A New Trigonometry

[Pancho Villa]

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've enjoyed the first chapter (pdf).

I am also going to buy the book. I have always had problems with trignometry. Nobody ever gave the definition of sine, cosine, and tagent to me, and I had to figure it out on my own. When I did, I found an absurd circular logic to them. cos (angle) = adjacent/hypotenuse. It was just a ratio that was universal - certainly it worked but I found the solution somewhat...brute-forcing your way to the solution. This seems to hold great potential.

Trig has really kept me away from the higher mathematics (which I am interested in greatly) because of all of the stuff I wrote above. Now I am going to purchase this book and see if its methods are any better.

Edit: It is so refreshing to read a the beginning of a math textbook and have the author explain "We are going to write things this way, because it will be useful to you when you go on to higher math to think of the concept in this manner."

Edited September 26, 2005 by Pancho Villa

[Felix]

Thanks for posting this. It looks very promising.

I'll buy the book when the next semester starts. It is well written and it looks like a real breakthrough in understanding trigonometry. At least it is a new angle () to look at the problem.

[HaloNoble6]

Very interesting. I wonder, however, of what use this would be for those of us who have already mastered "classical trigonometry."

[Pancho Villa]

Very interesting. I wonder, however, of what use this would be for those of us who have already mastered "classical trigonometry."

Well, I have "mastered" classical trigonometry, which is to say I passed a course in it. But future courses (calculus in particular) depend rather heavily on knowledge in this area. It has been painful for me to use this knowledge, which I have always had a feeling was broken, to learn these otherwise interesting subjects.

Your mileage may vary.

[HaloNoble6]

Well, passing a course is not what I had in mind. I guess I had a very good trig teacher in high school that explained the concepts well. Though the mathematics is more complex than in this new approach, I don't think the concepts in classical trig are necessarily difficult. I at least have certainly found no difficulty with classical trig and its concepts, and I've taken several high level math courses beyond calculus. At first glance I think this new approach will be more helpful to new-comers to mathematics, since it's meant to make the mechanics of mathematics easier.

[xavier]

One of the main motivations for this new system is to avoid square roots, as stated by the author, but then in the example problem on page 16, you have to take a square root to convert back to distance...not only that, you need the quadratic formula! (Top of page 16.)

Thanks for posting this. It looks very promising.

I'll buy the book when the next semester starts. It is well written and it looks like a real breakthrough in understanding trigonometry. At least it is a new angle () to look at the problem.

It certainly is a new angle to look at the problem, and it may help newcomers learn, but it certainly is not a breakthrough in trig. One can easily convert back and forth from this "new" quadrance/spread system and the classical angle/distance system. Doing so does not give you new insight/information. For example, the Laws of Rational Trig on page 10 can be re-written in classical terms, which is covered in high school math, so none of these laws are new.

Very interesting. I wonder, however, of what use this would be for those of us who have already mastered "classical trigonometry."

Not a whole lot. Ink marks on a page are concretes. Ideas such as distance/angle are abstract. So are quadrance/spread. But both those systems have a common underlying concept, namely, the nature of 2-D geometry.

In fact, relying on this system may be disadvantageous, if you are interested in advanced calculus, since they are all taught with sines and cosines. This is just convention.

[Felix]

It certainly is a new angle to look at the problem, and it may help newcomers learn, but it certainly is not a breakthrough in trig. One can easily convert back and forth from this "new" quadrance/spread system and the classical angle/distance system. Doing so does not give you new insight/information. For example, the Laws of Rational Trig on page 10 can be re-written in classical terms, which is covered in high school math, so none of these laws are new.

You say, in effect, that a new way of looking at things is just for newcomers. I have to object. A new system of representation is all mathematics is about. Mathematics is logic plus representational systems. A matrix is nothing new if you know how to solve equations. It's just a new way of looking at things. And you can transform the matrix back to classic equations. But you are completely missing the point of this.

A new way of looking at this, i.e. a new representational system is a major breakthrough.

