On 8/21/09, Fred lunnon <fred.lunnon@gmail.com> wrote:
... And in a different --- doubtless equal fatal --- direction, into posing this generalisation: given 4 angles A,B,C,D, what polynomial in the squares of their sines gives a condition (necessary only, since the signs remain unspecified) that A+B+C+D = k \pi ? WFL
How embarrassing --- the quartic for the sum of 3 angles, with SABC -> SD. At least the result is more symmetric than before: omitting isomorphic terms, - 40*SA*SB*SC*SD - 16*(SA^3*SB*SC*SD + ) + 16*(SA^2*SB^2*SC^2 + ) + 8*(SA^3*SB*SC + ) - 16*(SA^2*SB^2*SC + ) + 24*(SA^2*SB*SC*SD + ) - 4*(SA^3*SB+SA^3*SC + ) + 4*(SA^2*SB*SC + ) + 6*(SA^2*SB^2 + ) + (SA^4 + ) But 5 angles is getting a tad heavy --- can anyone spot a pattern? [From _two_ data items, that is --- 0,1,2 angles are sadly untypical.] WFL