Posted on 09/18/2005 8:41:47 AM PDT by cloud8
Mathematics students have cause to celebrate. 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.
Established by the ancient Greeks and Romans, trigonometry is used in surveying, navigation, engineering, construction and the sciences to calculate the relationships between the sides and vertices of triangles.
"Generations of students have struggled with classical trigonometry because the framework is wrong," says Wildberger, whose book is titled Divine Proportions: Rational Trigonometry to Universal Geometry (Wild Egg books).
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."
Wild Egg books: http://wildegg.com/ Divine Proportions: web.maths.unsw.edu.au/~norman/book.htm
Source: University of New South Wales
I don'k know. My HP-48 gives me sines, cosines, and tangents out to 12 decimal places. Any of your cheap scientific calculators that I've ever seen will give you better than 6 decimal places. I find Dr. Wildberger's remarks about accuracy suspicious.
Trig is easy these days what with calculators and everything. I remember when you had to do it without calculators. Now THAT was challenging. Still, once you got it then it wasn't so hard.
Anyway, I never took math again beyond the requisite stats courses. I am perhaps the only PhD in existence who has never taken calculus at any level. I've never once needed it.
No and there is nothing obviously new either. His "fundamental" laws are just the law of sines and law of cosines reexpresed in an arcane manner to obscure the simplicity of the trigonometric concepts.
There is nothing at all elegant about this. He surplants simple and easily grasped concepts with concepts that are not intuitive. Everyone understands distance, despite is unfounded assertion to the contrary - like in it is a 15 minute walk to school or the modern version - Mom, you expect me to walk all that way to school ?!. And angle is easily grasped, just as easily as spread, which is merely an analog of the sine of the included angle.
This is pure and utter sophistry - and dangerous sophistry.
The guy seems to have something against square roots. While he may like to use the square of the distance across a rectangle rather than the distance as fundamental, speeds, fuel consumption, tire wear, etc. are proportional to the square root of his terms.
Likewise, his stuff doesn't seem to tell us much about de Moivre's identity.
Worst of all, his methods would make Fourier series far more complicated.
On a cheap calculator, it is.
Nice link.
In fact, his two fundamental concepts, Quadrance and Spread, are straightforward ways of hiding some of the computational complexity underlying the ordinary trig functions, which functions derive ultimately from the infinite series for the exponential function. The 'Quadrance' of two points is just the square of the distance between the points. And the 'Spread' between two (not necessarily distinct) lines is just a ratio of two Quadrances (which ratio ends up being the square of the sine of the angle between the lines—yes, such an angle is not unique, but that's okay).
So ordinary trigonometry is lurking just beneath the surface of Wildberger's 'rational trigonometry'. For example, we find that what he calls his 'Spread law' for triangle A1A2A3 is just the square of the customary Law of Sines. Similarly, his 'Cross law' is just the square of the Law of Cosines (what he calls the 'cross' is just the square of a cosine).
His approach has advantages—perhaps it's less computationally challenging to beginners; it generalizes easily to different kinds of fields, including finite fields; etc. But his approach also has disadvantages—angles are additive, spreads are not; the ordinary trig functions will still have to be learned at some stage by those intending to take higher math courses since they occur in vital places in calculus and have wide applications throughout physics, chemistry and the various engineering disciplines, including computer programming; etc.
From a slightly more philosophical perspective, what Wildberger says on p. 20 about the 'vagueness' in the foundations of modern mathematics and the 'logical deficiencies' of mathematical analysis is just wrong. And his claim that no axioms are required to do his rational trigonometry is also mistaken. He's assuming as given the field of real numbers, but this field is defined by a collection of axioms (almost every type of object in modern mathematics is so defined). Surely he knows this, so why he says otherwise is a mystery.
Enough.
Another issue is that square roots are not uniquely defined. This means his spread function is defined only over a and of 0 to 90 degrees, rather than -90 and 90. I can uniquely define a point in a 2D frame as r sin theta. Defining it as r^2 * sin^2 theta introduces a four-fold redundancy.
Trig is the only course I took in High School or College that I truly hated, and still do decades later.
Years ago I read a book called "Overcoming Math Anxiety" that was about exactly that. In addtion, there are common words that have other than the common meanings when used in math and science. That is part of the reason that some just "get it" and some don't.
Surely some new-ager made the Chief a Princess in order to make it less threatening...
May all your asymptotes be straight.
Right you are.
The only (slight) advantage I look for (as a programmer by trade) is maybe something that will work in a computer program a little easier than the sin function. Maybe his stuff would be relevant in doing something useful with fewer computer cycles.
"Rational trigonometry replaces sines, cosines, tangents and a host of other trigonometric functions with elementary arithmetic."Huh?
Post VersaTRIG Slide Rule????
Great, who's going to cosine for my next loan?
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