's not pure trigonometry.
3) There is an exception: It doesn't work for isosceles triangles.
I like the history you are providing in recent days. is this related to your earlier poll and will you discuss more mathematical history? It's definitely interesting to know, especially considering how despite different cultures and different times, mathematics connects us all. I remember seeing depictions of 'Pascal's Triangle' from a Chinese mathematician and Ancient Indian Mathematician. Even without knowing the language, I can understand it as Pascal's triangle.
The "gameplay vs lore" Meme absolutely sums up maths perfectly
While doing maths in general is.... "Fun", the history that maths itself has is insanely interesting and intricate. Throughout hundreds to thousands of years, humans have advanced in this language, the language of truth itself
Media blowing up with nonsense: Zimba already did the "impossible" (itself a questionable statement by Elisha Loomis) nearly 15 years ago.
Honestly, I still don't really get what is so special about this proof. Of course, it's a very unique proof and those teenagers are apparantly brilliant. But the 'never done before' part of it is not entirely clear to me. A statement in a book that some kind of proof is impossible is not the same as a real mathematical conjecture.
So I can't really tell if the media is blowing things up, but I'm suspicious.
The media but also other RUclipsrs are clueless on this.
The majority of this "proof" is showing the trigonometry identity:
sin(2 alpha) = 2*tan(alpha) / ( 1 + tan(alpha)^2 )
But this is already a known fact and has nothing to do with proving the Pythagorean theorem.
Replacing tan(alpha) with a/b gets you pretty close to the end result.
Nevertheless, these students discovered this by themselves and that is quite an achievement.
its because its a bunch of kids. Why can't you just let them feel like they did something cool? Imagine if you were a highschooler who discovered something and online everyone is just saying how you are a fraud and your work isn't important.
You all are obviously jealous because instead of being positive you are going out of your way to find something negative to say. Its so sad.
@@Brad-qw1te I agree that it's cool, but I don't see how it merits media coverage to this extent.
@@Brad-qw1telol don't waste your energy on toxic people they are always there in our lives . I mean if the scientific community acknowledges that this is something to celebrate then some nonentities comes and dispute that what can you say? You should just let them swallow their toxicity alone
The last proof involving just the circle is (at 16:21s) the best and elegant proof of fundamental trig identity w/o using "pythagoras theorem"
The funny thing is that neither of them wanted to pursue mathematics in college but instead, if my memory is correct, environmental engineering and biochemistry. Of course both are math heavy but only math. Also, I believe it originated out of school competition and not just a paragraph in a textbook.
Burnham there are lots of hateful phd and math grads in the comments. 2 high school teens born in in the late 2000s just crushed them .
Well... Actually this proof uses not only trigonometry. It uses also limits and similarity. And you can easily prove the theorem using similarity only. :-)
Premislao, ok then provide us with the actual alternate solution.
@@PHlophe there was a proof published in 2009 using trig (Jason Zima) look it up
I fear the only people who know how to follow the logic of such a proof already understand the flaws in the two young girls' explanation.
@@bd12843 but yet you can't provide evidence yourself. Jealousy and rage are eating you inside.
@@PHlophe Hi PHlophe, please consider a right triangle ABC, with the right angle at C. Let AB be the hypotenuse, so AC and BC are the legs.
Now draw the altitude from C to AB, which will intersect AB at point D. Then AB + BD = AB.
This will create two smaller triangles, triangle ADC and triangle BDC, both of which are SIMILAR to triangle ABC by angle-angle criterion (angle ADC = angle ABC and angle DAC = angle BAC)
Triangle BDC is similar to ABC for the same reasons
Since ADC and ABC are similar, we can form the ratio of sides AD / AC = AC / AB (1)
and since BDC and ABC are also similar, we have the ratio BD / BC = BC / AB (2)
Now ADC and BDC have the same altitude CD. So that it looks like the pythagorean theorem, let AC = a, BC = b, and AB = c, which gives AD = a^2 / c and BD = b^2 / c after cross multiplying and substituting ratios (1) and (2)
Then AD + BD = a^2 + b^2 / c
Since AD + BD = c, we have
a^2 + b^2 / c = c, or
a^2 + b^2 = c^2
I'm not sure how I was able to do that through all the jealousy and rage eating me up inside, but it looks like I deserve a Fields medal :)
7:46 If alpha + beta is 90 degrees then you're assuming the euclidian parallel postulate, which is equivalent to the pythagorean theorem. Seems to me that the students are implicitly assuming that pythagoras is true in this proof, but I might be wrong.
well he just assming that all angles in a right triangle add up to 180 degrees right? if that implies pythagorean theorem, then idk
The Pythagorean Theorem are derived from Euclid's Postulantes, not the inverse.
