The usual trigonometry is overly complicated, inaccurate and logically dubious. This is the first of a series that shows you a better way---rational trigonometry!
We show how the basic notion of rational trigonometry---quadr ance---arises from the geometry of the ancient Greeks. The little-known sister theorem to Pythagoras features prominently, and is closely related to a theorem of Archimedes.
Rational trigonometry is applied to solve four examples of practical problems, concerning a flagpole, a ladder, a kite and the distance from a point to a line.
Angles have their origin in astronomy and spherical trigonometry. Here we introduce the rational alternative, called spread, and give examples from ISO paper sizes to the faces of a dodecahedron.
Rational trigonometry can be used to solve surveying problems, usually more simply than the current way. This video gives three examples: finding the height of a mountain, Regiomontanus' problem, and spreads over a right triangle.
We derive from first principles the main laws of rational trigonometry, using the concepts of quadrance and spread to replace the usual distance and angle. Most everything works out much simpler.
Complex numbers are here explained using geometry and their intimate connection with dilations and rotations. Pure rotations are related to the parametrization of the unit circle.
Cartesian coordinates allow us to talk precisely about points and lines, parallel and perpendicular, and quadrance and spread---the two main concepts from rational trigonometry.
Heron's formula, originally due to Archimedes, is here recast in a simpler and more natural form. And we prove it, using one of the basic laws of rational trigonometry.
We derive some of the most fundamental facts about a circle using rational trigonometry---the Subtended spread theorems and the Equal products theorem.
Using rational trigonometry we develop Menelaus' and Ceva's theorem and some related results, namely the Law of Proportions, and the Alternating spreads theorem.
The usual unit circle is best described by rational parameters, not transcendental ones. This approach is much older, and connects with Pythagorean triples, along with rational trigonometry.
This video reconciles two different definitions of the spread between two lines. It also shows why spreads are generally superior to angles in a Cartesian framework.
![WildTrig6: Heron\](http://img.youtube.com/vi/hooQuHLS-kk/2.jpg "WildTrig6: Heron\")
