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Trigonometry Redefined without Sines And Cosines
Spy der Mann writes "Dr. Norman Wildberger, of the South Wales University, has redefined trigonometry without the use of sines, cosines, or tangents. In his book about Rational Trigonometry (sample PDF chapter), he explains that by replacing distance and angles with new concepts: quadrance, and spread, one can express trigonometric problems with simple algebra and fractional numbers. Is this the beginning of a new era for math?"
The "New" refers to South Wales, rather than the uni. Might wanna read a bit closer next time.
As per the article .. Dr Wildberger is from UNSW, the University of New South Wales .. in friggin' Australia. South Wales is somewhere all together different. But as always people don't RTFA
I am Slashdot. Are you Slashdot as well?
``Your high school required every student to take Trig?''
Yes. But then, I live in the Netherlands. Our system is different from the US; instead of (basically) lumping everyone in the same school and making sure they all pass, we send people to different types of high school, based on how they perform on a test at the end of primary school. I went to the highest types of high school, where you get at least 3 years of math IIRC; I took math for the full six years of the program. In the other other types of high school, you get less math because (1) they last shorter, and (2) they tend to focuse more on practical issues than on theoretical ones.
Please correct me if I got my facts wrong.
The concept of spread is actually pretty straight-forward. Basically, given any two lines L1 and L2 that intersect at a single point O (parrallel lines are too trivial), spread is, informally, a function of their 'shortest quadrance (distance^2) apart'. Formally:
... use, I believe, Taylor series expansions.
(1) take any point A on the first line L1. Denote qudrance between O and A is Q.
(2) compute the shortest distance between A and the other line L2 and square it for quadrance (call the quadrance R)
(3) spread between L1 and L2 = s(L1,L2)=R/Q
Calculation of (1) and (3) is trivial. Calculation of (2) isn't so bad either (if you have a coordinate system -- but you can always add one). I believe that it basically involves a vector dot-product for a projection and then an application of the Pyth. Thm. using quadances.
The beauty is that you can do this by hand! In classical trigonometry, you practically need a calculator to handle angles and you'll likely end up with an irrational number somewhere that you'll approximate to a rational one. In a world of rational numbers, quadrance and spread give you rational numbers back! Now THAT's accuracy. In fact, you get rationals of polynomials with rational coefficients.
Basically, we've been spoiled by the advent of calculus and computers. Classical trigonometry is hard. The mesurement of an angle actually requires the computation of limits, and our modern calculations of COS, SIN,
For purposes of surveying (though IANA Surveyor so I'm sorry if this sounds ignorant), a machine that measures spread instead of angle and a calculator that inputs distances (and converts to quadrances) is the biggest change. As two lines become more separated, spread increases just like angle, though not at the same rate (probably at a rate of something like cos or sin).
Of course you can express all of it using SINs and COSs, but that's not the point. The real question for us in the engineering discplines is how it will effect our use of complex numbers. What we have now is fairly convenient, but I wonder what this has to offer? Unfortunately, they didn't provide the PDF for *that* chapter.
Wildberger has put a cleaner theory underneath the kind of geometry game engine developers use. This may turn out to be useful.
Lately I've been doing robot motion planning, which has too much unnecessary trig in it. With enough work, it's often possible to derive a trig-free solution to some of the key problems. Better ways to think about trig-free solutions will help.
If you would RTF Sample Chapter, you would see that this is exactly what Wildberger has done: redefined trignometry in terms of "rise/run" ratio ("spread") and the pythagorean theorem ("quadrance").
So your complaint basically boils down to this: "carpenters don't need to know trignometry, they only need to know Rational Trignometry".
No, it would make learning Calculus all the more painful. He admits in his first chapter that the transcendental trignometric functions "cannot be understood without a better understanding of calculus". The same can be said in reverse. His "prettification" of geometry, while simplifying trigonometric calculations, makes general geometry and calculus more difficult.
For example, instead of working on a Euclidean affine coordinate system, by using "Quadrance" as he calls it, the coordinates would not be translation invariant, and you will be forced to attach a non-trivial measure to make integrals work out. So while the integrand might be simplified in the trigonometric identities, you will end up, instead of integrating over "dx", over something like "1/sqrt(x) dx", which hardly makes the integral any more appealing.
Engineers also speak PDE, only in a different dialect.
Not to be a pain, but actually you only need exp(ix) = cos(x) + i*sin(x)! Since exp(-ix) = cos(x) - i*sin(x) (just remember sin is odd, cos is even), you can multiply 1 = exp(ix)*exp(-ix) = cos^2(x) + sin^2(x). So the first formula is actually encapsulated in the second, which is ALL of trignometry!
Parent may be "4, Interesting", but nonetheless is factually incorrect.
He didn't replace distance with angles, nor is one of the two "the fundamental element of trig" - they are together the fundemental elements of (standard) trigonometry.
What he actually did was that he replaced distance with distance squared and angles with sine squared.