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

[Graymatter]

I taught myself math, so I like it. But you have to spend a lot of time looking for the right books.

[Moonman62]

I am a math major in case you two couldn't tell. Math is wonderful, but I think for most people it takes a really great teacher to get past the silly words and funny symbols.

[stylin19a]

oh great...now what do i do with all these x's & y's i've hoarding ?

[elfman2]

Carl Graves teaches trig?

[punster]

I did hear about a mathematician that got pi eyed.

[InterceptPoint]

Take a look at Chapter 1 of the book. What I see is a complex replacement of sin/cos/tan with what amounts to the Pythagorean Theorem. You have to give up the concept of angle and distance and replace it with "quadrance" (distance squared) and "spread" (don't ask) and you still (obviously) have to be able to do square roots to get answers. So he makes you give up the intuitive idea of distance and angle just so you don't have to push the SIN, COS or TAN buttons on your calculator. Instead you have to push the SQR button several times.

This is progress?

[Publius6961]

UC

[Publius6961]

I taught myself math, so I like it. But you have to spend a lot of time looking for the right books.

Hogben...

[mtbopfuyn]

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

And yet he's left me still scratching my head since he failed to explain the five main rules.

[cloud8]

> New math was great (for those continuing in math and science).

SMSG math was utterly confusing.

> Ever try casting out 9's?

Check digits. That's not math...it's magic :)

[Texas Eagle]

Mathematics students have cause to celebrate.

Wow. That oughta be a party.

[Paladin2]

Then why does casting out 7's work in octal (base 8 for those who took new math)?

[Final Authority]

for later read

[Paladin2]

Move to polar/spherical coordinates.

[RBroadfoot]

Back when I was a boy,
We didn't have calculators,
or computers ...

We had slide rules,
And trig function tables,
With print so small that you needed a magnifying glass,
AND WE LIKED IT!

[Who dat?]

No s~t? What’s his name? Where did he teach?

His name was Joe Berland and he taught at Chabot College.

For some reason I wanted to take a class but it required you either having taken their trig class or go through some sort of trig test as a prerequisite.

I had taken a cheesy trig/intro trig type class years before in high school so I didn’t feel confident that I could pass a trig test so I decided to go ahead and take it there. The one that fit my schedule was the one he taught.

Just dumb luck on my part but he was an excellent instructor. Just about every semester the new crew at the junior college paper would re-discover that he was (or had been?) a professional musician.

http://www.miriamcutler.com/MIRarticles_fourteen.htm

(Don’t know when that was written but evidently he can play keyboards, accordion and clarinet – evidently at some point they reunited to do some sort of “play” (second paragraph from end))

[null and void]

Ever try casting out 9's?

He's out to get you swervie...

[JasonC]

What horsefeathers. You need to understand sines etc to understand any periodic phenomenon, it is hardly just about triangles. Then you need it to understand families of orthogonal functions, expansions, frequency spectra, light, fields, yada yada yada. This is just dumbing math down for nabobs who'll never learn enough of it to tie their shoes.

[fzx12345]

Hooray for new math,
New-hoo-hoo-math,
It won't do you a bit of good to read math.
It's so simple,
So very simple,
That only a child can do it!

---Tom Lehrer

[KC_for_Freedom]

I suspect those rules are "obvious to the casual observer".