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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?"
Perhaps Dr. Wildberger is trying to take geometry off on a weird tangent.
If only he could redefine Calculus to use simple algebraic expressions.
is 42
I did a stint as a Maths teacher, and it was hard enough trying to convince the kids that it was worth learning Trigonometry then. They'll be even more determined to be ignorant if they hear of this.
Aide-toi, le Ciel t'aidera - Jeanne D'Arc.
He does this the year after I take Algebra II/Trig. Bastard.
Note to mods: I'm probably being sarcastic.
The "New" refers to South Wales, rather than the uni. Might wanna read a bit closer next time.
I'm a failing engineering student at Iowa State University- or at least I was.. My failing point: math. I can out code some of the instructors in my classes, but I can't do math or memorize functions and derivitives of trigonometric functions.
If this book pans out, it would ultimately change Calculus for the better and might allow me to pass my classes- I wonder if I figured out how to apply this book to my classes and came up with the correct answers despite having worked the problems with his methods, if I'd still pass the class?
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
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This sounds promising, but I have two educational concerns: 1. Is this just a dumbed-down version of trig? ...and on the opposite end of the spectrum...
2. Would this lead to bombarding students with too much math as the requirement shifts from alg/calc/trig to alg/calc/trig/trig2?
Just wait until someone reimplements many software functions that rely on sine, cosine, and tangent, using these new concepts, and get a patent on them.
Inconceivable!
Erm, I actually read the sample chapter, and one thing I don't get: What did he do that is so revolutionary?
He redefined a side of a triangle with a Quadrance - a square of distance. He claims this removes the square root, but guess what? d^2 has the same effect.
He also defined a spread, which is the relationship between two lines. The catch? It is PROPORTIONALLY EQUAL to an angle. 90 degrees is 1, 0 degrees is 0, 45 degrees is 1/2.
I haven't read the full book, but from what I can tell, all this is doing is redefining the constraints of trigonometry, causing nice even numbers to be used.
ahh Sin= Op/Hyp
Cos = Adj/Hyp
Tan = Op/adjacent.
By as someone who's done some surveying it (which uses angles and distance), its easy to see why people use angles and distance. They are fairly easy to measure, so usefull for plans etc..
Replacing angles and distance with abstract quadrance and spread is exchanging one difficult thing (tan,cos,sin) for another (quad, spread)
Quandrance = distance ^2
Spread hard to see.
I am wondering if this could be used to make faster calculations
in raytracers and 3D engines by using integer numbers.
You're wrong.
It's not that trigonmetric functions are hard to learn, it's that they rely on transcendal function like sine/cosine which are calculated numerically (as a taylor power series expansion, for example) thus are only approximations to the true values (an accurate number would require the calculation of an infinite series, which isn't practical in given time/space).
The article clearly states that: "Advanced mathematical knowledge, such as linear algebra, number theory and group
theory, is generally not needed." (to use this method)
I think that having a percise, simple (polynomials, rational fractions) alternative to current methods in eucleadian trigonometry, is very welcome.
Wouldn't it be nice to be able to calculate angles and distances without having to use a calculator (for sine/cosine calculations)?
Sigs are for the weak.
The professor seems to have found an interesting way to present trig - however, it should also be noted that he does actually rely on sines as an underlying concept.
I read the article and part of hist concept of spread for an angle is the ratio (in a right triangle) of the side opposite the angle to the hypotenuse of the triangle. As I recall from my high-school geometry classes, though, this is exactly how the sine is calculated. (The cosine is the adjacent side divided by the hypotenuse.)
The interesting thing about his approach, though, is that he defines the concept of spread so that there's no need to refer to the underlying sine concept and the calculations are all (relatively) simple algebra - squaring and addition and subtraction. I would like to read some more and play with it a bit - the fact is I still have bad dreams about those damned trig identity tables, which I never really successfully felt comfortable with!
Interesting.
