You can be trained to solve the most challenging problems in the most prestigious mathematics Olympiads in the world. But there are no training programs to create alternative approaches from which you can derive all the most important theorems in a discipline. No one invented inversion in the middle of a math competition.
#math #halfanglesmatter #symmetrymatters #halfangleapproach
#halfanglesmatter
This is Luzia's proof of the Pythagorean theorem via half-angle formulas: https://www.cut-the-knot.org/pythagoras/Proof109.shtml
I shared my essay "The Theoretical Importance of Half-Angle Formulas" on Reddit: https://www.reddit.com/r/math/comments/16ojbnd/the_theoretical_importance_of_halfangle_formulas/
#math #geometry #trigonometry #halfanglesmatter #symmetrymatters
Problem 1315 from Gogeometry follows from GM / DG = cos(0.5 * γ) / cos(0.5 * α) and MN / EN = cos(0.5 * α) / cos(0.5 * γ), where α is the angle at A, and γ is the angle at C. Since DM = ME, the conclusion is easily deduced. You can find the detailed steps here: https://geometriadominicana.blogspot.com/2023/08/solution-to-problem-1315-of-gogeometry.html
Seems legit.
If the half-angle formulas aren't considered fundamental, then how did Viète manage to derive his formula for π using them? And why is it that the Weierstrass substitution, hailed as the most ingenious substitution in the world by Spivak, is actually based on those very formulas? If these half-angle formulas lack significance, then how on earth have I successfully derived the formulas of Heron, Brahmagupta, and Bretschneider, as well as the laws of cosines, sines, and tangents, the Mollweide formula (or perhaps we should credit Newton), the angle bisector formula, the inradius formula for mixtillinear incircle, and the sum and difference of angle identities, not to mention Euler's remarkable triangular inequality, all stemming from these supposedly non-fundamental formulas? And if that's not enough, how have I managed to extend the Pythagorean trigonometric identity (with over 2000 years of history), the Mollweide formula (or Newton's, if you prefer) (with over 300 years of existence), and even the half-angle formulas themselves (likely over 2000 years old) into broader generalizations? Still skeptical? Well, I invite you to click on the link below and see for yourself.
Method vs trick according to George Polya
ChatGPT agrees that my half-angle approach is a method rather than a trick.
#newmethod #halfangleapproach #math #geometry #trigonometry #georgepolya #halfanglesmatter #symmetrymatters #ChatGPT
Earlier I had said that the half-angle formulas are an essential ingredient in the derivation of Viète's \(\pi\) formula. According to Eli Maor, Viète's formula marks the beginning of mathematical analysis, and Jonathan Borwein calls its appearance "the dawn of modern mathematics".
The work described in the link below is another example of the centrality and versatility of the half-angle formulas (although the authors mistakenly call them double-angle formulas), a centrality that transcends the field of geometry and trigonometry in the plane.
https://www.sciencedirect.com/science/article/pii/S0021904513001159?via%3Dihub#br000045
#math #geometry #trigonometry #halfanglesmatter #symmetrymatters
My wake-up call for the extraordinary explanatory power of half-angle formulas has reached 3,000 views on MathSE. Are the half angle formulas more fundamental than the Pythagorean theorem or the law of cosines?
#math #trigonometry #geometry #halfanglesmatter #symmetrymatters
Yo defendiendo que las fórmulas de medio ángulo son más fructíferas que la ley de cosenos.
But what excites me the most about these formulas is their potential to revolutionize the way we teach trigonometry to high school students. By introducing the half-angle formulas, we can help students better understand the development of the formulas that derive from them. This is where the half-angle formulas truly shine and can make a significant impact on math education.
GeoDom: The Half-Angle Formulas: A Powerful Tool for Trigonometry Students http://geometriadominicana.blogspot.com/2023/03/the-half-angle-formulas-powerful-tool.html?m=1
#math #trigonometry #halfanglesmatter #education #newapproach
Viète's formula for \(\pi\), the Weierstrass substitution and my approach based on half-angle formulas are three examples of how useful it can be, when you think about mathematics, to obey certain symmetries.
Had I lived in the 18th century, it is likely that my approach to metric relationships in Euclidean geometry, based on half-angle formulas, would have been ubiquitous in high school textbooks, articles, and reviews on the subject nowadays.
Initially I had found an extension of Lami's Theorem for cyclic quadrilaterals, but on the PhysicsSE forum, Philip Wood was dissatisfied and challenged me to generalize Lami's Theorem to a general quadrilateral. So I did it.
Link: https://math.stackexchange.com/questions/4490209/generalizing-lamis-theorem
I wonder how you could adapt this meme to the generalized Pythagorean Identity?
\[ad\sin^2{\frac12\alpha}+bc\cos^2{\frac12\gamma}=(s-a)(s-d).\]
Link: https://geometriadominicana.blogspot.com/2022/06/a-generalization-of-pythagorean.html?m=1
Like Josefsson, I'm almost sure Euler would have loved my way of approaching metric relationships in Euclidean geometry using the half-angle formulas.
My generalization of Newton's formula has been accepted for publication 😁. The generalized formula in the image extends Newton's formula to cyclic quadrilaterals. The link below contains Newton's original formulas.
Trigonometry has given us much more than number theory. Let's make trig great again!
@johncarlosbaez In the image you can see why. BTW, the standard half-angle formulas can be obtained from the ones in the image by substituting from the law of cosines (which is also another way of writting the half-angle formulas). So they are different forms of the same thing. If you have time, also take a look of this: https://hsm.stackexchange.com/questions/15010/the-role-of-symmetry-in-mathematics-and-the-half-angle-formulas
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