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| Trigonometry No More |
| 2005-09-18 |
| ...A University of New South Wales academic, Dr Norman Wildberger, has rewritten the arcane rules of trigonometry and eliminated sines, cosines and tangents from the trigonometric toolkit. What's more, his simple new framework means calculations can be done without trigonometric tables or calculators, yet often with greater accuracy... Dr Wildberger has replaced traditional ideas of angles and distance with new concepts called "spread" and "quadrance". These new concepts mean that trigonometric problems can be done with algebra," says Wildberger, an associate professor of mathematics at UNSW. "Rational trigonometry replaces sines, cosines, tangents and a host of other trigonometric functions with elementary arithmetic." "For the past two thousand years we have relied on the false assumptions that distance is the best way to measure the separation of two points, and that angle is the best way to measure the separation of two lines. "So teachers have resigned themselves to teaching students about circles and pi and complicated trigonometric functions that relate circular arc lengths to x and y projections – all in order to analyse triangles. No wonder students are left scratching their heads," he says. "But with no alternative to the classical framework, each year millions of students memorise the formulas, pass or fail the tests, and then promptly forget the unpleasant experience. "And we mathematicians wonder why so many people view our beautiful subject with distaste bordering on hostility. "Now there is a better way. Once you learn the five main rules of rational trigonometry and how to simply apply them, you realise that classical trigonometry represents a misunderstanding of geometry." |
| Posted by:Anonymoose |
| #17 Trig isn't that difficult, it is a high school level subject that you use for the rest of your life. This would really have to be something special to make me even the least bit curious about it. |
| Posted by: bigjim-ky 2005-09-18 23:41 |
| #16 I finally got to his Chapter 1, and you're right, Darrell: simplicity isn't the first thing that comes to mind. And the theory seems a bit sterile, though it might have some links to number theory via pythagorean triples. And the whining about "traditional theory" is generally the sign of the crank. |
| Posted by: James 2005-09-18 21:31 |
| #15 I read Chapter 1 and was not impressed by any "simplicity": http://web.maths.unsw.edu.au/~norman/papers/Chapter1.pdf |
| Posted by: Darrell 2005-09-18 20:51 |
| #14 "Trigonometry is not only about distances and angles. It's useful in modelling repetitive, or periodic phenomena." Yeah, like Fourier Transforms for signals work. Does this mean we really didn't track all those Russian subs these past decades? |
| Posted by: Dave 2005-09-18 20:35 |
| #13 It sounds like some simpified rules and rote rules such that will allow you to work with triangles, but are totally worthless for all the other uses of trigonometry. Useful for paving stone layers and dress pattern makers, and that's all. |
| Posted by: Jackal 2005-09-18 20:25 |
| #12 Ima hafta go readup on this new math before I buy into it---mathematical equivalent of cold fusion, LOL! But if we have no sines and cosines, how will we do a Fourier series? Repeating waveforms, and all? |
| Posted by: Alaska Paul 2005-09-18 19:01 |
| #11 "For the past two thousand years we have relied on the false assumptions that distance is the best way to measure the separation of two points, and that angle is the best way to measure the separation of two lines. sheech! obviously a tape measure is...iffin you don't have an experienced eye. |
| Posted by: Red Dog 2005-09-18 18:52 |
| #10 "For the past two thousand years we have relied on the false assumptions that distance is the best way to measure the separation of two points, and that angle is the best way to measure the separation of two lines. Moron. |
| Posted by: gromgoru 2005-09-18 18:22 |
| #9 http://en.wikipedia.org/wiki/Rational_trigonometry Published by "Wild Egg" hm, Wildberger + nest egg? $80 to find out. The wikipedia article presents a trivial reformulation, but I'd have to see the book. The traditional trigonometry presentation leads to circular functions and they in turn provide an entry to complex numbers. These fields are just as important to modern math as the mensuration of triangles. Another Wolfram? |
| Posted by: KBK 2005-09-18 18:21 |
| #8 well, I was trying not to get too into it.... |
| Posted by: Frank G 2005-09-18 18:19 |
| #7 SOHCAHTOA, baby. Once you have that, you've got it all. |
| Posted by: Robert Crawford 2005-09-18 18:12 |
| #6 Trigonometry is not only about distances and angles. It's useful in modelling repetitive, or periodic phenomena. It's also useful for integration (calculus, trig substitutions). Well, let's wait and see. Quaternions ceded to vectors, maybe this is something similar. |
| Posted by: Rafael 2005-09-18 18:05 |
| #5 sines, cosines, etc are only a measure of ratios between sides on X-Y axes. I fail to see how it could be made easier....guess the accuracy could be dumbed down to 45 degrees then you'd almost always be able to guess.... |
| Posted by: Frank G 2005-09-18 17:36 |
| #4 I can't get at the reference the article points to, so I've no clear idea what he's talking about. But the trig functions are so central (and so easy) that I don't see them being replaced. I wonder what he wants to replace exp(ix) with? (BTW, nobody I know uses secant/cosecant etc, just ratios of sine and cosine. Too hard to remember all those names.) |
| Posted by: James 2005-09-18 17:14 |
| #3 Heh... I was thinking about orbital mechanics... Wow, is right! |
| Posted by: .com 2005-09-18 16:19 |
| #2 I'm not a mathemetician but I'm fairly intelligent layman (I did minor in math lo these many years ago). My first response, after incredulity wore off, is what does this mean for all those branches of physics that have to do with trig? If I remember correctly there are sines and cosines and secants etc. all over the place in particle physics and astrophysics. Wow. |
| Posted by: AlanC 2005-09-18 15:50 |
| #1 Now that's fascinating... I might buy the book, though I have no actual need, just out of profound curiosity. |
| Posted by: .com 2005-09-18 14:49 |