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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?"
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.
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.
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
Spread is NOT proportionally equal to an angle. 30 degrees is 1/4 and 60 degrees is 3/4
spread is the square of the sine of an angle.
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`
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.
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 think it kind of depends on what you do for a living. If you work at Wal-Mart, you probably don't need it. If you are and Physicist or Engineer (like my wife and I), you probably do. And thats hard to know in High School
Everyone complains how trig is not useful, but perhaps its useful because it is hard and is one of the few things left in schools today that actually mentally challenges students.
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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I take it you taught math in elementary school (K to 5th) then, as your point is completely wrong. For a physicist or computer scientist, the principles of trigonometry are invaluable. All those games you might play. All those electronic devices (cell phones, tv, etc) you use on a daily basis. Much of the theory used to devise how they could possibly work was done with *gasp* trigonometry and to a greater extent, calculus.
Simply because you choose a profession does not use it, does not mean it doesn't have value.
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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Well you certainly aren't working in animation or writing simulations, or writing AI programs, or code for robots, or doing any kind of graphics conversion, or audio programming or making any kind of games with your "programming"(I'll stop here, but I could go on and on). I would guess with your attitude toward math you're really not a programmer, you probably just tie stuff together that other people have written with your own code or scripts. You use libraries rather than write them. Not trying to insult what you do, but there's a lot more to programming than that, and it does take math.
And you are wrong about algorithms. Algorithms is math. No ifs, ands or buts about it.
No, Thursday's out. How about never - is never good for you?
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.
No. Absolutely not. People need a basic understanding of this stuff, because it is sop important to the things which make modern society work. People need to know enough to be critical of obviously dumb assumptions, at the very least. You need to know that your contractor is screwing you over by quoting you for more than twice the square-footage than you actually have; and it's amazing how many people can't even handle Pythagoras.
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.
-- Waht? Tehr's a preveiw buottn?
But its pretty easy to know that you only have a slightly greater chance of being a physicist than you do of being a profesional basketball player. You dont see us trying to train our kids to be basketb... Oh shit. Yep were fucked. They will end up at Wal-Mart.
Luckily its a great store for Physici...
Do you need a cart sir?
I think you underestimate just how much I just dont care.
Those black boxes are the reason that while I was relatively good at math, I sucked at trig, which screwed me when I got to calculus. I had always thought that when learning math, I could follow the steps to a solution which lead to an understanding as to why it worked. Black boxes, as you described it, do not do this.
Someone hates these cans.
I wasn't taught trig functions as black boxes. We learned right from the start that they're the ratios of the various sides. Once you understand that, it's easy to know which function to use to find which side or angle, and why. Identities were just s easy: they're just formulas that don't depend on the angle; they're right for any angle, so you can use them to simplify equations. Trig was fun, and I was good at it, but that might be because my teacher understood how to explain it instead of simply demanding rote memorization.
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