Unexpectedly Intriguing!
September 21, 2005

Tired of trigonometry? Having trouble keeping track of all those sines, cosines, tangents, cotangents, et cetera, so on and so forth?

Norman Wildberger of the University of New South Wales in Sydney, Australia may have done away with all of those in rewriting the Euclidean geometry that we've all come to know and dread in high school math classes. Here's the scoop from the UNSW's news release announcing the publication of his book Divine Proportions: Rational Trigonometry to Universal Geometry, now available from Wild Egg Books:

This text introduces a new and simplified approach to trigonometry and a major restructuring of Euclidean geometry. It replaces cos, sin, tan and all those other transcendental trig functions with rational functions and elementary arithmetic. It develops a complete theory of planar Euclidean geometry over a general field without any reliance on 'axioms'. And it shows how to apply this new theory to a wide range of practical problems from engineering, physics, surveying and calculus.

The news release spells out the potential impact for math students:

Trigonometry has for centuries been difficult for students to learn. Part of the reason is that the foundations of the subject are problematic, relying on vague pictorial definitions of the key concepts of angles and trigonometric functions. That means students rarely come to grips with the logical structure of the theory. Rational trigonometry is much simpler, relying on the fundamental quadratic notions of quadrance and spread, and with basic laws that can be derived completely rigorously using only elementary algebra.

As it happens, quadrance is equal to the square of the distance between two points and spread is equal to the square of the sine of an angle. Here are some extracts from the text that provide more detail of the scope of Wildberger's rewriting of the rules of trigonometry:

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