Rational Trigonometry

Rational trigonometry is a proposed reformulation of traditional metrical planar (and solid) geometry - which includes trigonometry - by Prof. Norman J. Wildberger in his book Divine Proportions: Rational Trigonometry to Universal Geometry (2005). The book is purposefully critical of traditional mathematics. It eschews transcendental operations entailed in the use of distance (taking square roots) and angle (evaluation of infinite polynomials) calculations central to current methods, in favour of purely algebraic ones. The change is accomplished by replacing ordinary distance with its square ('quadrance') and angular separation of lines by the sine ratio of quadrances in a right triangle ('spread') that, in turn, corresponds to the square of the usual sine ratio.

Following this approach of only using rational equivalences, much of Euclidean geometry is rebuilt without making assumptions of the underlying field. This is also the Universal Geometry aspect of Rational trigonometry: the claim that most results from classical geometry will be applicable, and possess analogs, over any field (not of characteristic two) not just the field of rational numbers.

The three main laws of trigonometry (pythagoras' theorem, the sine law and the cosine law) are substituted with rational analogs and augmented by two further laws: one relating the quadrances of three collinear points and one relating the spreads of three concurrent lines (for a total of five main laws).

Wildberger holds a Ph.D. in mathematics from Yale University, and taught at Stanford University from 1984 to 1986 and at the University of Toronto from 1986 to 1989; he is currently an associate professor of mathematics at the University of New South Wales, Australia.