@@rohangeorge712 The parallel postulate is that a line can be drawn parallel to anyother line, which if you remember 7th grade math, is used to prove the angle sum property.
Since it is a postulate, it is allowed to be used. All other proofs also use the parallel postulate, they literally HAVE TO.
1) It's not the first trigonometric proof of the Pythagorean Theorem.
2) It's not pure trigonometry.
3) The same proof could be done without any trigonometric equations at all. It's only superficially trigonometric.
There is a separate proof for the case of isosceles triangles, it's in the first few minutes of the video. Seems like lots of commenters literally didn't even watch it...
At 12:18 you have expressions for all three sides of the big triangle, which is a right triangle. You can show that the sum of the squares of the two shorter sides is equal to the square of the hypotenuse. That proves the pythagorean theorem without needing to go back to the original triangle.
I had thought the same
he needs to prove that a^2 + b^2 = c^2, just because the sum of the square of the two shorter sides is equal to the hypotenuse, that doesn't prove a^2 + b^2 = c^2. that would just prove the case for the super large triangle, which could just be a coincidence.
@@rohangeorge712no it wouldn’t because all the triangles are similar are form a right triangle cause of corresponding angles. It wouldn’t be a “coincidence”
@@prod_EYES yea nvm ur right mb
@@prod_EYESTechnically yeah, it would prove the Pythagoras Theorem but I guess they did that because the main objective was to prove specifically 'a²+b²=c²'.
I don't know if I missed it in the video; but I was a tad unconvinced, at first, that the similarity factor of all of the waffle-triangles would be the same. They ARE definitely similar, but I think it just needed a short explanation as to how you can tell the similarity factor for all of the triangles would be the same.
Your old side "a" becomes your new side "b" (ie, your short side length from the larger triangle is now your shorter side length of the smaller triangle). Based on the similar triangles, the ratio of the short side to the long side is a/b. That's basically it. Every time you want to calculate the next unknown perpendicular length, you would use this ratio (its also the tangent ratio!), so your similarity scaling factor is a/b.
After typing this out, I think this was touched on, but I hope this extra explanation is helpful for anyone who might have been initially unconvinced.
Congrats to these high students. I would not say they used trigonometry to prove the theorem. Instead, they used calculus plus a clever geometry construction to prove it, which is a great effort. BTW, I never heard such comments from mathematicians that a particular tool is IMPOSSIBLE to solve a specific problem.
1) It's not the first trigonometric proof of the Pythagorean Theorem.
2) It's not pure trigonometry.
3) There is an exception: It doesn't work for isosceles triangles.
When it comes to lost and/or missing proofs, I want you to reflect on how much knowledge has been lost to time because our own destructive tendencies.
Perhaps even more amazing than these students coming up with this never-before-discovered proof, is that the proof they came up with is relatively understandable by lay people. Great job!!
Wow! When I first heard about this proof from my mom, who hardly knows any math, by the way, and she told me that the students used trig to prove PT, naturally I was quite skeptical, but after watching this video, I now see that this is indeed a remarkable proof! Congrats to Ne'Kiya Jackson and Calcea Johnson! I wish I could have done something half this great when I was in high school!
It's possible to prove the law of cosines without using the Pythagorean theorem. The proof can be found on Wikipedia for example. Now using the law of cosines on a right triangle just gives the Pythagorean theorem as a special case.
This story and the proof that comes with it are nothing but brilliant, thank you!
1:27
I wonder, why they came up with this rather strange statement. The sine theorems in (right) triangles can easily be deducted purely from the definition of the sine, just like it's done in the video. Also it seems quite obvious that sin alpha = a/c will hold if and only if gamma=90° .
So anyway I just googled this book... it's from 1940. Not that people couldn't have known better back then, but it is quite imaginable that nowadays such books would be checked much more thouroughly by many more people. So that might explain it in conjection with someone being a bit too excited about the topic.
It's called a geometric series and scaling.
It's infinitely repetitive.
It says a^2 + b^2 = C^2 because a^2 + b^2 = c^2 over and over and over again for infinity.
