Rational Trigonometry

Rational trigonometry is a proposed reformulation of metrical planar and solid geometries (which includes trigonometry) by Canadian mathematician Norman J. Wildberger, currently a professor of mathematics at the University of New South Wales. His ideas are set out in his 2005 book Divine Proportions: Rational Trigonometry to Universal Geometry. According to New Scientist, part of his motivation for an alternative to traditional trigonometry was to avoid some problems that he claims occur when infinite series are used in mathematics. Rational trigonometry avoids direct use of transcendental functions like sine and cosine by substituting their squared equivalents.[2] Wildberger draws inspiration from mathematicians predating Georg Cantor's infinite set theory, like Gauss and Euclid, who he claims were far warier of using infinite sets than modern mathematicians.

For example, the unit square diagonally bisected is typically described as having angles 45º, 45º, and 90º; side lengths of 1, 1, and root-2, and an area of 2. In Rational Trigonometry, this same triangle has spreads 1/2, 1/2, and 1; quadrances of 1, 1, and 2, and a quadrea of 4. The advantages are obvious: almost everything is algebraic (not transcendental), computable, solvable, practical, and clear.

Because of its reliance on algebra – not infinite operations – Rational Trigonometry can be performed over a general field not of characteristic 0, making it highly extensible. It makes plain non-Euclidean geometries, especially relativistic geometry. It also connects with important polynomials very close to the Chebyshev polynomials.

Identites
Triple Quad Formula: Three points $$A_1$$, $$A_2$$, and $$A_3$$ lie on the same line exactly when

$$ (Q_1 + Q_2 + Q_3)^2 = 2(Q_1^2 + Q_2^2 + Q_3^2) $$

Pythagoras' Theorem: Two lines $$l_1 \equiv \langle a_1:b_1:c_1\rangle $$ and $$l_2 \equiv \langle a_2:b_2:c_2 \rangle$$ are perpendicular exactly when

$$ Q_1 + Q_2 = Q_3 $$

Spread Law: For any triangle with non-zero quadrances

$$ \frac{s_1}{Q_1} = \frac{s_2}{Q_2} = \frac{s_3}{Q_3} $$

Cross law: For any triangle

$$ (Q_1+Q_2-Q_3)^2=4Q_1Q_2c_3 $$

Triple spread formula: For any triangle

$$ (s_1 + s_2 + s_3)^2 = 2(s_1^2 + s_2^2 + s_3^2) + 4s_1 s_2 s_3 $$

Videos
Rational Trigonometry Quiz 1 Rational trigonometry: Solutions to Quiz (1) Medians, altitudes and vertex bisectors Trigonometry with finite fields (I) Trigonometry with Finite Fields (II) Trigonometry with Finite Fields (III) Highlights from triangle geometry (I) Highlights from triangle geometry (II) Spread polynomials Pentagons and five-fold symmetry Applications of rational trig to surveying (II) Stewart's theorem What size ladder fits around a corner? Trisecting angles and Hadley's theorem Polar coordinates and rational trigonometry Introduction to Projective Geometry Projective geometry and perspective Projective geometry and homogeneous coordinates Lines and planes in projective geometry Affine geometry and barycentric coordinates Affine geometry and vectors The cross ratio More about the cross ratio Harmonic ranges and pencils The fundamental theorem of projective geometry Conics via projective geometry An algebraic framework for rational trigonometry (I) An algebraic framework for rational trigonometry (II) How to learn mathematics Einstein's special relativity: an introduction Red geometry (I) Red geometry (II) Red geometry (III) Circles in red geometry Green geometry (I) Green geometry (II) Pythagorean triples An introduction to chromogeometry Chromogeometry and Euler lines Chromogeometry and the Omega triangle Chromogeometry and nine-point circles Proofs in chromogeometry Triangle spread rules Triangle spread rules in action Acute and obtuse triangles Proofs of the Triangle spread rules Rational trigonometry Quiz (#2) Hints for solutions to Quiz (#2) The 6-7-8 triangle Barycentric coordinates and the 6-7-8 triangle Squares in a pentagon Trisecting a right triangle Euler's Four Point Relation What is geometry really about? Determinants in geometry (I) Determinants in geometry (II) Determinants in Geometry (III) Spreads, determinants and chromogeometry (I) Spreads, determinants and chromogeometry (II) Spreads, determinants and chromogeometry (III) Coloured spreads and generalizations (I) Coloured spreads and generalizations (II) Geometry with a general dot product The general rational laws of trigonometry Maths education and rational trigonometry (I) Rheticus and 17th century trig tables Maths Education and Rational Trigonometry (II) Maths Education and Rational Trigonometry (III) Rational trigonometry and mathematics education (IV) The true role of the circular functions Understanding uniform motion: are radians really necessary? Turn angles, continued fractions and approximate geometry Isometry groups in planar geometry The group of blue planar isometries The Euclidean rotation group and complex number multiplication The geometry of Euclidean reflections and rotations (grounded!) Half-slope and the full group of Euclidean isometries
 * 1) Rational Trigonometry/0 - An Invitation to Geometry
 * 2) Rational Trigonometry/1 - Why Trig is Hard
 * 3) Rational Trigonometry/2 - Quadrance via Pythagoras and Archimedes
 * 4) Rational Trigonometry/3 - Spread, Angles and Astronomy
 * 5) Rational Trigonometry/4 - Five Main Laws of Rational Trigonometry
 * 6) Rational Trigonometry/5 - Applications of Rational Trigonometry
 * 7) Rational Trigonometry/6 - Heron's Formula Viewed Rationally
 * 8) Rational Trigonometry/7 - Solving Triangles with Rational Trigonometry
 * 9) Rational Trigonometry/8 - Centers of Triangles with Rational Trigonometry
 * 10) Rational Trigonometry/9 - Laws of Proportion for a Triangle
 * 11) Rational Trigonometry/10 - Geometry of Circles with Rational Trigonometry
 * 12) Rational Trigonometry/11 - An Invitation to Geometry
 * 13) Applications of rational trig to surveying (I)
 * 14) Cartesian coordinates and geometry
 * 15) Why spreads are better than angles
 * 16) Rational parameters for circles
 * 17) Complex numbers and rotations