Rational trigonometry?, Replaces Sin, Cos, Tan with arithmetic

This sounds very exciting:

http://www.physorg.com/news6555.html
http://web.maths.unsw.edu.au/~norman/book.htm

Would any mathmaticians or other people with insights care to comment?

Thanks...

I must admit it sounds very interesting, and does away with the need for trigonometric functions, but some of the ideas are slightly unfounded - he says concepts such as an angle is hard to understand, however i think you'll find most people will argue the opposite - that spread is a much more complicated concept. Also the idea that trigonometric functions are hard to define and require complicated calculus - take A level maths and you'll have to learn how to define them. Furthermore the ideas that 24 and 156 degrees are somehow equivalent because they give the same values in trigonometric functions is wrong - even if you can specify with ac (acute) or ob (obtuse). What happens if you have two obtuse angles that both give the same spread? they are indistinguishable. It would make a big difference to you if your clock hand had moved 617 degrees instead of 257, but rational trigonometry says they're the same(?)

I doubt that in its current form it will be making it into any curriculum etc, and whether it is better at solving complicated problems more accurately will have to be tested, but since normal trigonometry has given accurate enough answers up until now - and given that there is no such thing as 100% accuracy in real life - it remains to be seen if the extra accuracy is actually required by anyone.

http://www.physorg.com/news6555.html

This is interesting, but what about things that are naturally transcendental, like say a Fourier transform? I"m trying to comprehend how this would affect the other uses of Trig (other than getting a grade in high school). One thing I liked is the elimination of the square root for a lot of operations, which often brings in complex numbers where they are not needed. I suspect that the square root construct is responsible for what I think might be a couple of basic errors in physics, but that"s just me. For other things (I do digital sig processing for a living) they"re pretty cool, but cooler yet when you can fold them out of what you"re doing by some symmetry.

joeearl: Wildberger addresses two of the points you make in the first chapter of his book: The first (removable) ambiguity arises only because of the fundamentally projective approach he takes and the second because such distinctions of periodicity belong to mechanics, not geometry. What he says about his "rational geometry" being more natural and general is quite right too (at least in hindsight and from the perspective of modern geometry). I certainly wish I'd been taught from the outset to think in terms of intersecting lines having spreads of 1/2 when they were half way from parallel to perpendicular etc. - perhaps as well as the (sometimes) more complicated and analytically based angle system. I haven't read the whole book yet and perhaps it will prove a too difficult approach for students but it looks to me like Wildberger may have provided a foundation for modernising elementary geometry, and if nothing else, I expect to be able to play Doom4 without having to buy a new computer ;-)

From the viewpoint of a student with difficulty learning math concepts (odd, I suspect, on these forums), I'd just like to say that I'm ecstatic that someone is looking into making them more comprehensible to those of us with less apptitude in the area

Of course, my hope is that this is everything everyone hopes it to be and that it is implemented into the curricula of schools before my kids hit high school/college. If it makes it easier to learn, then I believe it's worth investigating - even if the accuracy rate is only minimally better.

Hi All,

Gee... I hope this is easy otherwise it is going to be an impost on students. The "real world" uses angles and distances. There needs to be something really "good" to make this stuff be acceptable. Look at all the literature that has angles and distances, is this to be somehow superseded now? Who is going to rewrite all this in the new format? Is this another thing to learn without any revolutionary benefit like Cuisenaire Rods?

Cheers

Of course it's good to have book on use of Pythagorean theorem to deal with arbitrary triangles, just why call it "rational trigonometry"?
Indeed, it may be the case that school devotes too little time for Pythagorean theorem - i seen people getting stuck using trigonometry where they would easily be able to solve problem using Pythagorean theorem instead and avoiding angles.

Speaking from the point of view of a physicist (to be clear, a sophomore in college, not a professional), for me math is mainly a means to an end, be it an answer, or understanding. Some equations make more sense in certain forms, even though other forms are mathematically equivalent, they just aren't as clear--they don't convey the physical concept as well. I'm hoping that, even if this doesn't catch on in mainstream schools and replace traditional trigonometry, it could be used to help facilitate understanding by presenting ideas in a simpler or more direct way. Also, no one will argue that it's always a good thing to have one more mathematical trick up your sleeve.

The 'Real World' does not use angles and distances. Some of us percieve and relate to the world in this way and have found excellent ways of mapping that perception in ways that predict the results of further world observations. Understanding that this is the case is what enables pioneering scientists to move beyond our current constructs to a more complete understanding of what is being observed, rather than a more complete application of our current way of understanding it.

It's aabout time somebody figured this out!
I'm an intuitive problem solver, and have always known trigonometry was somehow wrong!

honestly, I have no idea what on earth he is talking about. I am in an Honors Trig class right now and have had no problems at all understanding it, noone has. If anything, the biggest problem is forgetting to switch from degrees to radians mode on our calculator.

The problem with the "rational trigonometry" system is that the simplicity of the operations is bought with nonlinearity in it's basic concepts. If I divide a line of length 1 in half, the length of its two halves is 1/2 each; whereas the quadrance of the two halves is 1/4 each. If I connect two lines of quadrance 4 and 9 respectively end-to-end, I get a line with a total quadrance of 25. For a system that promises simplicity, that seems to be a remarkably non-contrived example that requires implicitly switching back to distance to get the answer.

Quoting joeearl: "I certainly wish I'd been taught from the outset to think in terms of intersecting lines having spreads of 1/2 when they were half way from parallel to perpendicular etc."

This is an example of the dangers of a nonlinear system of measurement. What the poster says is true, but it's almost coincidental. If we bisect an angle with spread 1/2, we don't get two angles of spread 1/4, we get two angles whose spreads are just a bit over 1/7 (specifically 0.146446609406...). So while spread 1/2 is halfway between parallel and perpendicular, halfway between 1/2 and parallel (0) is not .125, but .1464... Again, that's not a contrived example, and it seems to me to be a high price to pay for the purported benefits of "rational trigonometry".

-Bryan Alexander, M.S. Physics

I have not read the book and have no comment on its value. However, I do have a comment on the notion that we could somehow dispense with teaching the trig functions in high school or college. Trig functions are not just about figuring triangles. Fourier Transforms have been mentioned in another post; there are many mathematical concepts that make use of trigonometry and have no obvious connection to triangles. Engineers and many other practical users of mathematics rely on these functions for many basic calculations. Is it possible to supplant the trig functions with something else, in such a way that it would generalize to all the other current uses of trigonometry in higher math? If so, how do we make the transition? Can we really have one generation of engineers who understand and write things in terms of sine and cosine, and another generation immediately following who have no knowledge of these things? Or one branch of the field that uses one set of definitions and another branch that can't communicate with the first because it uses another set of definitions? I won't say it's impossible to establish a new paradigm, nor that it should not be done. Just that it seems extremely difficult. Proceed with caution before eliminating something so basic from the mathematics curriculum. If it really is a good idea, work it from the top down--get the graduate schools on board first, then the undergraduate schools, etc. so the reliance on the old system can be weaned in an orderly manner.

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[B]I am another who has difficulity understanding math concepts. I also enjoy math immensely and have worked successfully in technology for many years.
Having new concepts to explore, it seems to me, will be fun. Who knows what fresh thinking might come out of this, no matter if it is workable or not.

figures, they would come up with this after i get my degree