It's was also already done in a previous proof:
John Arioni "Pythagorean Theorem via Geometric Progression"
There are literally hundreds of trigonometric proofs of the Pythagorean Theorem.
Every trigonometric function is only a ratio of 2 geometric knowns.
EVERY geometric proof to the pythagorean theorem can be expressed in terms of trigonometric functions.
This strikes me as more of a geometric proof than a trigonometric one. Most of the steps in Case 2 are using trig functions to add extra steps to simply taking ratios of similar triangles. Nonetheless, it is a very nice and creative proof.
I love the new proof using infinite series …
I think i recall learning the first alternate/other proof in high school, - a new proof - genius! these students are legends in high school. Thanks everyone for shining light on this!
There is another trigonometric proof I have been taught at school. The only thing you need is to come up with a shape of an outer square with the side a+b, inner square with side c and the rest of space filled with 4 equal right triangles rotated 4 ways (0, 90, 180, 270 degrees). Essentially, you put those in the corners of a big square. Then you write the area of the big square two ways and voila - very easy;)
You lost me but i trust you
The thing that is still missing is that it is BY DEFINITION what sin and cos mean, and they are based on right triangles. There is nothing new in the “proof” here, except to take as defined one thing vs another, neither of which are independent (of course not, otherwise Pyth theorem would not be true).
Sin and cos can be defined outside of triangles using Taylor series
1) It's not the first trigonometric proof of the Pythagorean Theorem.
2) It's not pure trigonometry.
3) There is an exception: It doesn't work for isosceles triangles.
math is not based on true reality, it will never be completely proven or real.
@@kirkb2665Why 2 and 3?
@@needtasteingames This Math PhD explains it better than me:
Is a Trigonometric Proof Possible for the Theorem of
Pythagoras?
Michael de Villiers
RUMEUS, University of Stellenbosch
CONCLUDING COMMENTS
To get back to the original question of whether a trigonometric proof for the theorem of Pythagoras is
possible, the answer is unfortunately twofold: yes and no.
1) Yes, if we restrict the domain to positive acute angles, any valid similarity proof can
be translated into a corresponding trigonometric one, or alternatively, we could use
an approach like that of Zimba (2009) or Luzia (2015).
2) No, if we strictly adhere to the unit circle definitions of the trigonometric ratios as
analytic functions, since that would lead to a circularity.
Can someone explain why exactly the other proofs told at the end of the video are not considered trigonometric proofs for the Pythagoras theorem, and why does this new proof special?
I also want to know
More views
1) It's not the first trigonometric proof of the Pythagorean Theorem.
2) It's not pure trigonometry.
3) It comes straight out of a calculus book. It's a geometric series and the math is a Taylor series.
4) Homework questions phrased like "Find the length of the zigzag line in the triangle" are solved using this "proof".
I saw this proof when it first came out last April. A very elegant proof and good on the students.
By the way, the last proof that you showed is also one of the fundamental equalities in basic trigonometry, taken nearly straight from the definition of cosinus.
Did you know that the Pythagorean Theorem or the Equation to the Unit Circle is embedded within the simplest of all arithmetic equations: 1+1=2? And with that, so are all the definitions, identities and properties of Trigonometry. If you can construct a Unit Circle you already have the Pythagorean Theorem and from that Unit Circle you can construct a Right Triangle and this is why the Trigonometric Functions have a Pythagorean Identity. The equation 1+1 = 2 is the unit circle located with its center (h,k) located at (1,0). Each unit value of 1 is a radii and their sum of 2 is the unit circle's diameter.
In order to understand this proof, you can not think of 1 and 2 as being a scalar value. You have to think of them as being unit vectors with the operation of addition (+) as being a linear transformation of those unit vectors. When you look at them within this context, then it becomes clear as to why the cosine also has a direct relationship to the dot product.
I could even go one step deeper than this. All of these properties are actually embedded not just within this arithmetic or algebraic expression but they are also embedded within the identity properties of both addition and multiplication a+0 = a, a*1 = a, as well as in the equality or assignment operator as in a = a. If you don't think so, then consider the following equation: y = x. This is a specific case of the slope-intercept form of y = mx+b where b is the y-intercept and m is the slope of the line defined as rise over run which can be solved by m = (y2-y1)/(x2-x1) where (x1,y1) and (x2,y2) are any two points on that line. This can be simplified to dy/dx. With the case of y = x, the intercept is 0 as this line crosses through the origin (0,0) and the slope here is 1. The line y = x bisects the first and third quadrants at 45 and 225 degrees or PI/4 and 5*PI/4 radians. At 45 degrees or PI/4 radians, both the sine and cosine of those functions are equal, sqrt(2)/2 . This is also tan(45) or tan(PI/4). Within the linear equation in the slope-intercept form we can substitute m = dy/dx with sin(t)/cos(t) = tan(t) where t, theta is the angle between the line y=mx+b and the +x-axis. And this can be derived or constructed simply from the line y = x which is equivalent to x = x which is equivalent to x+0 = x and/or x*1 = x.
x | y
====
.. | ...
