## rational trigonometry overview

This new theory dispenses with the usual complicated circular functions cos and sin and tan and their many complicated relations, and establishes a trigonometry based entirely on elementary arithmetic and algebra. The essential quadratic aspects of geometry are brought to the fore, with the key concepts of

Because of the rational form of the basic laws, the theory extends to general fields, opening up numerous new areas of investigation for number theory, algebraic geometry and combinatorics. A lovely new family of orthogonal polynomials called

*quadrance*and*spread*replacing distance and angle. These are much easier to work with, leading to polynomial relations requiring generally only quadratic equations to solve, so this is a huge opportunity for mathematics educators to simplify things for learners.*Divine Proportions: Rational Trigonometry to Universal Geometry*shows how to solve even complicated geometric problems without tables or calculators, with more accurate answers. Many engineering, surveying and physics problems are solved in this book, illustrating in detail the advantages of this new approach. For large scale problems, rational trigonometry harnesses modern computational power more efficiently. The two subjects of geometry and periodic circular motion are cleanly separated; it is neither necessary nor desirable to have a theory which merges the two.

Because of the rational form of the basic laws, the theory extends to general fields, opening up numerous new areas of investigation for number theory, algebraic geometry and combinatorics. A lovely new family of orthogonal polynomials called

*spread polynomials*extends the theory of special functions to general fields.Using rational trigonometry as a basis, a new form of metrical Euclidean geometry is constructed, incorporating most of high school geometry, but extending in exciting new directions. This new

*Universal Geometry*contains similiar triangles, Heron's formula, centroids, circumcenters, orthocenters, cyclic quadrilaterals and Brahmagupta's theorem, circles, parabolas, the Euler line, tangent lines and tangent conics, nine point circles, and much more---all over a

*general field.*

Examples are taken not only from the usual decimal numbers, but also finite fields and the complex numbers.

*Divine Proportions*argues that there is a good reason why the standard high school mathematics curriculum turns so many people off mathematics, often permanently. The reason is that the mathematics being currently taught is logically weak and overly complicated for what it accomplishes. It confuses the study of triangles with the subject of uniform circular motion. These are separate areas of mathematics, requiring different concepts and techniques.

To understand the geometry of a triangle, one doesn't require transcendental trigonometric functions such as sin, cos, tan and their inverse functions. Until now, there has been no sensible alternative to focus the minds of educators, even though most students intuitively understand that the theory is too complicated. This book finally addresses the failure of modern geometry to win the minds of young people, and provides the mathematical foundation for a dynamic and elegant new approach to teaching trigometry.