#halfangleapproach — Public Fediverse posts

When a calculus student ends up solving this integral using partial fraction decomposition (PFD) or integration by parts (IBP), what they have actually done is successfully applied a classical algorithm. It’s like solving the Rubik’s cube by following an algorithm you found on the Internet: the heavy lifting of inventing PFD and IBP was done centuries ago by Bernoulli, Leibniz, and Gregory.

That said, the USM is, as far as I know, the only published general scheme that, for this kind of radical–rational integrals, systematically reduces the integrand to polynomial-type functions (Laurent polynomials, to be more precise). There’s no need to set up complicated PFDs or wrestle with sec³, which can be tricky with IBP. That’s what I call solving a problem, if you’ll allow me a bit of self-promotion.

You can find a first draft (I’ll soon upload a second version with benchmarks and some improvements) of the USM method on arXiv: https://arxiv.org/abs/2505.03754

#calculus #math #symmetrymatters #halfangleapproach #euler #integration

When a calculus student ends up solving this integral using partial fraction decomposition (PFD) or integration by parts (IBP), what they have actually done is successfully applied a classical algorithm. It’s like solving the Rubik’s cube by following an algorithm you found on the Internet: the heavy lifting of inventing PFD and IBP was done centuries ago by Bernoulli, Leibniz, and Gregory.

That said, the USM is, as far as I know, the only published general scheme that, for this kind of radical–rational integrals, systematically reduces the integrand to polynomial-type functions (Laurent polynomials, to be more precise). There’s no need to set up complicated PFDs or wrestle with sec³, which can be tricky with IBP. That’s what I call solving a problem, if you’ll allow me a bit of self-promotion.

You can find a first draft (I’ll soon upload a second version with benchmarks and some improvements) of the USM method on arXiv: https://arxiv.org/abs/2505.03754

#calculus #math #symmetrymatters #halfangleapproach #euler #integration

When a calculus student ends up solving this integral using partial fraction decomposition (PFD) or integration by parts (IBP), what they have actually done is successfully applied a classical algorithm. It’s like solving the Rubik’s cube by following an algorithm you found on the Internet: the heavy lifting of inventing PFD and IBP was done centuries ago by Bernoulli, Leibniz, and Gregory.

That said, the USM is, as far as I know, the only published general scheme that, for this kind of radical–rational integrals, systematically reduces the integrand to polynomial-type functions (Laurent polynomials, to be more precise). There’s no need to set up complicated PFDs or wrestle with sec³, which can be tricky with IBP. That’s what I call solving a problem, if you’ll allow me a bit of self-promotion.

You can find a first draft (I’ll soon upload a second version with benchmarks and some improvements) of the USM method on arXiv: https://arxiv.org/abs/2505.03754

#calculus #math #symmetrymatters #halfangleapproach #euler #integration

Hey, I just generalized the 𝐥𝐚𝐰 𝐨𝐟 𝐜𝐨𝐭𝐚𝐧𝐠𝐞𝐧𝐭𝐬! Take a look at it on my blog: https://geometriadominicana.blogspot.com/2025/08/generalization-of-law-of-cotangents.html

#math #geometry #trigonometry #halfangleapproach #symmetrymatters

Hey, I just generalized the 𝐥𝐚𝐰 𝐨𝐟 𝐜𝐨𝐭𝐚𝐧𝐠𝐞𝐧𝐭𝐬! Take a look at it on my blog: https://geometriadominicana.blogspot.com/2025/08/generalization-of-law-of-cotangents.html

#math #geometry #trigonometry #halfangleapproach #symmetrymatters

Hey, I just generalized the 𝐥𝐚𝐰 𝐨𝐟 𝐜𝐨𝐭𝐚𝐧𝐠𝐞𝐧𝐭𝐬! Take a look at it on my blog: https://geometriadominicana.blogspot.com/2025/08/generalization-of-law-of-cotangents.html

#math #geometry #trigonometry #halfangleapproach #symmetrymatters

Hey, I just generalized the 𝐥𝐚𝐰 𝐨𝐟 𝐜𝐨𝐭𝐚𝐧𝐠𝐞𝐧𝐭𝐬! Take a look at it on my blog: https://geometriadominicana.blogspot.com/2025/08/generalization-of-law-of-cotangents.html

#math #geometry #trigonometry #halfangleapproach #symmetrymatters

Hey, I just generalized the 𝐥𝐚𝐰 𝐨𝐟 𝐜𝐨𝐭𝐚𝐧𝐠𝐞𝐧𝐭𝐬! Take a look at it on my blog: https://geometriadominicana.blogspot.com/2025/08/generalization-of-law-of-cotangents.html

#math #geometry #trigonometry #halfangleapproach #symmetrymatters

USM really does slash the rote burden, chiefly because one handful of exponential/hyperbolic identities replaces a patchwork of separate trig, inverse-trig, radical and Euler recipes.