-3 | -3
-2 | -2
-1 | -1
0 | 0
1 | 1
2 | 2
3 | 3
... | ...
And this is just an simple expression or equation which doesn't even involve function composition. The reason this works is because the Pythagorean Theorem A^2 + B^2 = C^2 for all tense and purposes is equivalent to or a simplified form the equation to a circle (x-h)^2 + (y-k)^2 = r^2 where (x,y) is any point on the circumference (h,k) is the center of the circle and r is it's radius. So going back to the expression 1+1 = 2 (full circle) the tail of the first vector of (1,0) starts at (0,0) and it's head ends at (1,0). The addition operator (+) translates this vector in the same direction (+) by convention to the right by its magnitude of (1) since the second operand of the equation is also a unit vector. The result of this linear transformation (translation) now has the 2nd vector starting at (1,0) and its head is at (2,0). The center of the unit circle is (1,0) and both points (0,0) and (2,0) are on the circumference of the circle. We can substitute these in to the equation of the circle (x-h)^2 + (y-k)^2 = r^2 == (2-1)^2 + (0-0)^2 = 1^2 which == 1^2 = 1^2 = 1 which is the radius of the unit circle and is also the unit vector. The diameter of the unit circle is 2r which is 2*1.
Therefore when you see any term in any mathematics of 2x or 2n, you actually have a solution to a given circle where x or n is its radius.This is also why the distance formula between any two points is directly related to the equations of a line as well as the Pythagorean Theorem. You don't need a "Triangle" to have a "Pythagorean Theorem". You only need a Line Segment. With that, everything within mathematics is related from basic arithmetics to algebra to geometry to trigonometry to linear algebra to calculus, and so on...
These are all possible simply just because we can Count or Enumerate. This is why I appreciate numbers and mathematics.
that's great.
I can dig it. You put a wonderful deal of effort into this reply and the relations are mutual. Excellent work!
But.
Trigonometry: tri. You need more than one line segment. In the circle, the perimeter is the so-called "continuous line", drawing not a boundary, but a measurable distance away from some point, for all points swept out by the so-called "full rotation" about the point... another line, here, delineates the distance r to some point p.
I think you are implicitly assuming the parallel postulate here. The sum angles in a triangle don’t have to be 180 degrees, but if they are, then you basically assuming thing that make the Pythagorean theorem true
What about it
The parallel postule and the Pythagorean theorem are equivalent
So all proofs must assume or imply something equivalent to the postule, else you have a proof that's independant of to it
Here's another way to prove it with trigonometry:
d/dx (cosx) = -sinx and d/dx (sinx) = cosx is proven without pythagoras.
We say that:
(sinx)^2 + (cosx)^2 = f(x), for some function f
DIFFERENTIATE BY X TO GET:
2sinx(cosx) + 2cosx(-sinx) = f'(x)
0 = f'(x) --> f(x) = C, where C is a constant
(sin0)^2 + (cos0)^2 = 1 --> C = 1
--> (sinx)^2 + (cosx)^2 = 1
@MindYourDecisions I appreciate that you try to be historically sensitive. I really do. But there is a lot of value in naming conventions. If a large portion of people know the thing under the same name, the Pythagorean theorem, then that's an achievement all on its own. And it's not worth giving that up for the chase of "Who did it first".
The simplest is to take the limit as spherical trigonometry approaches plane trignometry. Use the formula cos A = cos B cos C + sin B sin C cos(a - b). A B and C are the sides and a, b, c the vertex angles of a spherical triangle - all dimensionless. In the limit A, B, and C are very small and cos A -> 1 - 1/2 A^2 and sin A -> A to highest order. Let the angle a-b be pi/2, then we get (1 - 1/2 A^2) = (1 - 1/2 B^2) (1 - 1/2 C^2) - working out to highest order, A^2 = B^2 + C^2. This proof does not presuppose the theorem in any way, just the power series expansions for sin and cos and the angular geometry of a sphere. It works in any number of dimensions as well.