You can formulate equations in a new manner and see new patterns, new relationships that weren't apparent before. Like the squareroot of seven in one of the examples.

I wonder what many formulas of advanced physics look like in this new system. Maybe this can launch a new idea for a new understanding of physics. It may bring new ideas for speeding up computer chips.

Hell, who knows what a gift a new representational system can be.

As far as I am concerned, this is just brilliant and I will definitely read the book and try to formulate some known equations in the new system. The worst thing that can happen is I get a deeper understanding of the fundamentals.

Remember: Galt found a new understanding of what energy is. What did you think he found? He found a new way of looking at old things and therefore found new things. This is the power of representational systems.

Edited September 26, 2005 by Felix

[woolcutt]

I too have read the first chapter and as I am teaching part of a Trigonometry class here at UH will claim to have some expertise in the field. The beauty of this new system is that any of the old formulas involving the relatively arcane trigonometric and inverse trigonometric functions are done away with.

The point was NOT to do away with squre roots and irrational numbers (which is unfortunately suggested by the name "Rational Trigonometry"), but to do away with complicated functions like Sin(x) and ArcTan(x) which are hard to explain to someone without the use of calculus and power series. For instance its hard to calculate the value of ArcTan(19/20) without a calculator (in fact would be very difficult to explain to the average trigonometry student and almost impossible to explain the concept of why that calculation works), but this calculation does come up in relatively simple trigonometry problems.

[Note: I have in mind the method of approximation by power series, if my particular example is easily worked out by hand forgive me]

The book replaces the concepts of "distance" and "angle" with "quadrance" and "spread". Instead of deriving formulas involving distances and angles (i.e. Laws of Sines and Cosines), the book shows us the relations between quadrance and spread which turn out to be (relatively) simple polynomial equations. This means that instead of using approximations, students can use the quadratic formula (something that is pretty easy to understand) and get EXACT ANSWERS. This alone would be such a boon for the average TA grading papers (like me), and would save hours of time hunting through HW scrawl looking to see if a wrong answer is the result of not understanding the concepts or a rounding error early on in the problem.

Thats my two cents, I like the first chapter and look forward to reading the rest. There are implications for much of higher mathematics too, but I only wanted to comment on the "basic" trigonometry aspect right now.

[LauricAcid]

From just my quickskim of the pdf chapter, this looks pretty good. I'm putting it on my shopping list.

[xavier]

After reading your post, Felix, I re-read chapter one, and I must admit that it's more facinating than I originally gave it credit for.

However, I still stand by my assertion that going from angles/distances to spread/quadrances is just a change in convention, and would not call it a major breakthrough in math. It is more akin to changing from cartesian coordinates to polar coordinates than going from scalar equations to matrices.

The new spread/quadrance system has advantages and disadvantages over the classical system, just like the polar coordinates have over the cartesian coordinates. For example, as woolcutt mentioned, an exact answer in one system may be only an approximation in another. But by and large, they really solve the same class of problems.

Going from a list of equations to a matrix is different. You can do things, like take the inverse of a matrix, that has no counterpart in individual scalar equations. I would call this a major breakthrough.

Even so, the change from angles/distances to spread/quadrances is no where near as pronounced as going from cartesian to polar. Quadrances is just the distance squared, surely this alone will not spur any new research in math/science/engineering, besides it has already been used before. I guess I was disappointed in the definition of spread. When I took trig, I immediately understood the relevance of the ratio, and the functions sine/cos/tan are just additional conventions. Besides, as mentioned, the inverse trig functions exist, which converts you back to the square root of the spread. I find it hard to believe that a researcher in math/science has his potential limited, just because he forgot to use the arctan function.

I'm interested in reading the other chapters in this book, but I'm still skeptical. I suspect that it will be a re-hash of my education in math, with the same laws, conclusions, applications, only with new notation.

