The End of SIN (cos and tan)

taken from:http://physorg.com/news6555.html
also see:http://web.maths.unsw.edu.au.nyud.net:8090/~norman/papers/Chapter1.pdf

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

Very neat.

I think this is kind of like a quaternion versus an euler. I can visualize eulers to a degree, but have a hard time visualizing quaternions, however doing complex manipulation with quaternions makes a lot more sense.

i am glade that this theory come true! i was a math student, i give up because of imaginary numbers... math is not imagination but real calcs

newton was right...
einstein was right...
but now, stephen hawking explains the universe, and show us that both was not totally right!

thats the way cience grow. we suppose 1+1 = 2, and now we know that 1+1 is not allawys 2

i am glade that sin is dead!

There is just one thing I can say to this:

!

Yeaah... but...

If you want to describe something rotating at a constant rate, how is that weird spread thing varying? With angles you have angle = some_constant * time (as anyone who has played with llTargetOmega will know). What is the relationship between spread and time?

Going to be horrible I bet.

Ok, I take it back, it isn't that horrible. The spread turns out to be equal to sin(t). So you still need trig to do this. There is no escape. Resistance is futile.

It appears that this is a forthcoming book, not an existing book, and will be the first one produced by the company.

I would be a bit leary of this, it might be vaportext from a vanity press.

The sin is being used as the angular measure, so it hasn't been eliminated, just disguised.

If you define the measure of two line segments using quadrance, and you want to stick the two line segments together and find the resulting quadrance, you can't just add the quadrances; you have to find the square of the sum of the square roots of each quadrance.

So the irrational numbers the author seems to hate along with trig functions may have been disguised in some places but they reappear in other places. If you want to use one fype of unit in one place where it makes the math look easy and the other type of unit where that unit makes the math look easy then you will have to be converting back and forth between two, which means one will still need to know both types of units and how to manipulate them, so people will still need to deal with sin, cos, and tangent, and not be intimidated by irrational numbers and square roots and cube roots and such.

Aside from the mathematical merits of the idea, there is also reason to question the validity of the idea that trig was some kind of barrier to understanding that when eliminated would create a big improvement in math education.

The sad truth is that most adults that I come into contact with can't solve simple fractions, don't know what the dividend, denominator, and quotient are, can't do even the simplest of functional geometric tasks such as scaling up a rectangle so that it maintains the same shape while changing size. The problems with the math teaching system in the part of the world I am at all familiar with starts at the very beginning of the process, and trying to solve the problem by keeping the kids from rooting around and committing sin in high school is too little and too late.

I looked at the sample chapter one, and this impresses me as horse-doodoo.

What's everybody got against trig anyway?

Jess

I haven't looked at the chapter yet, but the article sure sounds interesting. What I would be most interested in is, what the potential ramifications for graphics computations are? From the little I know about how computing trig functions work, I know that accurately computing sine etc. is computationally extensive (thus slow) and that in 3D rendering, faster approximating functions are often used instead of true trig functions. So, could this approach help increase, for instance, rendering speed?

I read chapter one with interest, only to be disappointed. While it may be fine for some instances of carpentry, imagine redefining modern analog electronics this way.

I cannot think of a more difficult endeavour, with the possible exception of using Roman Numerals to describe the value of pi.

Perhaps the proponents wish to lay a foundation for a post-apocalyptic era without $5 calculators.

Mathematics is a language spoken to people and machines. These concepts merely isolate a victim within an obscure mathematical framework, likely with dire social implications.

Quaternion trigonometry already accomplishes even that 'goal' more effectively.

Like the last sentence!

I agree whole heartedly with this statement, and as a teacher of adult numeracy I can provide a wedge of data to support your statement from personal experience and UK wide surveys instantly, from any country you care to choose with a little digging.

I would say, however that trig does provide another barrier to those that pass the first few hurdles. One of the things that seems odd in maths and not present in other subjects is the number of new fields and barriers that arise over time. You get to grips with counting and along comes multiplication (usually OK) and division (often confusing). Get to grips with them and along comes fractions. Get to grips with them and we get decimals and percentages. Get to grip with them and we get stats, trig etc. Get to grips with them and we get calculus. Then imaginary numbers, then...

Biology, although it gets more complex, seems to follow on. Chemistry too. (They're all subjects I teach to top end high school level or higher) whereas maths seems like a whole new set of rules and ideas each time.

But I've ordered the book, it will be interesting to see what it's like at the end of the day and if people are interested I'll report back.

I think the advantages of this stuff boil down to what many people know already - that it is often possible to work with sin(t) and cos(t) directly to solve your problem, without ever needing to find the angle t itself. Certainly I think most computer graphics people know about this already (because the trig functions are computationally slow, and so work to avoid using them).

first of all.. kudos to so much of SL being able to hold a conversation on math!!!!