@13:45 That is awesome.
Excellent work by these two young scholars!!! Now lets get the proof added to Algebra, Geometry, Trigonometry, & Calculus texts. This is good for young people to see they too can add to our math base. Most classes at the HS level are boring, uninspired, and cover nothing new or interesting. Students need to see they can connect to advanced mathematics and that math when connected to actual practical problems is useful, exciting and very much connected to their lives.
This proof is awesome. My biggest question is how did these two students specifically collaborate on such a unique proof?
And also, why are they unable to explain it..?
@@CrowsDoMath Are they unable to explain it?
@@CrowsDoMath They can explain it.
I think it was part of a competition
Using trig on a triangle with side a, b, c and angle A.
a = c*cosA. b = c*sinA, a^2 + b^2 = c^2*(cosA)^2 + c^2*(sinA)^2 = c^2*((cosA)^2 + (sinA)^2) = c^2. I used only trignometry.
You was using (cosA)^2 + (sinA)^2 = 1, which is one result derived from Pythagorean Theorem. It is mentioned at 1:17.
@@tranhadminthich That can be derived from the expressions for sin and cos in terms of e^iA and e^-iA. What are they if not trig?
I was taught in grad school the Greeks did not have algebra. The Greeks had a straight edge and a compass, so using trigonometry with algebra seems slightly anachronistic. I would be really blown away to see the proof using just a straight edge and a compass!
On the other hand, my hat goes off to the two high school students. Great work.
Uncle Tony, we should praise the ladies for doing what 99% truly can't . but it is sad that older dudes in the comments are salty because 2 girls not only came up with something new and they did so with 2 separate solutions of their own.
the need for a < b was for the similarity between the triangles ?
This is a clever construction, but I would not be surprised at all if some previous mathematics student (undergrad or grad) made a similar observation that such an infinite series could be constructed to provide a demonstration of this famous theorem.
I'm happy to see that a HS student had the patience and determination to explore this approach because that is how mathematics is done in the professional level, just with higher complexity of concepts/techniques and typically a much longer period of time.
There is already an infinite series proof by John Arioni. Johnson's "waffle cone" can be found in many calculus textbooks. Jackson's proof was a copy of another that can be found at cut-the-knot (proof 60) by B. F. Yanney and J. A. Calderhead.
i've could have derived that, but I screwed up my knee
Did you take an arrow to the knee
The infinite geometric series part is a pre-existing proof.
They just slapped a couple of triangles on top of it and it just happened to work.
Makes me wonder if you could do the same with other known proofs.
I am surprised no one has thought of this before for 2000 years. It was so satisfying when I realized how they were going to attack the proof. Good job!
I derived this myself 6 or 7 years ago, but I assumed it was already known.
@richardcheney6964 sure you did. Did u also discover that E=mc^2??
Well, the thing is that there are other proofs that use the same law of cosine, are much simpler and do not rely on limits. Kudos to the students for their creativity, but they have just reinvented a crooked wheel.
@@kyriakosphilitas6950 Jason Zimba did 14 years ago though, and you don't even know who he is. Why did the the news of the first to discover it not make it to you but this news did? Most likely because they are black, female, high school students. If they read this paper he wrote beforehand it is less impressive.
forumgeom.fau.edu/FG2009volume9/FG200925.pdf
@@kyriakosphilitas6950 He most likely did considering how the first officially known and approved modern FULLY TRIGONOMETRIC proof of Pythagoras was published in the early 2000s by Jason Zimba so... it is clear that these student's didn't really come up with anything new. As a matter of fact, Zimba's proof was taught in highschool to us and is written in my book, which I still have in my posession to this day.
15:17 in our maths textbook similar proof is given only difference is instead of sin it relies on rules of similar triangles
You're quite right. The use of trigonometry in this proof is quite unnecessary. The use of sines is simply a different way of expressing the similar triangles and adds nothing useful to the proof.
I'm an engineering student and I don't know who said no proof existed before. I've proven it with a basic method that takes 2 minutes to explain with 1 square
Is that andrew taint as your pfp?? ewwww
@@endxofxeternity you know it looks good
Even with everything it's really really hard to follow but hell I had a hell of a time with plain geometry so this was practically impossible for me to follow
My math teacher literally proved it to us using trigo like 10 years ago and nobody cared cuz we assumed it was common knowledge and a waste of time 💀💀💀
99% of trigonometry is dependent on a²+b²=c². If your teacher used something like sin cos tan, that's not REALLY considered proof. For the same reason you can't have two essays that cite eachother as a source.