[Felix]

Going from a list of equations to a matrix is different. You can do things, like take the inverse of a matrix, that has no counterpart in individual scalar equations. I would call this a major breakthrough.

That's what I hope for rational trigonometry. Of course, I will only know once I read the book.

[woolcutt]

It would appear that the benefits of the new system are two-fold.

Doing away with the Trigonometric functions allows students to solve equivalent problems and eliminates the pedagogical difficulty of explaining what the trigonometric functions are. This not a breakthrough as it stands, but a nice refinement of the system.

The interesting application to higher mathematics (which DOES have the potential to influence new research) is that since the fundamental tools of Rational Trigonometry are polynomials instead of infinite power series, we can use them to analyze the geometry of different mathematical objects besides the real number plane. Polynomials are fundamental to all field structures, but power series may be more meaningful in one field than another (i.e. may have a limit in one field but not another).

Of course I haven't had a chance to read this part of the book... So I don't know exactly how far the author intends to push this analogy.

[tnunamak]

So what about the places in math where you still need sin and cos, outside of trig?

For example... e ^ix = sin(x) + cos(x)

How do you represent that without sin and cos?

[Haud Rex]

There's more to this story than pragmatic concerns about ease of calculation. This is a partial restatement of geometry. For example, the author asserts that quadrance has primacy over distance and that the spread relation has primacy over the trigonometric ratios. Indeed, to define spread he uses only the concepts of point, line and perpendicular/parallel. Spread can thus be discovered by examining only the two lines and no other entities. Of course, this can be done with "classical" trig, but only by "skipping ahead".

[Unconquered]

So what about the places in math where you still need sin and cos, outside of trig?

For example... e ^ix = sin(x) + cos(x)

How do you represent that without sin and cos?

e^ix = cos(x) + i * sin(x). You missed the i multiplier on sin(x).

All of the conventional functions can be represented by various power series. The one usually taught at an elementary level are Taylor series, but there are more sophisticated methods such as Chebyshev polynomials. When you punch a number into your calculator or a computer program to compute sin(x), cos(x), e^x, etc., the computer is simply using a power series which computes that function to a certain level of precision.

e^x is one of the simplest to represent in power series form:

e^x =(approx) 1+x+x^2/2! + ... = sum(x^n/n!, n=0...N)

This page has more of them, including for sin and cos.

Note that you should be able to see how the power series representation of (cos(x) + i sin(x)) should termwise equal e^ix, if you substitute (ix) for (x) in the series expansion for e^x.

[woolcutt]

So what about the places in math where you still need sin and cos, outside of trig?

For example... e ^ix = sin(x) + cos(x)

How do you represent that without sin and cos?

You don't have to represent it without sin and cos (but you can!). Its not like this new book is going to destroy any old relations or formulae--it will only aid in the development of new mathematics. If you learned your sines and cosines fantastic! I have a strong feeling that when we wake up tomorrow, the old formulae will still check out. But there is nothing mystical or necessary about scratch marks on paper--only the relations they represent (necessary, NOT mystical). To be bound by tradition is, well, archaic.

Edited September 30, 2005 by woolcutt

[peoplater]

I looked at the first chapter and it does not appear any simpler than regular trigonometry. It replaces 2 undefined terms for two other undefined terms. I do not see how this leads to any simplifications. The amount of work involved in getting answers from data is still the same, and in one example he got two answers and decreed that the first one was the obvious choice. His explanation did not hint at why the first one was the obvious choice. If anyone here buys it and actually goes through with it and learns the material then please post a review for the rest of us. As it stands I would not buy this book.

Edited November 13, 2005 by peoplater

[Hal]

Explaining e ^ix = sin(x) + icos(x) in terms of powerseries rather than angles would be really really silly since you'd miss the whole point of thinking in terms of polar coordinates. The conceptual idea of the x in e^ix being the angle of rotation on the Argand diagram is too powerful to do without, and I'm not sure how you'd justify it from a purely powerseries point of view.

