### Algebra Eats Trignometry

If trigonometry and algebra got in a fight, who would win?

Ever since the ancient Greeks codified these disciplines, the answer to this question has been unknown. Now it would seem we have an answer. And things don't look good for old trig.

This story at physorg.com caught my eye because how often do you see an incredible breakthrough in a subject you were taught in JUNIOR HIGH SCHOOL? You know when that happens, something major has gone down.

But if the claims being made in a new book are true, trigonometry classes the world over may be looking at dark days. An Australian mathematician has re-analyzed trigonometry starting from the most basic premises, and has discovered that we've been working with a flawed framework all this time. Trig doesn't need sin, cos, tan or any of those other weird little keys on your calculator. Nope, nor does it have to be as hard as it is. It only has to be as hard as algebra. (I always found trig and geometry a walk in the park compared to algebra, so this is not personally the greatest news.)

As an avid - though amateur - mathophobe, I'm in no position to evaluate the following statements for correctness, but the new book by NJ Wildberger claims:

The new form of trigonometry developed here is called rational trigonometry, to distinguish it from classical trigonometry... An essential point of rational trigonometry is that quadrance and spread, not distance and angle, are the right concepts for metrical geometry (i.e. a geometry in which measurement is involved). Quadrance and spread are quadratic quantities, while distance and angle are almost, but not quite, linear ones. The quadratic view is more general and powerful...

When this insight is put firmly into practice, a new foundation for mathematics and mathematics education arises which simplifies Euclidean and non-Euclidean geometries, changes our understanding of algebraic geometry, and often simplifies difficult practical problems.... although the actual definitions used in this text are independent of distance, angle and the trigonometric functions. They are ultimately very simple, based on finite arithmetic and algebra as taught in schools.

New laws now replace the Cosine law, the Sine law, and the dozens of other trigonometric formulas that often cause students difficulty.... The derivation of these rules from first principles is straightforward, involving some moderate skill with basic algebra. The usual trigonometric functions, such as cos θ and sin θ, play no role at all.

Rational trigonometry deals with many practical problems in an easier and more elegant fashion than classical trigonometry, and often ends up with answers that are demonstrably more accurate. In fact rational trigonometry is so elementary that almost all calculations may be done by hand. Tables or calculators are not necessary, although the latter certainly speed up computations. It is a shame that this theory was not discovered earlier, since accurate tables were for many centuries not widely available.

The mind (at least my mind) reels. I wonder what else we've gotten wrong? I'm hoping some English professor will check in soon reporting the long-theorized existence of the verbless sentence. (If so, then I happy.)

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