Divine Proportions: Rational Trigonometry To Un... 【2024】

s=QoppositeQhypotenuses equals the fraction with numerator cap Q sub o p p o s i t e end-sub and denominator cap Q sub h y p o t e n u s e end-sub end-fraction The spread ranges from indicates parallel lines and indicates perpendicular lines. 3. Apply the Main Laws

Rational Trigonometry is a purely algebraic alternative to classical trigonometry that replaces distance and angle with and spread , allowing for exact geometric calculations without transcendental functions.

(Q1+Q2+Q3)2=2(Q12+Q22+Q32)open paren cap Q sub 1 plus cap Q sub 2 plus cap Q sub 3 close paren squared equals 2 open paren cap Q sub 1 squared plus cap Q sub 2 squared plus cap Q sub 3 squared close paren : The rational equivalent of the Sine Law: Divine Proportions: Rational Trigonometry to Un...

), a dimensionless ratio that measures the "separation" between two lines. Unlike angles, which are circular, spread is a rational function. For a right triangle with quadrances Q1cap Q sub 1 Q2cap Q sub 2 , and hypotenuse Q3cap Q sub 3

Q=(x2−x1)2+(y2−y1)2cap Q equals open paren x sub 2 minus x sub 1 close paren squared plus open paren y sub 2 minus y sub 1 close paren squared 2. Replace angle with spread Angles are replaced by ( (Q1+Q2+Q3)2=2(Q12+Q22+Q32)open paren cap Q sub 1 plus cap

s1Q1=s2Q2=s3Q3the fraction with numerator s sub 1 and denominator cap Q sub 1 end-fraction equals the fraction with numerator s sub 2 and denominator cap Q sub 2 end-fraction equals the fraction with numerator s sub 3 and denominator cap Q sub 3 end-fraction

: These laws work in any coordinate system, including those used in Einstein's Special Relativity (Minkowski space). ✅ Answer Replace angle with spread Angles are replaced by

with purely algebraic concepts. By avoiding irrational numbers and infinite series, it allows for exact calculations over any field, not just the real numbers. 1. Replace distance with quadrance