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" has an initial capital for a reason

[Bewbewbew]

The "New" refers to South Wales, rather than the uni. Might wanna read a bit closer next time.

UNSW .. not South Wales

[OzPeter]

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

Re:Don't worry...

[RAMMS+EIN]

``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.

Re:Don't worry...

[mdwh2]

Yes, but simply because there are professions which use it doesn't mean it has to be taught to everyone. I agree that school has to provide a certain common base, but I don't really think trigonometry belongs in there. A class teaching how to automate common computing tasks would have been more useful to more people, I imagine.

True - part of the problem is that (at least here in the UK, I don't know how the US works) Maths is compulsory (until 16), along with English and Science. With optional subjects, you can presume people taking them may want to work in those areas, so it is important to teach accordingly.

With compulsory subjects, I agree that the compulsory bits should be only those which everyone needs in everyday life. In maths, I'd say that things like understanding statistics are more important than trigonometry (consider how often statistics are given in the news and so on, and how many people misunderstand them).

So ideally you'd have "core maths/english/science" then a separate set of classes instead for those who choose to take all of that subject.

But here's the problem: I suspect that most schools won't have the resources to teach two sets of those subjects; it may be simpler just to do things as they are now. (Plus as someone else points out, you may not know what you want to do when you are 14 - or they may not realise just how many jobs may require an application of maths - so it's good to teach it anyway)

Re:This reminds me of a test in grad school

[HuguesT]

Hi,

If you have some experience in solving integrals of that sort, the substution x = t is pretty standard.

In this case letting x = tan t is very productive. Working through the algebra one finds that (TeX notation)

Just remembering $\tan = \sin/\cos$ and $\cos^2 t + sin^2 t = 1$, on can work out the following:

We have $1/(1+x^2 = 1/(1+\tan^2 t = \cos^2 t$

Also $dx = 1/cos^2 t dt$, therefore

\[
\int_0^a \frac{1}{1+x^2} = \int_0^{\tan^{-1} a} 1 dt = \tan^{-1} a
\]

So you don't have to remember the form of the integral but you do have to remember how to do a variable substitution in an integral, though, as well as some classical tricks.

Re:Why are there 360 degrees?

[damiam]

Because 360 is divisible by a lot of numbers, making it easier to work with than say, 359. It's really pretty arbitrary, which is why mathematicians use radians for most serious purposes.

Re:SOHCAHTOA and abstract survery results

[controlguy]

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:
(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, ... use, I believe, Taylor series expansions.

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.

Re:Well, not exactly

[moonbender]

Look again. The circle is there, but only to show a similarity to the previous examples. The radius of the circle is irrelevant, and he only uses one point on the circle - in other words, the circle is totally unused and you could use any point on any of the lines.

Very nice. Makes sense to a game programmer

[Animats]

Most of the relationships Wildberger explains are well known to those of us who write physics engines, or the more geometrical parts of game engines. Trig functions are too expensive to use in inner loops, and their corner cases are annoying. If at all possible, everything is done with linear operations on vectors, matrix multiplies, and quaternions. These operations not only go fast, they parallelize; all 16 multiplies of a 4x4 matrix multiply can be done simultaneously, and every modern graphics card has the 16 multipliers necessary to do that.

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.

Re:Not just physicists or engineers use trig....

[mrbnsn]

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".

Trig is not hard, it's just taught REALLY badly

[hagbard5235]

Trig should be about a 1 to 2 week topic in school. If instead of having students memorize endless identities you simply teach them 1 (Eulers equation) and show them how to easily derive the rest then it becomes pretty trivial.

Euler's equation:

e^(i*x) = cos(x) + i*sin(x)

Need a double angle formula? No problem.

e^(i*2*x) = cos(2*x) + i*sin(2*x)
e^(i*2*x) = (e^(i*x))*(e^(i*x))
                    = (cos(x) + i*sin(x))*(cos(x) + i*sin(x))
                    = (cos(x))^2 - (sin(x))^2 + i*2*cos(x)*sin(x)

So you can clearly see that

cos(2*x) = (cos(x))^2 - (sin(x))^2
sin(2*x) = 2*sin(x)*cos(x)

All of the trig identities fall out as simply through simple algebraic games when you start with Euler's equation. It also makes looking at the calculus of trig pretty trivial, greatly simplifies the study of waves, etc. I cannot for the life of me understand why everyone doesn't teach it this way.

Re:Now ...

[omega_cubed]

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.

Or you could just sketch the functions

[Dioscorea]

and figure out the derivatives that way.

Re:Now ...

[bennigoetz]

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!

Re:No sines and cosines?

[emandres]

This is an interesting enough concept, but the math involved with it would require a bit more algebra than I knew when I learned the trig basics. Also, this doesn't seem like it would have much practical application in calculus. Anyone who's ever taken calculus beyond just the basics can tell you that it is a pain in the butt integrating and deriving rational functions. Unless his replacements for sine and cosine, etc, are all related in the way they are in classical trig, it would be a nightmare trying to do the simplest of integrations, like proving the sine is the antiderivative of cosine.

Parent is factually incorrect

[Jesus 2.0]

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.

Re:Or you could just sketch the functions

[Anonymous Coward]

If only I had mod points I would have modded you up.

Everyone else in this entire article and all the postings seem to be suggesting memory tricks to avoid any understanding of the trig.

Your approach is the one I use, and it really gives you a feel for what the functions actually represent and the meaning behind them. Thank you for giving people that advice.