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
In rational trigonometry, we do not use "distance" (which often involves square roots). Instead, we use ( ), which is the square of the distance. For two points
), 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 Divine Proportions: Rational Trigonometry to Un...
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
: Replaces the Triangle Inequality. For three points to be collinear, their quadrances must satisfy: s1Q1=s2Q2=s3Q3the fraction with numerator s sub 1 and
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
(Q1+Q2−Q3)2=4Q1Q2(1−s3)open paren cap Q sub 1 plus cap Q sub 2 minus cap Q sub 3 close paren squared equals 4 cap Q sub 1 cap Q sub 2 open paren 1 minus s sub 3 close paren Why This Matters : You never need to use a calculator for 2the square root of 2 end-root . All results are exact fractions. Unlike angles, which are circular, spread is a
: Computers are much faster at adding and multiplying than calculating trigonometric series.