Sigh. My id isn't prime. 2 2 2 2 2 3 5 313
As a math teacher, it may not be as obvious to you, but as someone who learned trigonometry in school and isn't a math teacher now I can say for certain. When people say they'll never use that in the real world, they're absolutely right.
Now sometimes while slogging through my day, paying bills, shopping, working and things like that I sometimes say "Man I really should calculate the cosine of this electric bill", but as of yet I haven't been harmed by not actually doing that.
What happens when kids get to math subjects where trigonometric functions are used for more than just calculating the dimensions of geometric figures? How does this "spread" thing represent angles greater than 180 degrees without redefining what you are measuring, and how does this really make a persons life any easier unless someone tells you the spread as in the textbook chapter? If his explanation is to be taken as any sort of indication of how to measure the spread, then you might as well just walk off the dimensions of what you're measuring because you'll have to do that anyway to calculate it. How will you integrate problems that call for constant rotation using spread? This seems like trading a little bit of pain now for a lot more down the road, and I pray that it won't catch on in the US. If students are really having this much trouble with this subject then it should be introduced earlier and in smaller portions, not ignored. The last thing we need is for someone to take another slice out of the already anaemic math programs in our primary schools.
`which fortune`
Imagine if we'd been using "quadrance" and "spread" for years - and then some bright spark suggested calculated using sines and cosines. Which would people see as easier?
I don't usually advocate this kind of behaviour, but the chapter is actually well worth reading. Quadrance is a neat hack - use the square of distance instead of distance to eliminate some nasty square roots, but spread is a much more interesting concept. The notion of spread removes the dependence upon circles to define relative direction, which removes a lot of complexity from trig.
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Even kids who go into trades like carpentry would benefit from a knowledge of the fundamentals of trig. Laying out angles for roof rafters, staircase stringers, etc....
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It looks like all that is being done is removing squareroots and negative numbers.
quadrance is the square of the distance
spread is the square of the sin angle
If you want to teach trig this way, it will certainly be easier to learn. But then, your students will not appreciate or understand square roots and negative numbers.
It is highly unlikely to lead to any revolutionary new algorithms. It appears that you are changing an angle to something rational, because all most of the popular sine values have square roots in them. However do not be fooled by this, it is just as hard to calculate spread as it is to calculate the sine of an angle. (except in a few cases).
I was taking a real analysis class in my first semester of grad school. I did a good job on the first question, figured everything out, and got that the answer was the integral of 1/(1 + x^2).
I got no points at all for the question - despite solving the parts relevant to the class - because I didn't know off the top of my head that the integral of that is the arctan function.
I love abstract math but I hate trig.
My dad is a machinist, and that job is very heavy on trig. If you're trying to figure out how to calculate a correct taper or how to calculate threads, you need to know trig.
If you implement a geometrical function whose input is a bunch of lengths and their ratios and whose output is a bunch of lengths and their ratios then you frequently shouldn't need any trig in your function, instead you should be using algebraic functions (+, -, *, /, nth root etc.). But many people find it easy to solve their problem by converting a bunch of stuff to angles using trig and then converting back. In fact, I optimised someone's rendering code recently by literally looking for every line of code where he'd used trig and replacing it, where possible, with the algebraic version. It worked well and speeded up the code massively. In addition I uncovered what you might call bugs. For example at one point this guy did a 180 degree rotation by converting to polar coordinates, adding M_PI to the angle and converting back. Rotating (x,y) by 180 degrees gives (-x,-y). The guy who'd written it was 'trig happy' and should have stopped to thing about the geometrical significance of what he was doing rather than just doing the obvious thing. If this book encourages people to use trig less it might be a good thing. But you don't need to talk about 'quadrance' and 'spread' to use the principles I'm talking about.
But of course there are times when you need trig. I'd hate to see the guy differentiate rotations without trig.
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Here is what he does: He replaces the distances by its square, and calls it a squandrance. He replaces the angle by the square of its sine and calls it a spread.
Ok, the relations between the squandrances and spreads of a rectangular triangle are simpler than those between its sides and its angles -- they are just simply obtained from pythagoras' theorem.