Article (draft): https://arxiv.org/abs/2505.03754

#math #calculus #integral #new #euler #arxiv #halfangleapproach #symmetrymatters

USM really does slash the rote burden, chiefly because one handful of exponential/hyperbolic identities replaces a patchwork of separate trig, inverse-trig, radical and Euler recipes.

Article (draft): https://arxiv.org/abs/2505.03754

#math #calculus #integral #new #euler #arxiv #halfangleapproach #symmetrymatters

USM really does slash the rote burden, chiefly because one handful of exponential/hyperbolic identities replaces a patchwork of separate trig, inverse-trig, radical and Euler recipes.

Article (draft): https://arxiv.org/abs/2505.03754

#math #calculus #integral #new #euler #arxiv #halfangleapproach #symmetrymatters

The first and most complete unification of classical integration techniques ever! Say hello to USM.

Draft article: https://drive.google.com/file/d/12DayP6cD1VwDIZCL-nMlcaNH2XUwHfAy/view?usp=drivesdk

#math #calculus #halfangleapproach #symmetrymatters

The first and most complete unification of classical integration techniques ever! Say hello to USM.

Draft article: https://drive.google.com/file/d/12DayP6cD1VwDIZCL-nMlcaNH2XUwHfAy/view?usp=drivesdk

#math #calculus #halfangleapproach #symmetrymatters

The first and most complete unification of classical integration techniques ever! Say hello to USM.

Draft article: https://drive.google.com/file/d/12DayP6cD1VwDIZCL-nMlcaNH2XUwHfAy/view?usp=drivesdk

#math #calculus #halfangleapproach #symmetrymatters

The first and most complete unification of classical integration techniques ever! Say hello to USM.

Draft article: https://drive.google.com/file/d/12DayP6cD1VwDIZCL-nMlcaNH2XUwHfAy/view?usp=drivesdk

#math #calculus #halfangleapproach #symmetrymatters

The first and most complete unification of classical integration techniques ever! Say hello to USM.

Draft article: https://drive.google.com/file/d/12DayP6cD1VwDIZCL-nMlcaNH2XUwHfAy/view?usp=drivesdk

#math #calculus #halfangleapproach #symmetrymatters

The USM, the Dominican method 🇩🇴, has relegated Euler substitutions to mere historical relics. Modern integration has a new name.

Method draft: https://drive.google.com/file/d/12DayP6cD1VwDIZCL-nMlcaNH2XUwHfAy/view?usp=drivesdk
#math #calculus #integrals #method #euler #halfangleapproach #symmetrymatters

The USM, the Dominican method 🇩🇴, has relegated Euler substitutions to mere historical relics. Modern integration has a new name.

Method draft: https://drive.google.com/file/d/12DayP6cD1VwDIZCL-nMlcaNH2XUwHfAy/view?usp=drivesdk
#math #calculus #integrals #method #euler #halfangleapproach #symmetrymatters

The USM, the Dominican method 🇩🇴, has relegated Euler substitutions to mere historical relics. Modern integration has a new name.

Method draft: https://drive.google.com/file/d/12DayP6cD1VwDIZCL-nMlcaNH2XUwHfAy/view?usp=drivesdk
#math #calculus #integrals #method #euler #halfangleapproach #symmetrymatters

The USM 𝐢𝐧𝐜𝐨𝐫𝐩𝐨𝐫𝐚𝐭𝐞𝐬, 𝐞𝐱𝐭𝐞𝐧𝐝𝐬, 𝐣𝐮𝐬𝐭𝐢𝐟𝐢𝐞𝐬, and 𝐬𝐮𝐫𝐩𝐚𝐬𝐬𝐞𝐬 Euler's substitutions. In Euler's substitutions, the choice of signs based on the domain must be made manually, whereas in the USM, the supporting theorems prescribe which sign to use according to the domain. Moreover, the USM shows that Weierstrass substitutions and the use of complex exponentials for integration are merely two sides of the same coin. The USM not only 𝐮𝐧𝐢𝐟𝐢𝐞𝐬 these two techniques into one, but also 𝐠𝐞𝐧𝐞𝐫𝐚𝐥𝐢𝐳𝐞𝐬 them.

