New trigonometry is a sign of the time

physorg.com ^ | September 16, 2005

[cloud8]

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

[megatherium]

My favorite trig mnemonic is "Oscar Had A Heap Of Apples" (opposite/hypotenuse, adjacent/hypotenuse, opposite/adjacent, which codes for sine, cosine and tangent respectively).

[metmom]

"Once you understand it's just ratios, it takes all the "mystery" out of it."

You are so right.
I never took trig in high school and managed to get through calc withou it in college, with the help of an excellent prof. It wasn't until I homeschooled my kids and had to teach them trig out of a text book that I learned it myself and realized it's simplicity. It's just fractions! I couldn't believe it.

[tarheelswamprat]

The only tune I knew by Oingo Boingo was one that ended up on the Fast Times at Ridgemont High soundtrack in the 80’s (“Goodbye, goodbye” or something like that) .

How about their appearance in Rodney Daingerfield's movie "Back To School". I believe they did a song called "Dead Man's Party".

[SC Swamp Fox]

In Navy/Marine Corps tech schools we taught it:
Sally Can Tell Oscar Has A Hairy Old A##!

Sally Oscar/Has
Can A/Hairy
Tell Old/A##

To this day, I still remember that Tangent = Old A##

[Flashlight]

ok now I get it.

now I'm gonna write a book that replaces sin(x) with exp(ix)-exp(-ix)/2i, cos(x) with exp(ix)+exp(-ix)/2, and tan with sin/cos. exp(x) is easily calculated using the Taylor series 1+x+x**2/2!+x**3/3!...

but seriously folks, somebody should shoot that stupid sohcahtoa stuff.

sin, cos, and tan aren't three functions, they are essentially only two - sin and cos are the same function, they just start at a different place (it's like making a distinction between the split end and the flanker in football - yes they have to line up correctly, but essentially they're the same thing: they're both wide recievers).

can't explain everything here of course, but it's easy to intuitively grasp the concepts of sin and cosine by knowing that cosine is the "projection" onto a surface and sin is the "height" (multiplied by the length of the segment that's being "projected") - this is a whole lot easier to see and understand than to explain in writing here.

the point is, to really "understand" this stuff, the student should just be able to look (or visualize) at a problem like "if a stick of length 4 is sticking out of the ground at a 34 degree angle, what's the distance from the top of the stick to the ground?" and just "know" that, well sin is the height and the stick is length 4, so the answer is 4*sin(34 degrees).
anyone who resorts to formulas like hypotenuse/ajacent or whatever may end up with the correct answer but they're lacking the essential intuitive knowledge that will serve them as they progress in trig and calc.

[velocityguy]

Ofcourse a great teacher is always good. But I strongly believe Math is one of those subjects that can be easily learned even without a great teacher. The only thing that one needs to have to learn math is the willingness to use some brain ....unlike other subjects.

Just in case you missed my Profile,an fyi.... I have a Ph.D and I have extensively used Probability and real Analysis in my research.

[metmom]

We found the right books to be Saxon Math. It goes all the way through to Calc and Physics. I know there are some out there who don't like it but it got my daughter a 740 on the math section of the SAT test. I loved it because it's so simple in it's presentation and easy to understand. I learned a LOT of math. All that stuff they tried to cram down my throat in high school suddenly made sense. As a matter of fact, so many homeschoolers in my old district used it and had such good results on their standardized tests that the school district adopted the Saxon for several grades worth.

[Graymatter]

Hogben---after my time, but if she's good I can always use another algebra book. My youngest is at that stage now. Thanks for the tip!

[BADROTOFINGER]

This sounds kinda cool. Please add me to your CE ping list, FA, and thanks!...JFK

[skinkinthegrass]

I wonder when this will make it into surveying equipment, if ever...Civil Engineer Ping

I just got used to the "old" rules. :D
..Is My HP Calculator useless now?

[Dick Vomer]

OhMiGod, they killed Pricess SohCahToa!

Sine=Opposite/Hypotenuse

Cosine=Adjacent/Hypotenuse

Tangent=Opposite/Adjacent

She Offed Him ,Candy And Honey, To Offset Acid...

