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

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