However, two way more fundamental relations suddenly become horribly difficult: Say going from town A to C I go from A to B then from B to C. A to B is 5 squandrance, B to C is 3 squandrance, how much squandrance is it from A to C? No, not 8, but 5 + 3 + 2*sqrt(8). The simple addition of distances becomes a square-root function...
Also, say I turn left by 30 degree, and then do that again. Guess what, I turned left by 2*30 = 60 degrees. However, if you were doing this in spreads, you don't want me to tell you the answer. I think there is a reason why sailors have been using angles and not spreads...
As for engineers: Why do they have to learn trigonometry? Not for geometry, you can just do that with coordinates. But because it turns up in waves everywhere, which in turn turn up everywhere. (Aside: the mathematical reason is that sine and cosine are the solutions of one of the most fundamental differential equations.) For example, the voltage of an AC current will depend proportionally on a sine fucntion. The angle you substitute in sine will be proportional to time. The spread you would have to substitute in a rational geometry function instead is NOT proportional to time, but to the square of the sine of time. Too bad my stopwatch measures time in seconds and not in the square of its sine...
Will his approach lead to faster computations? Of course not. Whereever it would help, people of course already knew last year already (and maybe even, gosh, in 2003) to use some trigonometric substitution....
I am a high school mathematics teacher and I train students for mathematics competitions. I think most of you are missing the point of Dr. Wildberger.
Dr. Wildberger is not trying to redefine trigonometry, he is simply trying to give it a new perspective and hopefully, the new perspective would allow new insights into new methods of solving trigonometric problems. Protesting that memorizing the trigonometric functions as side adjacent over side opposite, etc., etc., is very easy and intuitive ignores the fact that in analytic geometry, that is not even how the trigonometric functions are defined!
Yes, really! For example, the sine of theta is defined in analysis as the y component of the radial vector from the origin to a point in a circle of unit radius whose arc distance from the x-axis is theta. The cosine of theta is defined similarly but this time taking the x-component. From this two simple definitions, the entire panoply of the trigonometric identities can be usefully derived!
The analytical definition is certainly not intuitive and not easy to memorize for a high school student! The side opposite, side adjacent trick is just that, a trick that is useful sometimes and certainly useful enough for high school mathematics but it is not a very useful definition as far as analysis is concerned.
For example, computing the derivative of the cosine function is not easy to understand if you restrict the definition of cosine to side adjacent over hypotenuse! Not to mention the fact that most students think there is magic involved in the computation of the trigonometric functions because the method of computation is not in their textbooks. It is only when one studies the calculus that the methods for computing the trigonometric functions are explained!
Dr. Wildberger has an idea that he thinks will make trigonometry more intuitive and I hope he is really onto something here. It would certainly help me with my students. I have read only the downloadable first chapter of the book and the idea is intriguing. Waving off Wildberger's new ideas without reading the entire book and without understanding the mathematics of trigonometry is just tragic.
In times like this I always remember the architect (I forgot the name, help me out here please) who refused to accept an architecture medal because the society that was giving out the medal invited Prince Charles to hand out the medal. That architect said, "I refuse to accept a medal from a person who believes that our grandfathers already know everything there is to know about how to build buildings and that there is nothing we can ever add to that knowledge anymore."
Just my two cents.
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.
you still get a degree!
Bert
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.
Ironically, you've actually harmed, not helped, anyone who hadn't read the article. The main point of this work is that distance AND angle are the wrong things. The quadrance (square of distance) and spread (effectively, the square of the sine of the angle -- though help points out 'angle' is a handwaving concept to begin with) should be the fundamental elements and makes the trigonometry meaningful and easy. That quadrance and spread are (independently) functions of distance and angle is trivially obvious, but it is exactly the difference that is his point that makes the math so much simpler.
Perhaps a more specific example for those programmers out there. No more need for lookup tables or Taylor expansion for trig functions unless a human at some point needs an angle value at which case it can be converted in the final step before output. (Internally, programs would never need them.)