Divine: Proportions: Rational Trigonometry To Un...

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

Rational trigonometry simplifies classical laws into polynomial forms that are much easier for computers and students to manipulate:

(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...

: The rational equivalent of the Cosine Law (using "cross"

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 with purely algebraic concepts

), 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

(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

: Replaces the Triangle Inequality. For three points to be collinear, their quadrances must satisfy: