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18 Sep 2005 - 02:00

## weird new trig

Two friends, completely independently, sent me links tonight about a new book that redefines trigonometry without sines and cosines and pesky irrational numbers. Slashdot says it could be a completely new and much simpler formulation of trigonometry, and the beginning of a new era in math.

There's a pdf of the first chapter available online; I read far enough to see the definitions of his two main concepts, quadrance and spread (replacements for the concepts of distance and angle). They are (respectively) the square of the distances between two points, and the square of sine of the acute angle between them. That's as far as I've gotten.

When you say your prayers tonight, pray that the educational establishment doesn't get wind of this.

### (request from Catherine)

Carolyn and I don't normally barge into each other's posts, and I'm only doing so because I want to make sure people see this.

I've never taken trigonometry, and will need to do so before I tackle calculus.

If folks have text recommendations, I'd appreciate hearing them. Thanks

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-- CarolynJohnston - 18 Sep 2005

So this is a bad idea???

I've never taken trig, and I'll need to take it to advance to calculus--so I need advice!

-- CatherineJohnson - 18 Sep 2005

It looks like Mary Dolciani includes trigonometry in her Algebra 2 book.

-- CatherineJohnson - 18 Sep 2005

"When you say your prayers tonight, pray that the educational establishment doesn't get wind of this."

Why? What's wrong with it?

-- KtmGuest - 18 Sep 2005

What's wrong with it?

-- the fear is presumably
that hundreds of years worth of published work
(and an army of well-prepared users of the "old" trig)
will be jettisoned in favor of the "21st century"
approach: it's not that new ideas are threats
(intrinsically), but rather that old ideas are threatend
(in fact).

if the new way is better, it'll come out
eventually without propaganda -- we'll just use it
because it's easier: nobody needed a grant
to make the "floor" and "ceiling" notations
replace the once-common (and still, what do you know,
universal in textbooks) "bracket" notation
(i refer of course to [x] := "the greatest integer
less than or equal to x"; i can't draw the "floor"
but it looks vaguely like |_ x _| ...).

-- VlorbikDotCom - 18 Sep 2005

Actually, there is a real mathematical problem with using "quadrance" (squared distance between points) as a measure of distance that this guy (with his background) should know. Maybe he does know it and deals with it somehow, I haven't read that far ( will say though that I doubt it). But here it is, anyway, for those who care.

Serious math digression

Any measurement of distance between points has to have three properties. If these three properties don't hold, then our intuition falls apart. The weirdest hyperbolic-geometry notions of distance still satisfy these 3 criteria; they are very basic.

1. (reflectivity) the distance from a point to itself should be zero.
2. (symmetry) the distance from A to B should be the same as the distance from B to A.

Now here's the one that bears a little thinking:

3. the distance from A to C should not be any greater than the distance from A to B plus the distance from B to C (the triangle inequality).

The 'normal' distance formula that has been used in trig forever -- |A-B| -- involves a square root. This author is trying to get rid of the square root sign, plain and simple, by using the 'normal distance squared' as his notion of distance. The problem with doing that is that distance squared doesn't satisfy property 3, the triangle inequality.

An example: draw a triangle with vertices A, B and C with the short sides AB and AC both equal to 3, and the 'long side' BC equal to 5. The squared distances are 9, 9 and 25 respectively, and 9+9 is not greater than 25. So the triangle property is not satisfied.

So 'quadrance' is a crummy notion of distance.

-- CarolynJohnston - 18 Sep 2005

I am entering this explanation in the Book-style index NOW.

-- CatherineJohnson - 18 Sep 2005

No, you don't need to take Trig before taking Calculus. They're completely unrelated. You can skip Trig entirely if you want to.

There's a reason why Trig is required before Calculus. Trig, among other things, gives you some down-to-earth examples of functions which are not simple algebraic formulas. Most students don't realize that that's what they've been given, but they have.

There is danger here. Those teachers who want to get to Calculus quickly or who are thinking that Calculus is the more important subject will teach Trig completely from the function-theoretic point of view. While that is an important part of Trig, it is a beautiful subject in its own right which can be taught completely without reference to functions.

Unlike Calculus, I've used Trig many times in engineering applications.

-- BernieJohnston - 19 Sep 2005

"if the new way is better, it'll come out eventually without propaganda" -- VlorbikDotCom?

Possibly, but that doesn't justify propaganda against the new way IMHO.

"Actually, there is a real mathematical problem with using "quadrance" (squared distance between points) as a measure of distance that this guy (with his background) should know." -- CarolynJohnston?

He doesn't call it a "measure of distance", or a "notion of distance". Quadrance is obviously not a distance function. Your "problem" does not arise in his presentation.

-- KtmGuest - 19 Sep 2005

Possibly, but that doesn't justify propaganda against the new way IMHO.

Skepticism is warranted. I see no real need for this 'new trigonometry formalism', and if it's quackery and gets picked up by the mainstream educational establishment, it has the potential to do a lot of harm.

He doesn't call it a "measure of distance", or a "notion of distance". Quadrance is obviously not a distance function.

Then what is it? Explain, please.

-- CarolynJohnston - 20 Sep 2005

From the write-up for the book;

"...with the key concepts of quadrance and spread replacing distance and angle."

It sounds like quadrance is distance. If it is the same as a^2 + b^2 (without the square root), then it sounds familiar. People doing computer graphics often need to compare distances, which can be done more efficiently by not taking the square root. This is nothing new.

I read the book write-up(hype), but was left with the question:

What's the problem?

Also, my impression is that this is really not something new - maybe its formalism, but it sounds much like what people doing computer graphics or geometric modeling do to avoid calculating trigonometric functions. Everything in computer graphics has to be optimized and I have 5 books on my shelf (Graphics Gems I-V, 3000 or so pages) that go into these short-cuts in detail. Just like there are explicit, implicit, and parametric equations, different approaches have different trade-offs.

Trig functions really aren't difficult, so what's the problem?

-- SteveH - 20 Sep 2005

Wow, Steve, you have all the Graphics Gems books?

That's really heavy, man.

-- CarolynJohnston - 20 Sep 2005

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