Sine Wave Harmonic From Pythagoras' Theorem

Primary, dependent variable for any alternatively contiguous, time evolved dimension

( 1 ) x = θ , ω * t , 2 * Pi * f * t

Algorithm derived directly from Pythagoras' Theorem

( 2 ) c [ i ] ^ 2 = a [ i ] ^ 2 + b [ i ] ^ 2 ,

( 3 ) a [ i ] ^ 2 = x ,

( 4 ) b [ i ] ^ 2 = y ,

( 5 ) c [ i ] = a ,

Sine wave equation

( 6) y , sin ( θ ) = a ^ 2 - x ^ 2

( 7 ) a ^ 2 = c ^ 2 - b ^ 2 ,

( 8 ) a = ( c ^ 2 - b ^ 2 ) ^ 1/2 ,

( 9 ) a = sin ( θ )

Sine wave function

( 10 ) A f ( x ) = a ^ 2 - x ^ 2

Unit (as well as contiguous) sine wave as alternative proof of equivalency of (9) and (10). (11) repeats itself continuously and contiguously.

( 11 ) A , f ( x ) = ( ( a ^2 - 2 * ( x - 1 /2 ) ^ 2 ) ^ 1/2 ) ^ 2

+ ( a ^2 - 2 * ( x - 3/2 ) ^ 2 ) ^ 1/2 ) ^ 2 ,

Sine wave equation applying only Cartesian variables

( 12 ) A = sin ( x )

Either (10) or (11) when correctly programmed, according to Cartesian variables, rather than conic sections, will map X to Y as sin ( x ).

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