USM: https://geometriadominicana.blogspot.com/2024/03/integration-using-some-euler-like.html

#math #calculus #integrals #technique #new #halfangleapproach #symmetry

The USM is superior to traditional methods!

Using the USM’s IV transformation formula, from the summary at the link below, you convert integrals of this type into polynomial-type integrals. The traditional method reduces it to the integral of csc³, which would require complicated reduction formulas or integration by parts. The third Euler substitution reduces it to the integral of √(t² + a²), which also requires memorizing a standard formula or integrating it from scratch. None of this is as simple as integrating a three-term polynomial-type function, which is taught in the first few weeks of any integral calculus course.

Check out the technique here: http://geometriadominicana.blogspot.com/2024/03/integration-using-some-euler-like.html
#math #calculus #integration #USM #newmethod #halfangleapproach #symmetrymatters

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

I gave this integral as an answer at MathSE.

\[\int_{0}^{1}\left(\frac{5}{2} \left((x - \sqrt{x^2 - 1})^{2i} + x^4\right)-1\right)\,dx=e^\pi.\]

The integral yields \(e^{\pi}\). I don't know why the LaTeX isn't displaying properly.

Link: https://math.stackexchange.com/questions/122693/is-there-a-definite-integral-that-yields-e-pi-or-e-pi-in-a-non-trivial/4861138?fbclid=IwAR3CPXtBoNDY3yc5a1HDAgfWN7QFAO2Ph91uvH7MUgSxSRjsq-t8TWYHWdY#4861138

#math #integral #eulernumber #halfangleapproach #symmetrymatters #calculus

An 'impossible' integral to solve (Mathematica fails):
\[\int_{2}^{3} \frac{{1 - \tan\frac{{\sec^{-1}x}}{2}}}{{1 + \tan\frac{{\sec^{-1}x}}{2}}}\sqrt{\tan\frac{\csc^{-1}x}{2}} \,dx\]
Unless you know my new technique:
https://mathoverflow.net/questions/463459/identities-that-simplify-tedious-integrals?noredirect=1#comment1203657_463459
#math #impossibleintegral #calculus #halfangleapproach #symmetrymatters

In my latest blog post, I have derived what appears to be a new formula for solving quadratic equations. Were you aware of it?

Link: https://geometriadominicana.blogspot.com/2024/01/trigonometric-formula-for-solving.html

#math #halfangleapproach #symmetrymatters

Do you know what I mean, right? Half-angle formulas are central! Their utility for you will depend on how ingenious you are. Look at Euler, learn from him, he is the master of all of us, someone once said.
A proof : https://m.youtube.com/watch?v=eYow30t9aQg&si=S8J4Kk43VgbybtXx&fbclid=IwAR1jf1tM-rgZMapC07vH0W6v7QMQDfM_EXDy_-7p4358XZFGASyZ8gnH1-Q

#math #halfangleapproach #symmetrymatters

Another theorem falls under the grasp of the half-angle approach: Burlet's theorem.

I give a generalization here: https://geometriadominicana.blogspot.com/2024/01/a-generalization-of-burlets-theorem-to.html?m=1&fbclid=IwAR1oQEwlKoQl2Ajv5BDoXdGHtUUew46WiB9iHqWALL4p1o58I8m8JdMzuT8

#math #geometry #halfangleapproach #symmetrymatters

I GOT PUBLISHED!!

The MATINF magazine has officially published my generalization of Newton's formula.

I am immensely grateful to Masters Leo Giugiuc and Costel Balcau for being instrumental in making this publication possible.

You can download and view the article on page 19 at the following link: http://matinf.upit.ro/MATINF9_10/mobile/index.html

#math #halfangleapproach #symmetrymatters

After discovering the central role that half-angle formulas play in relation to classical metric geometry, the natural question that arises is, why on earth are these formulas so useful? And my answer is symmetry, the same underlying concept in Galois theory or Noether's theorems.
#math #halfangleapproach #symmetrymatters

Context: https://www.reddit.com/r/math/comments/16ojbnd/the_theoretical_importance_of_halfangle_formulas/
#math #halfangleapproach #symmetrymatters #meme #mathmeme

The half-angle formulas are central! And as a picture is worth a thousand words...

If you don't know what I'm talking about, read my essay 'The Theoretical Importance of Half-Angle Formulas' and discover the most powerful formulas of all time.