Second of all.. I know that in some 3D accelerations "look up tables" are used to make finite precise computations (almost a pixilation of angles) for faster games.

Would this be a key to faster programing or is it only "better (arguably) on pencil and paper. IE does it have advantages in computation.

The idea isn't to get rid of trig.. the idea is to replace the calculus needed in trig with algebra.

-Adam

Where'd your decimal place go? ;P


I don't know what all the hubbub is about... Sure, there are different ways to look at things. If you are taught the trig functions correctly, you won't have any trouble with them. They're all just ratios--opposite over adjacent and so on. I think that at its heart, this is probably just the trig functions in disguise. And then, if you go to a school that uses this shiny new "Trig EZ", you won't understand anyone else when they talk.

Sine, cosine, and tangent are here to stay. Too many books have been written with them to change it now. It's like Esparanto-- That language that was going to be the new world standard, but no one cared about it, because they already HAD a language.

Of course, I always have enjoyed math, so my viewpoint might be biased.

I'd love a book (preferably with big print and lots of pictures, and maybe a sock puppet narrator) that could just tell me what the Hey trigonometric identities are for.

I had a tough HS trig teacher, but I followed along OK...until the day he started teaching identities.

From that day on, I might as well have been auditing a class in Klingon Epic Poetry. Fortunately, it was close enough to the end of the year that the subject didn't serve up much test material, but my grade suffered nonetheless.

I've never experienced such a sudden and thorough disconnect from any subject I've tried to study. Somebody, anybody...what are they, and/or why does Humanity care?

-- jj

Here's a basic example, and the others are really not very different:

sin(t)^2 + cos(t)^2 = 1

This just gives a relation between sin(t) and cos(t). If you know one, you can get at the other. Humanity cares when it knows one and want to know the other.

As for why it is true, remember that SOHCAHTOA stuff? sin(t) = opposite/hypotenuse, cos(t) = adjacent/hypotenuse. Like this:

  |\
  |t\
  |  \  H
A |   \
  |    \
  |_____\
     O

So suppose you decide that the hypotenuse is going to be of length 1. Then sin(t) = opposite and cos(t) = adjacent:

       |\
       |t\
       |  \  1
cos(t) |   \
       |    \
       |_____\
        sin(t)

Then that identity, sin(t)^2 + cos(t)^2 = 1^2 = 1 is just Pythagorus.

Ha!

Nice, but not quiiiite the irrational number... though I suppose any representation of values would do if you wrote out the series expansion.

Values for sin, cos, and tan are usually stored in lookup tables. Unfortunately, lots of graphics calculations can't be done beforehand. For example, rotation and translation in 3-dimensional space is one of the core functions of a 3-D application. Rotating a point in 3-space requires matrix math (or at least that's how I did it), and precalculating every possible rotation between every possible set of two points would be ridiculous. Chapter 1 does not go into rotations in any dimension, so I have no idea how you'd do them, or whether they would be faster or slower.

However, I can tell you one thing. If you have to sit there and take a bunch of squares and square roots to get the same values as you'd get from sin/cos/tan, it would be MASSIVELY slower than sin/cos/tan, since they can operate purely on lookup tables. And then, once you have those values, you STILL have to do matrix math, which is not the fastest thing.

The author claims that angles are the hardest thing EVER, and you have to use calculus to have the first clue what they are, etc. I really have to call bullshit on this one. Even infants have some basic understanding of angles; the circuitry is built right into our brains. Goldfish have that wiring (google "brain calculus" if you are interested.) I don't have to know calculus to know what an angle is, any more than I have to have a Ph.D in botany to tell you what grass is.

Perhaps most on-point, after skimming Chapter 1, I don't see how this is materially easier than "classic" trig. You still have to learn a lot of stuff before you can do anything useful with it. By the time you get done learning about quadrance (and I still don't get why he used that word, unless it's some allusion to quadratics) and spread, which is tied up in the unit circle (which, by the way, is the point of reference for sin/cos/tan), you've probably exerted as much energy as you would learning about sin/cos/tan, opposite/adjacent/hypotenuse, and S=O/H, C=A/H, T=O/A. (I find SOHCAHTOA is a pretty easy mnemonic.) If you found "classic" trig hard, I find it highly doubtful that you will simply pick up this book and magically "get" all of it. Not understanding these things is a problem of perception and analysis, not the stuff itself, so swapping in different stuff does not seem a viable answer.

Huns,

Small point of correction; Matrice math isnt that slow, if you have a modern CPU.

Starting with MMX, a lot of x86 extensions have added matrice functions at a processor level, infact MMX stands for 'Matrix Math eXtensions', although to be fair, MMX only allows integer matrice operations -- it took SSE[2] to add floating point matrice ops.



-Adam