View Full Version : Rational Trigonometry

Matthew

2005-Sep-18, 07:59 AM

Imagine doing trigonometry without sine, or cosine, or even tangents. Hard? Well an Australian professor has come up with rational trigonometry which uses pure algebra to create a more accurate analysis.

Click here (http://web.maths.unsw.edu.au.nyud.net:8090/~norman/book.htm) for the link to the book.

This looks interesting. Though it seems extrmely revolutionary (to me), I wonder how this is going to effect trigonometry. Is this going to become the defacto?

Taks

2005-Sep-19, 04:33 AM

given that sin, cos, etc. are just ratios of lengths of a simple geometric figure (right triangle), i'm not surprised such a concept is possible. it would be an interesting read, and i'm also surprised it has not been done before. good link (you aren't the author, are you? :) ).

taks

hhEb09'1

2005-Sep-19, 04:41 AM

Imagine doing trigonometry without sine, or cosine, or even tangents. Hard? Well an Australian professor has come up with rational trigonometry which uses pure algebra to create a more accurate analysis.

Click here (http://web.maths.unsw.edu.au.nyud.net:8090/~norman/book.htm) for the link to the book.That link says that the book is not yet available. I haven't tried the publisher link, to see if it has come out early:

Divine Proportions: Rational Trigonometry to Universal Geometry will hopefully be published this year by Wild Egg, an innovative new publishing company specializing in ground breaking mathematical titles. The first edition will be hardcover, 320 pages on 100% acid-free paper, with section sewn binding. It will be printed by BPA, one of Australia's foremost quality printers. The cover design is by Alex Snellgrove. It will be available online at http://wildegg.com at the end of September 2005.

PS: I did click on the link. The book will come out Sep. 20th. And it is the publisher's first book. And the publisher's director is the author of the book.

Matthew

2005-Sep-20, 06:34 AM

(you aren't the author, are you? :) ).

taks

It would be nice to be the author. Alas I am not. :doh:

There is a chapter excerpt available (Ch.1 natuarally). Click here (PDF) (http://web.maths.unsw.edu.au.nyud.net:8090/~norman/papers/Chapter1.pdf) for the excerpt.

montebianco

2005-Sep-20, 02:35 PM

I took a look. My impressions:

Certainly this is not a research contribution. It may be a good alternate way to teach trigonometry, particularly to people who will not go on to more sophisticated maths; I'll leave it to teachers to judge whether it really is. It does have a bit of a "Trigonometry for Dummies" feel about it.

The basic idea is to change variables. Length is replaced by quadrance, which is squared length, and angle is replaced by spread, which is the squared sine of the angle. It looks like much of the book will be about developing the laws of geometry and trigonometry in terms of these two measures instead of the more traditional angle and length. The motiviation appears to be that the author feels angle is a very unnatural measure (I disagree - see below), and that the relations between quadrance and spread can be expressed in terms of simple arithmetic operations, whereas those between angle and length cannot.

While certainly some problems become easier when expressed in terms of quadrance and spread, others become harder. Consider the movement of the moon about the earth. Assume you are standing on a point within the moon's plane of orbit around the earth, and that the moon's orbit is circular. If the moon has just popped above the horizon, how long will it take to reach a certain point above the horizon? If you measure this distance by angle, all you have to do is take the angle, divide it by the moon's angular speed, and that's the amount of time it takes. If you measure this distance by spread (as per the author of this book), you must subtract double the spread from one, find the inverse cosine of the result, and divide by twice the moon's angular speed. Which is simpler?

So this is not a development of new results, it is a presentation of known results in a different way. The alternate presentation will make some things easier. But it will make other things harder. Anyone who is comfortable with advance mathematics ought to have no problem finding the change of variables that simplies a particular problem; for those not so comfortable, maybe this book would be helpful.

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