Essay: https://drive.google.com/file/d/1jPrWxUhB1Tnsd5RH5UiMhwmkp2jCv11v/view?usp=sharing

#math #halfangleapproach #symmetrymatters #insight

High school teachers in Singapur are already using the half-angle approach.
#math #halfangleapproach #symmetrymatters

On every platform where I've shared my work on half-angle formulas, except for my personal blog and my social media, there's always been someone trying to sabotage my post. Recently, I posted my essay on Reddit. Although my post was initially accepted and garnered many positive votes, the supportive comment from RussianChattus was strangely pushed to the bottom and then deleted for several days until I complained in the comments. After my protest, RussianChattus's comment reappeared, but oddly enough, it was surpassed by a comment criticizing the lack of punctuation within parentheses, becoming the most upvoted comment. A hilarious situation!

#math #halfangleapproach #symmetrymatters

The Pythagorean theorem, merely a specific instance of the half-angle formulas, derives its remarkable utility from these very formulas.
#math #halfangleapproach #symmetrymatters

Akin to the ascent of a tall tree, the identification of fundamental theorems within a field is a critical task. This analogy draws parallels between understanding the core principles of a subject and recognizing the trunk of a towering tree when one endeavors to reach its pinnacle. While it is acknowledged that not everyone, particularly the more adept individuals in this metaphorical climb, may need to traverse the trunk directly, the importance of this identification becomes apparent, especially for the average person.

#math #halfangleapproach #symmetrymatters

In terms of understanding the body of metric relationships in classical geometry, I have far surpassed my predecessors, constructing a unified framework that will aid students in better comprehending what is happening here.
#math #halfangleapproach #symmetrymatters

The fact that I have been able to derive the most important theorems of classical geometry and generalize a good number of them without ANYONE being able to find a reference in 3 years is compelling evidence that the half-angle approach is more profound.
#math #halfangleapproach #symmetrymatters

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

This is Luzia's proof of the Pythagorean theorem via half-angle formulas: https://www.cut-the-knot.org/pythagoras/Proof109.shtml

#halfanglesmatter #math #trigonometry #halfangleapproach

Two things make my work with half-angle formulas unlikely: 1) it was developed by a Dominican amateur, and as everyone knows, the Dominican Republic is light-years away from having a strong mathematical tradition. In fact, we don't even have a single mathematical journal. 2) The work is innovative in a field that's over 2000 years old.

#math #geometry #halfangleapproach

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

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

The USM is superior to traditional methods!

Using the USM’s IV transformation formula, from the summary at the link below, you convert integrals of this type into polynomial-type integrals. The traditional method reduces it to the integral of csc³, which would require complicated reduction formulas or integration by parts. The third Euler substitution reduces it to the integral of √(t² + a²), which also requires memorizing a standard formula or integrating it from scratch. None of this is as simple as integrating a three-term polynomial-type function, which is taught in the first few weeks of any integral calculus course.

Check out the technique here: http://geometriadominicana.blogspot.com/2024/03/integration-using-some-euler-like.html
#math #calculus #integration #USM #newmethod #halfangleapproach #symmetrymatters

The USM is superior to traditional methods!

Using the USM’s IV transformation formula, from the summary at the link below, you convert integrals of this type into polynomial-type integrals. The traditional method reduces it to the integral of csc³, which would require complicated reduction formulas or integration by parts. The third Euler substitution reduces it to the integral of √(t² + a²), which also requires memorizing a standard formula or integrating it from scratch. None of this is as simple as integrating a three-term polynomial-type function, which is taught in the first few weeks of any integral calculus course.

Check out the technique here: http://geometriadominicana.blogspot.com/2024/03/integration-using-some-euler-like.html
#math #calculus #integration #USM #newmethod #halfangleapproach #symmetrymatters

𝗧𝗵𝗲 𝗳𝗶𝗿𝘀𝘁 𝗶𝗻𝘁𝗲𝗴𝗿𝗮𝘁𝗶𝗼𝗻 𝘁𝗲𝗰𝗵𝗻𝗶𝗾𝘂𝗲 𝗱𝗶𝘀𝗰𝗼𝘃𝗲𝗿𝗲𝗱 𝗶𝗻 𝘁𝗵𝗲 𝗖𝗮𝗿𝗶𝗯𝗯𝗲𝗮𝗻: 𝗶𝗻𝘁𝗲𝗴𝗿𝗮𝘁𝗶𝗼𝗻 𝘃𝗶𝗮 𝗲𝘅𝗽𝗼𝗻𝗲𝗻𝘁𝗶𝗮𝗹 𝘀𝘂𝗯𝘀𝘁𝗶𝘁𝘂𝘁𝗶𝗼𝗻

