Slashdot Log In
Trigonometry Redefined without Sines And Cosines
Posted by
CowboyNeal
on Sat Sep 17, 2005 08:32 AM
from the numbers-and-stuff dept.
from the numbers-and-stuff dept.
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?"
This discussion has been archived.
No new comments can be posted.
The Fine Print: The following comments are owned by whoever posted them. We are not responsible for them in any way.
Full
Abbreviated
Hidden
Loading... please wait.
No sines and cosines? (Score:5, Funny)
Re:No sines and cosines? (Score:5, Funny)
Parent
Wonderful! (Score:5, Insightful)
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.
Re:Don't worry... (Score:5, Insightful)
Parent
Re:Don't worry... (Score:5, Insightful)
Simply because you choose a profession does not use it, does not mean it doesn't have value.
Parent
Re:Don't worry... (Score:5, Insightful)
As an aside: I'm always amazed how many people who do sciences and other technical stuff are always interested in many things, like music, politics, aesthetics, social structure...but hardly any political science or sociology student has even a passing interest in the sciences. I'm starting to believe that the latter are just to stupid to realise how much of an impact those things have on their life.
Parent
Re:Don't worry... (Score:5, Informative)
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.
Parent
Not just physicists or engineers use trig.... (Score:5, Insightful)
Parent
Figures. (Score:5, Funny)
UNSW .. not South Wales (Score:5, Informative)
SOHCAHTOA and abstract survery results (Score:5, Insightful)
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.
Faster calculations ?? (Score:5, Interesting)
in raytracers and 3D engines by using integer numbers.
Yes, for some badly written code (Score:5, Interesting)
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.
Parent
Great for eighth grade, but ... (Score:5, Insightful)
I don't see how this is "easier" (Score:5, Insightful)
Most of you missing the point. (Score:5, Insightful)
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.
Re:Now ... (Score:5, Interesting)
Parent
This reminds me of a test in grad school (Score:5, Interesting)
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.
Parent
Re:Now ... (Score:5, Interesting)
Parent
Re:huh? (Score:5, Insightful)
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)?
Parent
Re:Redefinition? (Score:5, Insightful)
spread is the square of the sine of an angle.
Parent
Re:Wow (Score:5, Insightful)
If you want to be the kind of engineer that implements other engineer's ideas then, by all means, blow off your math classes. But if you want to be someone who your peers turn to when they need help, do yourself a favor and learn the math.
All of the engineering sciences are founded on math (this is espescially true of computer science). If you can out code your instructors, that means you can probably out math them too. What you are interpreting as an inability to memorize functions, is probably really just disinterest.
This disinterest may stem from a feeling that what you are studying has little utility, it may stem from a personal dislike of an instructor, it may stem from the notion that math geeks are all squares and smell funny.
Whatever the reason, you need to get past it. A thorough understanding of the math behind engineering will make your life MUCH easier on down the road.
Parent
Re:Wow (Score:5, Insightful)
And you are wrong about algorithms. Algorithms is math. No ifs, ands or buts about it.
Parent
Re:Wow (Score:5, Interesting)
But first, this small reminder:
sin x (vertical component)
cos x (horizontal component)
tan x = sin x over cos x
sec x = 1 over cos x
csc x = 1 over sin x
cot x = cos x over sin x
-> sctsct
Now we substitute these trig functions with simple symbols:
I = sin x (vertical component)
II = cos x (horizontal component)
III = tan x = sin x over cos x
IV = sec x = 1 over cos x
V = csc x = 1 over sin x
IV = cot x = cos x over sin x
Think of them as one, two, three, four, five, and six. Now it boils down to remembering simple combinations of numbers:
integral{ I } = -II
integral{ II } = I
integral{ III } = ln | III + V |
integral{ IV } = ln | IV + VI |
integral{ V } = ln | III |
integral{ VI } = ln | I |
Once you write down more of these combinations, you'll discover patterns in it and from there on it should be easier than ever to remember trig integrals. And when you know most of the trig integrals, you will know most of the trig derivatives, too!
If anyone is interested in some more documentation on this, then by all means contact me. I am in the process of writing these things down. They should be available some time next week on my homepage.
Parent
Re:Units? (Score:5, Insightful)
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.
Parent