@@josephwodarczyk977 Well, the thing is that there are other proofs that use the same law of cosine, are much simpler and do not rely on limits. Kudos to the students for their creativity, but they have just reinvented a crooked wheel.
Just draw in any right angled triangle the vertical to the hypotenuse c going through the opposite corner. Name the 2 segments a and b. With similarity follows a^2 + b^2 = c^2
No mathematician ever said it was 'impossible'. Nor has any professional mathematician been trying to do a trig proof in the past 2000 years. There is so much nonsense in this story that it makes my brain hurt.
Lol, found the hater.
@@IRanOutOfPhrases Has nothing to do with being a hater.
Found the skipped highschool math
@@abdulrahman01234 it's exclusively hate
@@IRanOutOfPhrases Triggered?
I love when a proof that seems to be spiraling away into unintelligible complexity suddenly is resolved into simplicity itself, seemingly out of the blue.
Bravo!
É incrível como na matematica existem coisas que ainda não descobrimos, e ainda não provamos
A matemática sempre me fascina pois ela é um alfabeto completo que podemos descobrir coisas com puro raciocínio e apenas lendo de forma alfabética, é legal resolver problemas nessa elegância de ler matematicamente, e por isso apoio totalmente olimpíadas de matemática, são incríveis
A matemática é cheia de incompletudes e apesar de parecer estressante, só me deixa mais fascinado com essa idéia
A matemática tem muita coisa ainda a ser tratada. Mesmo que demore 300 anos pra provar algum problema proposto por algum matematico maldoso. Kkkkk quem sabe mexer de verdade com a matemática, pode usufruir de tudo dela e dos problemas, mas também pode ser um cara chato que aterroriza todos do ensino médio com sua fórmula que ele cria para provar alguma coisa bem doida. Amo
While proofs like the one at 15:10 use trigonometry, it doesn’t really require it - you could do almost the same proof just using similar triangles and not mentioning sine. The student’s proof is arguably “more trigonometric”, since it uses the law of sines, while still not requiring anything beyond trigonometry.
except for the whole infinite limit, disgusting a calc proof as a trig proof
...................I'm gonna go watch Animation vs Math
proof of Pythagorean theorem involving only trigonometry:
assuming cos(b) != 0 (for b != Pi/2 )
cos(b) = cos(a-(a-b)) = cos(a)cos(a-b) + sin(a)sin(a-b)
=cos(a)cos(a)cos(b) + cos(a)sin(a)sin(b) +sin(a)(sin(a)cos(b)-cos(a)sin(b))
=cos(b)[ cos^2(a) +sin^2(b) ]
cos^2(a) +sin^2(b) = 1
the teenagers haven't proven anything that hasn't been known for thousands of years
the general formula for triangles is c^2 = a^2 + b^2 - 2ab cos theta, where theta is the angle between sides a and b
trigonometry (i.e., sin, cos, tan, etc.) depends on an understanding of Pi and of radians -- that the circumference of a circle is proportional to its diameter... once you have that then the trig functions are straightforward to understand: the hypotenuse of an inscribed triangle grows with the radius of the circle and so does the "sine" and the "co-sine" of the arc generated by the chord (which are the lengths of an imaginary bow, with the bow itself being the circles arc, etc.)
The true master equation of planar geometry is Norman J. Wildberger's Cross Law, which is the cosine law squared (=1-sin²), out of which Pythagora's theorem is a limit case when the spread ("sine squared") between two sides of a triangle is 1. The other limit case is when a spread of a (degenerate) triangle is 0, hence about 3 collinear points, as strong and as fundamental as Pythagora's theorem. This video is much wind for nothing, NJWildberger cracked geometry properly, without the sine qua none need for infinite processes 😊 Nevertheless, I enjoy the channel and content very much!! Keep up the good work 😁🙏🙋
Pythagoras' theorem is not part of trigonometry as it does not involve angles, but only sides...it is wrongly lumped with trigonometry because it's a triangle...