Edited November 13, 2005 by Hal

[y_feldblum]

In vector notation, which I prefer:

x y vectors, xy := g(x y) is the metric

Quadrance(x) := xx is the norm-squared or the interval from special relativity

Spread(x y) := 1 - (xy xy)/(xx yy) is an invariant vector comparison independent of the intervals of the vectors

Note

Cross(x y) := (xy xy)/(xx yy) is another invariant vector comparison independent of the intervals of the vectors

Cross(x y) + Spread(x y) = 1

Cross may be slightly simpler to use than spread, or slightly more complex, depending on the circumstance.

"Quadrance", "spread", and "cross" are all terms that Wildberger came up with.

[Quantum Mechanic]

Hello all. It's been a while since I've posted here, but I just came across this thread.

I'm currently presenting a 4-part introduction to Rational Trig for my senior level Seminar Course in pure mathematicis.

For those even a little interested in this stuff, let me tell you it has the potential to completely revolutionize the way we think about geometry. For those that have dabbled in the higher level maths (in particular geometry), they have no doubt come into the grandiose question of axiomatics, initially proposed by Hilbert. Wildberger makes some very controversial claims about these axiomatics, but it is my belief that he has potentially developed the framework to settle this debate once and for all.

Specifically, I am interested in the application of this so-called 'Universal' geometry to projective geometry and field theory. It really is a pleasure to learn about, and I highly recommend it!

I'm even thinking of a Master's thesis proposal related to the subject.

Best regards,

QM

[xavier]

For those even a little interested in this stuff, let me tell you it has the potential to completely revolutionize the way we think about geometry.

Could you explain this a bit more? Is it simply because the various trig functions have been reduced to a single entity that is more fundamental?

Edited December 4, 2005 by xavier

[Quantum Mechanic]

Could you explain this a bit more? Is it simply because the various trig functions have been reduced to a single entity that is more fundamental?

No, it's quite a bit deeper than that. These concepts of spread and quadrance aren't terribly interesting in normal Euclidean space. But when you apply them to completely arbitrary fields and vector spaces, some very powerful results follow. This generalization is called "universal geometry" and is the main subject of Wildberger's book. The first chapter is merely an introduction to rational trig, and is not to be understood as the point of the book.

[xavier]

But when you apply them to completely arbitrary fields and vector spaces, some very powerful results follow.

Sounds more and more interesting than I originally thought. I'm familiar with vector spaces, but not differential geometry or anything like that. Any book/class recommendations before I read Wildberger's book?

Edited December 6, 2005 by xavier

[dondigitalia]

Very interesting. I wonder, however, of what use this would be for those of us who have already mastered "classical trigonometry."

I just read the first chapter, and am definitely putting the book on my list of "things I want." I've always found "classical trigonometry" to be cumbersome. I would say I've mastered it, i.e. I don't have any real problems working with it other than a personal aversion. I cringe whenever I see a trigonometric function. I just plain don't like trig.

This system, however, seems far simpler. I'm interested to read more about these "powerful results" Quantum Mechanic mentions. At this point, I'm willing to accept the idea that classical vs. rational trig may be an issue of personal preference, but if it does open the doorway to some truly new stuff, then it could be an immense achievement.

[Quantum Mechanic]

Sounds more and more interesting than I originally thought. I'm familiar with vector spaces, but not differential geometry or anything like that. Any book/class recommendations before I read Wildberger's book?

According to Wildberger, one doesn't need more than high school math to understand the rest of his book. But keep in mind mathematicians have a terrible habit of thinking they are clearer at explaining things than they actually are.

I recommend strong algebraic skills, and at least a basic understanding in mathematical logic and proof writing. Along with a little faith (ie, accepting a proof as true when you don't really understand it), you should be able to get through no problem. And some chapters are quite unrelated to others (for instance, there's a chapter on calculating things like volumes in complex physical systems, and another devoted to projective goemetry), so one is able to skip over much of the material the first time through.

Well worth the rather pricey $80.

Regards

QM