Date: Tue, 23 Sep 2014 11:59:54 -0600 From: rcs@xmission.com The atan formulas can be understood as being about atan(N/D) = imag(log D+iN), and factoring D+iN into complex primes. You combine a few relations, and eventually wind up with log(i) or imag(log 1+i). You can go after new formulas by factoring complex numbers A+Bi with large A and small B. "Fortuitously", 239+i has norm 57122 = 2 * 13^4, while 5+i has norm 26 = 2 * 13, leading to pi/4 = 4 atan(1/5) - atan(1/239). ("Fortuitously" comes from 169, a square, turning up in the solutions of x2-2y2 = +-1 as (239,169). Maybe something similar could be made of 144 being a Fib, and being part of x2-5y2 = +-4 as (321,144)?) Can you do something similar with the arcsins? I notice that the implied base leg for the 11/43 case is sqrt(43^2-11^2) -> sqrt(32 * 54) -> 24 sqrt3. This could also be useful in simplifying your arcsin expressions, if you regard "factoring" as a simplification. Rich --well... arcsin(x) = -i * ln( i*x + sqrt(1-x*x) ) arsinh(x) = ln( x + sqrt(x*x+1) ) arcsin(x) = -i * arsinh(i*x) = arctan( x / sqrt(1-x*x) ) so, from this plus Rich's arctan(N/D) = imag ln(D+iN) we can conclude arcsin(N/D) = imag( ln( sqrt(D*D-N*N) + i*N ) ) where |N|<=|D| so then the natural factorizations happen within the complex numbers with squared-integer norms [where norm(A+iB)=A*A+B*B] and real and imaginary parts which both are constructible-algebraic numbers. This is a ring, but a pretty nasty ring.