Whoa.. that is quite a work! Thank you for making the video so we could understand it easier👍
I really wish math instructors would focus more on the details. I get that this is youtube, and there is a certain standard for the length of videos, but I would have loved to learn all the reasoning for example visually of all the connecting ideas within the equations. (and can literally be a second-long, you can throw a picture in with text that explains how you were able to move forward). I loved that I remembered how to simplify complex formulae, but even for youth, I can see it being extremely useful for reviewing material. Also, how much do I bet that length and rate of 'pause' in a video is the single most defining factor in how interested people are in your videos.
omg just start your own youtube channel already
very sceptical this hasn't been done before.
isn't sin() and Pyth connected?
This proof is implicitly using the flatness of euclidean geometry in that current form of the law of sines (there is a modified form in the other geometries), and is using this flatness in saying sin(alpha)=a/c etc. since this is directly derived from the law of sines. In general, this just simply is not the case. Therefor, this proof is using implicitly the assumption of a flat euclidean geometry, which is the geometry which assumes the parallel postulate, and the parallel postulate is equivalent to the pythagorean theorem. Therefor, the proof is starting with the assumption that the pythagorean theorem is true, albeit subtly and implicitly. It is a smart proof, but it is circular.
Jackson just copied a proof by B. F. Yanney and J. A. Calderhead.
Johnson just copied a proof from a calculus textbook.
15:06 This is such an elegant, easy to understand proof, I love it.
This discovery is truly like the saying " The treasure is not the goal but the Journey ".
"With the few screenshots we got, we can predict what they did to do this"
Proceeds to give a detailed explanation for how they did it
Reflective! Now to return to the first known constants, 1,2. 1/2. .5. And take a reflective look without limitation.
It is not an impossible discovery. All we have is another proof of the Theorem, which is nice for the students, but other than that, not a remarkable discovery that goes into history books. The reason why this is "hyped up" is because the Pythagorean Theorem relates to a lot of people through their high school math. So easier accessible for general audience. The proof of Fermats theorem by Wiles was far more ground breaking than anything else but then...who knows Fermat anyway?
Since when infinite series is "trigonometry alone"? There is another branch of mathematics that gives as the sum of it, it's called calculus. That was NOT a proof using "trigonometry alone".
I read the story in the Guardian and am glad to have watched your explanation of their clever approach
They used trigonometry to prove that x = 2ac/b (see ruclips.net/video/juFdo2bijic/видео.html) , but they could have proven it by the proportionality of sides in similar triangles (in fact, that's what Pythagoras used). In other words, it is really just one more proof of the theorem. Anyway, it is great for high school students.
It is, first, a circular reasoning test... pun subtended.
Using the definition of sin to prove sin is literally circular reasoning. This is a big gut check: drop nuts and call a spade a spade.
That’s innovative. Never would have thought of this. Very nice
If (a = b) @ 1/1 ratio
(a^2 + b^2) = c^2
(2Sine(90)^2 + Cos(90)^2) = 2Tan(45)
So, we can conclude
sq.root. (a^2 + b^2) =
(Sq.root of 2)Tan(45)
I'm confused... you say the students found the impossible: A trigonometric proof of the Pythagorean Theorem. Then you proceed to show it, and it uses infinite iterations of smaller triangles. That's fine. But then in the "More Proofs" section of this video you show a different, much simpler trigonometric proof. Wasn't that simpler proof already known?
you are right. People like to think about these things as if they're mystical so they fall into the trap of fruitless erroneous imagination (and also fall under the illusion that they are smart and very special)
The part that I don’t understand is how does the geometric series count under trigonometry? I would’ve considered that to be a more algebra or calculus concept
Presh said “Pythagorus”! I’ve been waiting years for this day.
as
In this "proof" sin(alpha) is just the ratio a/c where a is the side opposite alpha and c is the hypotenuse. So we don't need to mention the sin function and in that sense, it's not "trigonometric". The "proof" assumes the ratio properties of similar triangles, which themselves are proved from the Euclidean axioms. Euclid gives a far simpler ratio proof. He also gives an area proof, but this depends on further axioms to do with areas.
I'll give you that this proof technically doesn't use trigonometry, though I don't know why you put proof in quotes. But in your same line of reasoning, nobody uses trigonometry because it's all ratios anyway.
@@mike1024. OK the quotes were unnecessary. The point with the ratios is yes, they provide definitions of sin, cos, tan etc, but they don't tell you how to calculate them. In fact, the functions were tabulated from mediaeval times or before, in various places and by various methods, culminating in the modern series definitions. Anyway Euclid + trig functions is different from Euclid by itself, and necessary for astronomy, surveying and the like.
