Skip to comments.New trigonometry is a sign of the time
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 write trig functions in software for a living. This stuff might make some things faster, simply by avoiding trig functions.
You can do trig like stuff with matrix algebra, although I don't understand it except to use existing functions. It's much faster than trig functions. Maybe this stuff is based on a simplified matrix algebra.
It was CHIEF SohCahToa in my school....
God only knows how I passed trig, because I certainly don't!
Oh, and we'd have been given a big, fat F if we'd be caught with a calculator. They weren't allowed even in class, nevermind at a test.
> These new concepts mean that trigonometric problems
> can be done with algebra ...
I've been doing a lot of angles work in constructing a
new house, and frankly, I'd rather work with classic
trig than algebra.
In any event, it will be interesting to see this theory.
Unfortunately, the wildegg site is pure hype, and the
web.maths.unsw.edu.au site is non-responsive.
Math is the most realistic, most wonderful and the only subject that keeps my brain engaged .
Social Science, literature, pottery, music etc are all crap compared to math. Math is awesome.
SohCahToa got me through Trig and I passed on the info to my kid when he took Trig (somehow the teacher didn't mention it) and it got him through too, LOL.
Once you understand it's just ratios, it takes all the "mystery" out of it.
Yeah, I don't see how it could be done with greater accuracy. Trig is an exact science.
The first chapter of the book is here.
I was lucky enough to miss new math....
""And we mathematicians wonder why so many people view our beautiful subject with distaste bordering on hostility."
Yep...still have bad memories of Mr. Bigalow for high school geometry trying to stuff my mathematically empty head with theoroms and postulates, etc.
However, when I got to college I had a wonderful woman professor who we called "Dr. Marie"...I had her for "Mathimatical Statistics" ...she was an outstanding instructor and her's was the only math class I took in all of my academic history that made any sense at all. (I got an "A") (Also learned a ton about playing "Keno" using statistics).
To grasp the concept? Not much.
But I remember being an AA graduate in surveying and having a Structural engineer with a Masters join the company and doing a bunch of mapping calculations using pi as 3.14
Took me weeks to fix everything...
Wasn't that the volcanic island that blew up in the 1800's? ;)
Trig methods will still appeal more to the spatially-oriented than algebraic solutions.
I remember that vaguely.
But I know for a fact that it wasn't until last year that I was exposed to a fascinating and useful concept: digital root!
Where did he graduate from?
" we have relied on the false assumptions that distance is the best way to measure the separation of two points"
I didn't see any better assumption in his article. I googled up "quadrance" and "spread" and learned that that these are not linearly additive metrics. I suspect that this "new" trigonometry achieves simplicity by requiring complexity in the measurement of quadrance and spread. I can't think of anything simpler, or more intuitive, than using a ruler and a protractor.
Trig is elementary for folks with a bit of mathematical acuity. I really don't see the point of trying to devise a simpler trig. It's useless for most for those whose abilities are in other areas.
There was this tribe of Indians in the old west who had a custom that whenever a couple was expecting, that the brave would venture into the wilderness, slay a mighty beast and bring the skin back to wrap the newborn in. It was supposed to bring the power and spirit of the slain animal to the new member of the tribe. Well, one particularly fertile year there were three couples expecting. The braves all went out into the wilderness about the same time to do their solemn duty to their newborn and to the tribe. The first brave went out and slew a grizzly bear and proudly returned with the skin to wrap his new baby boy in. The second brave went out and slew a cougar and proudly returned with the skin to wrap his new baby girl in. The third brave went to a special place he knew in the wilderness (there were those who suspected it was very close to the zoo), and slew a hippo, which he skinned. When he returned, he discovered that his wife had given birth to twins, a boy and a girl, which he proudly wrapped in the hippo skin.
This just goes to prove that age old adage (or is it a theorem?) Ready?
The squaw of the hippopatamus is equal to the sum of the squaws of the other two hides!
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