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 wonder when this will make it into surveying equipment, if ever.
When I took Trig I walked 5 miles uphill in the snow to school and 5 miles uphill home.
To keep our fingers from freezing we would rapidly slide our slide rules....one student even ignited some parchment.
I wish YOU had been my trig teacher.
What you just wrote here on this FR thread made more sense to me than all the gobbledygook ever uttered by my math teacher way back when.
Regards,
The beginning stuff wasn't too difficult w/ a calculator.
My grammar book calls for using the apostrophe when pluralizing numbers or single letters [this sentence contains [three c's, e.g., and two 6's, and probably has very little to do with the 1960's]. I don't know why the usage does not extend to acronyms.
There were hand-held calculators when I took trig, but practically the only ones who had one were astronauts. I remember my parents buying one for me for my birthday and knowing that that $65 came very dear to them.
I don't think I'd ever received a $65 gift before.
Regards,
If one wants an understanding of this sort of stuff from a practical computing perspective, maybe one should talk to the people at id Software. Given the amount of math-bashing it does, I'm sure Quake uses a few tricks (and Doom before it may have used even more, since it runs without a mathco).
I am really bothered by the guy's attitude if the quotations here are indicative of it. There are no doubt some calculations that can be done more efficiently with his techniques than with 'normal' trig techniques. On the other hand, which is apt to make more of an impression on someone: using geometric models to show that the sum of the square of the sides equals the sum of the hypotenuse (by using dissection of squares), or telling students that the quadrance (whatever that 'means') of the hypotenuse equals the quadrance of the sides?
To be sure, there are times when quadrance can be useful. When sorting points by distance, for example, it's a lot easier to sort by quadrance than to take lots of square roots. But IMHO it is a mistake to eschew teaching students concepts that have intuitive real-world meanings in favor of ones that don't, even if the latter concepts are computationally easier.
I think the present author could contribute a chapter to the next edition.
OK. I've got a doctorate in math, and do research in geometry, and have looked at this stuff, and I've figured out what this guy is trying to accomplish.
It's a good idea.
It is NOT a good idea to teach INSTEAD of the standard geometry trigonometry, as long as all the rest of mathematics uses the standard functions. However, it IS a good way to teach the subjects of geometry and trigonometry themselves, as long as, at some point inthe course, the STANDARD definitons are introduced and their basic properties are proved.
Basically, the "quadrance" of a line segment is the square of its standard length, and the "spread" of an angle is the square of the sine of the angle. If you make these your fundamental quantities, most formulas and computations become simpler and their logical foundations become clearer. As long as, by the end of the course, students can translate freely between the two ways of looking at things, no harm is done, and a lot of conceptual and computational obstacles are avoided.
Almost every problem in standard trigonometry which requires a numerical solution must be solved by calculators or tables with messy approximations, even though the ultimate answer can be expressed in terms of the initial data by rational operations and square roots (and square roots are much, much, easier to calculate "by hand" than sines, cosines, and tangents). So Wildberger is onto something, though he's up against millennia of pedagogic tradition.
It sounds like the Frieden calculator we used in college chemistry in the early '60's.
Fortunately I grew up in a part of Florida where there was no snow and very few hills. ;-)
Doesn't this stuff sound an awful lot like vector cross products? Doing matrix maths on vectors can certainly be very useful (Quake does such maths in spades) but teaching it so early on seems premature.
BTW, if I were solving his example problem, I wouldn't use trig functions at all. I'd compute the area of the triangle, compute the altitude from that, and then have the slope of the upper-left edge. From that, I'd compute the intercept. The required length would follow naturally from that. No need for sines or other such functions (except that tan(45°)=1).
If this guy is going to substitute Linear Algebra for trig, may god help his students. That stuff is great for computers but it is h*ll on pencils. I can approximate trig functions in my head, the largest matrix I could . or x would be 2by2's.
With all due respect, you were a victim of bad math teachers. On the other hand Bill Gates, Steve Jobs, and all, had good New Math teachers. The New Math fueled our digital revolution. But asking run of the mill public school math teachers to teach it was like asking a gas station mechanic to fix a flying saucer.
You must have had a big bedroom.
OK but how do you get the sine of the angle to square without trig - just wondering.
I bet that calculator was very dear to you, both because you knew the sacrifice your parents had made for it, and because you knew it showed their faith in you and that they were trying to help you reach your goals. :-)
No kidding, and octal converts so easily into hex... Well, I didn't get to start playing with 360 System DOS until the 1980s, but thank goodness I wasn't colorblind... Being able to tell between yellow and green...
Mark
One use of trig is to transform equations into polar coordinate systems, where solutions are easier, depending on the geometry.
Can this be done using his approach?
IIRC, trig identities also were fundamental to much of calculus.
I remember the original "Bowmar Brain," a 4 function calculator that was quite expensive... There was a TI calculator with the 4 functions, plus square root and reciprical... I never quite got that one... Why dedicate a function for that when some other function could have been added... Somewhere around here, I've still got my HP41C. What a great calculator, but when I use a calculator today, the "equals sign" still confuses me! I've got a DOS graphics program that still works in WindowsXP of an HP11, so I can still get by.
Mark
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