@@pwmiles56 I see what you're saying! I'm aware that the trig ratios were originally used for astronomy hundreds of years ago, but I've never actually studied their history. It still might be worth doing! I have no idea how they calculated the values previously.
@@mike1024. There's an article on Wiki, History of Trigonometry. They used various formulae, in the process discovering some of them, e.g.
sin(a+b) = sin a cos b + cos a sin b
With this and the corresponding formula for cos(a+b) you could make a a general angle and b a very small one and kind of inch forward that way.
Bravo, students. Brilliant!!!
they are only tricking us into thinking, its a trigonometric proove. However its a graphical one, cause it only gives a name to a/c and call it sin(α). But at that point its just a name,and has nothing to do with trigonometry itselve.
I like how the proof has that iconic wedge shape. I wonder if the high-school students will get any offers from colleges.
Muito bom! Parabéns e obrigado!
What’s weird is that very same graph is printed in this old book I have about linear programming
It's not actually their proof. They took a known proof and just added a little bit of trig so it's not actually a trig proof. It's only superficially trig.
I thought this must have been done long time ago… It’s so simple that ordinary high school students should be able to understand it!!
They aren't the first to prove it this way. The video is misinformation
Everything is obvious after you know about it
I would guess because there wasn’t really any knowledge about infinite geomteric series back then, or infinite series in general. Those were concepts that were only really developed much later.
That would me my guess.
@@Ninja20704 But I’m referring to the hundreds of proofs published in the 20th and 21st centuries. And they come in all flavors.
@@wesleysuen4140 ok sorry for my misunderstanding
Technically you can prove the theorem with sin²x + cos²x = 1 without circuluar reasoning. You can prove sin²x + cos²x = 1 wothout the pythagoras theorem.
Pf:
Since sin x and cos x are continuous functions, so are the sum of their squares. Then, take the derivative of sin²x + cos²x, which is 0. Since sin²x + cos²x is continuous and has a derivative equal to 0, sin²x + cos²x is just a constant. Substitute in a good value for x:
sin²0 + cos²0 = 1. Since sin²x + cos²x is constant, it is equal to 1.
How’d you prove its continuity? The proofs I see require sin^2(x)+cos^2(x)=1
@@gealbert5737 We wanna prove lim x->a sin x = sin a. We use epsilon delta:
Using the difference of sines formula, sin x - sin a = 2 cos ((x + a)/2) sin ((x - a)/2). Thus abs(sin x - sin a) = abs(2cos((x + a)/2)sin((x - a)/2))
I once sat a test and I wrote
"therefore by Baudhayan a^2 + b^2 = c^2"
and I got 0/10
The teacher commented,
"It's the Pythagoras theorem, what is wrong with you?"
I don't believe you that the teacher subtracted 10 out of 10 points just because of a name. Maybe 1 point.
@@efi3825 its a joke lmao
I've always thought that the idea of a "trigonometric proof" was silly, as trigonometry and geometry are the same. Consider the law of sines, it may seem that its application here is somehow "escaping geometry by using trigonometry" when in reality its all the same. Consider a triangle with sides A,B,C and angles a,b,c (angle a opposite of side A and so on). Then let h_a be the height of the triangle when A is the base (also called the altitude off angle a) and h_b and h_c defined similarly. Then we can show that h_a/BC = h_b/AC = h_c/AB. We can do this using an area argument or a similar triangles argument. This relationship is equivalent to the law of sines, notice that if we were using trig sin(a) = h_c/B and so on for the other angles. It is less intuitive sure, but we could have used the geometric relationship and similar triangles to arrive at the same result, and I image that while it may be convoluted to do so we could "triggify" many geometric proofs. This does not take away from these girls unique approach involving the geometric series, which was something I'd never thought I would see in a geometry proof.
What makes me fascinated in mathematics is that even an 18 year old can make a groundbreaking discovery.
This is not groundbreaking in math. It is just a nice geometry problem plus a little bit of calculus. Groundbreaking works in mathematics generally shed light on a subject in a way that has never been considered before. This does not really come close to any revelation about Pythagoras' theorem, it is just another proof using limits, in fact you never have to mention sin and the proof stays the same. Groundbreaking work is stuff like the introduction of perfectoid spaces, or the development of the Lebesgue measure, central models, etc etc. These are brand new ideas that fundamentally change their subject.
New Calculus of John Gabriel is groundbreaking