S=O/H sine=Opposite /Hypotenuse

C= A/H cosign=Adjacent/Hypotenuse

T=O/A Tangent = Opposite/Adjacent

.....70's thing... ya'll wouldn't understand..hahahahhaha

[Fzob]

I found this "Why rational trigonometry?"

http://merganser.math.gvsu.edu/david/reed05/projects/halserogers/html/why.html

[BADROTOFINGER]

Yeah, but Oscar had a heap of apples...JFK

[Fierce Allegiance]

I don't really have one, but I guess I will have one now.

Civil Engineers Ping List
Fierce Allegiance
JimWforBush
TheSISUKid

I know I forgot some, but they'll let me know, I'm sure!

[No Truce With Kings]

I have to admit to being skeptical about his claims. A bald assertion such as the one the author is making is not necessarily the naked truth. I will have to get his book and see if it really *is* a breakthrough.

That said, people will have to check his claims before dismissing them. In the late 1700s a sailing captain with little formal education wrote a new approach to spherical trigonometry that allowed badly-educated sailors to understand it and use it for navigation. The man was Nathaniel Bowditch, and his book is *STILL* in print, recently having had its 200th anniversary of continuous publication.

[Moonman62]

Perhaps it is better then to have no teacher at all than to have a bad or mediocre one. Math wasn't natural for me. I had to keep trying over and over again and slogging through some poor instruction and one real jerk. When I finally got a professor who knew his stuff I did very well and enjoyed the subject.

[Towed_Jumper]

"I have bad memories of college freshman Algebra. The prof would chicken scratch the first few variables on the board..."

OK...that would have been frightening but... here's another of my math horror stories from College: I took a "Self Directed" Calculus class. It was a "move at your own pace course, where you'd study on your own then come in on Fridays to sit for the weekly test that covered whatever calculus module you had been studying. That wasn't so bad. The worse part of it is the university had hired Teaching Assistants (TAs), to be available during the week so you could come in and receive some one-on-one tutoring. The only problem was that all of the TAs were CHINESE grad students who's English sucked.

Now, try to imagine my situation: I hate math to begin with. I struggled with Math all through high school. Now I'm in college taking calculus on my own and not understanding very much of it at all. I go into see the TAs who can't speak English, and they proceed to try and "educate" me on some incomprehensible aspect of Calc in broken English...you get the picture.

The good news is that I stuck it out for all 16 weeks that semester and pulled a "B". The bad news is that to this day the sum total of all the calculus I know and understand would comfortably fit inside of a thimble.

Actually, it was our college class that began a student rebellion at the university to make sure the Math Department hired TA's who could speak English. That was many years ago. Both of my kids have gone (or are going to now), the same university and I understand that potential TA's now have to pass the TOEFL exam before being allowed to teach....thank God.

[mwilli20]

Can you please post a link to the Slashdot thread? Thanks.

[Excuse_My_Bellicosity]

What I'd really like to see is this guy's "new spherical coordinate system used in calculus", but I'm not going to pay $80 to find out. I'm betting that this is a bookselling gimmick.

[megatherium]

My assessment:

Wildberger happens to be an "abstract harmonic analyst," which is an area of mathematics devoted to the development of analogues and generalizations to harmonic analysis (what mathematicians call the area of math that is based on trigonometry, and which includes such important mathematical ideas as the Fourier transform or Fourier series). This is my own specialty as a mathematician. Wildberger is extremely good at abstract harmonic analysis. (Much better than me!)

This book, which I have only read the first part of, seems pretty solid to me. It might be oversold -- certainly it will not replace angles and ordinary trigonometric functions, especially in physics and engineering. But based on my initial impression, it is a very pretty piece of work. (Certainly, there are no obvious mistakes; Wildberger at the least is fully competent.) What the book is likely to do is keep a fairly small group of mathematicians rather busy for a few years, as they work out generalizations in Wildberger's framework of modern harmonic analysis. A worthwhile enterprise (exactly what we do in my biz) but again, not terribly likely to change the mathematical landscape from the point-of-view of your average engineer or physicist (and certainly not your average 11th grade trig student).

One detail of the trig system he proposes strikes me as simple yet clever. This is using the square of the distance instead of the distance between two points (his "quadrance"). This reminds me of the crucial insight by Fisher in statistics to use the "variance" instead of the standard deviation. The variance is simply the square of the standard deviation -- but what this accomplishes is that when one adds independent random variables (normally distributed), the variance of the sum is just the sum of the variances. (Standard deviations do not add across that way, you must use square roots.)