We introduce 𝗘𝘅𝗽𝗼𝗻𝗲𝗻𝘁𝗶𝗮𝗹 𝗦𝘂𝗯𝘀𝘁𝗶𝘁𝘂𝘁𝗶𝗼𝗻, which is a transformation that simplifies certain composite trigonometric integrals and offers an alternative approach to integrating through trigonometric, hyperbolic, and Euler substitutions. For a more detailed description of how this technique works, visit the blog post

'Integration Using Euler-like Identities' at https://geometriadominicana.blogspot.com/2024/03/integration-using-some-euler-like.html

Examples 2, 3, 7, 8, 9 and 10 illustrate how this method could be 𝗺𝗼𝗿𝗲 𝗲𝗳𝗳𝗶𝗰𝗶𝗲𝗻𝘁 than the traditional approaches.

#math #calculus #integration #newtechnique #complexnumbers #halfangleapproach #symmetrymatters

I gave this integral as an answer at MathSE.

\[\int_{0}^{1}\left(\frac{5}{2} \left((x - \sqrt{x^2 - 1})^{2i} + x^4\right)-1\right)\,dx=e^\pi.\]

The integral yields \(e^{\pi}\). I don't know why the LaTeX isn't displaying properly.

Link: https://math.stackexchange.com/questions/122693/is-there-a-definite-integral-that-yields-e-pi-or-e-pi-in-a-non-trivial/4861138?fbclid=IwAR3CPXtBoNDY3yc5a1HDAgfWN7QFAO2Ph91uvH7MUgSxSRjsq-t8TWYHWdY#4861138

#math #integral #eulernumber #halfangleapproach #symmetrymatters #calculus

I gave this integral as an answer at MathSE.

\[\int_{0}^{1}\left(\frac{5}{2} \left((x - \sqrt{x^2 - 1})^{2i} + x^4\right)-1\right)\,dx=e^\pi.\]

The integral yields \(e^{\pi}\). I don't know why the LaTeX isn't displaying properly.

Link: https://math.stackexchange.com/questions/122693/is-there-a-definite-integral-that-yields-e-pi-or-e-pi-in-a-non-trivial/4861138?fbclid=IwAR3CPXtBoNDY3yc5a1HDAgfWN7QFAO2Ph91uvH7MUgSxSRjsq-t8TWYHWdY#4861138

#math #integral #eulernumber #halfangleapproach #symmetrymatters #calculus

I gave this integral as an answer at MathSE.

\[\int_{0}^{1}\left(\frac{5}{2} \left((x - \sqrt{x^2 - 1})^{2i} + x^4\right)-1\right)\,dx=e^\pi.\]

The integral yields \(e^{\pi}\). I don't know why the LaTeX isn't displaying properly.

Link: https://math.stackexchange.com/questions/122693/is-there-a-definite-integral-that-yields-e-pi-or-e-pi-in-a-non-trivial/4861138?fbclid=IwAR3CPXtBoNDY3yc5a1HDAgfWN7QFAO2Ph91uvH7MUgSxSRjsq-t8TWYHWdY#4861138

#math #integral #eulernumber #halfangleapproach #symmetrymatters #calculus

An 'impossible' integral to solve (Mathematica fails):
\[\int_{2}^{3} \frac{{1 - \tan\frac{{\sec^{-1}x}}{2}}}{{1 + \tan\frac{{\sec^{-1}x}}{2}}}\sqrt{\tan\frac{\csc^{-1}x}{2}} \,dx\]
Unless you know my new technique:
https://mathoverflow.net/questions/463459/identities-that-simplify-tedious-integrals?noredirect=1#comment1203657_463459
#math #impossibleintegral #calculus #halfangleapproach #symmetrymatters

An 'impossible' integral to solve (Mathematica fails):
\[\int_{2}^{3} \frac{{1 - \tan\frac{{\sec^{-1}x}}{2}}}{{1 + \tan\frac{{\sec^{-1}x}}{2}}}\sqrt{\tan\frac{\csc^{-1}x}{2}} \,dx\]
Unless you know my new technique:
https://mathoverflow.net/questions/463459/identities-that-simplify-tedious-integrals?noredirect=1#comment1203657_463459
#math #impossibleintegral #calculus #halfangleapproach #symmetrymatters