On 8/16/09, Fred lunnon <fred.lunnon@gmail.com> wrote:
... My own yard-long program eventully disgorged a quartic in SABC with 504 terms in SA,SB,SC, after some obscure interaction between Maple expand() and simplify() was resolved. WFL
So for two angles [my earlier dodgy identity should have read]
SAB^2 - 2*(SA-2*SA*SB+SB)*SAB + (SA-SB)^2 = 0,
with SA -> sin^2(A), SB -> sin^2(B), SAB -> sin^2(A+B).
Er, make that a quartic in SABC with 71 terms in SA,SB,SC. To wit, With SA -> sin^2(A), SB -> sin^2(B), SC -> sin^2(C), SABC -> sin^2(A+B+C), 0 = SABC^4 + SABC^3 * ( -16*SA*SB*SC + 8*(SA*SB + SA*SC + SB*SC) - 4*(SA + SB + SC) ) + SABC^2 * ( + 24*SA*SB*SC + 16*(SA^2*SB^2 + SA^2*SC^2 + SB^2*SC^2) - 16*(SA^2*SB + SA^2*SC + SA*SB^2 + SA*SC^2 + SB^2*SC + SB*SC^2) + 6*(SA^2 + SB^2 + SC^2) + 4*(SA*SB + SA*SC + SB*SC) ) + SABC * ( - 40*SA*SB*SC - 16*(SA^3*SB*SC + SA*SB^3*SC + SA*SB*SC^3) + 8*(SA^3*SB + SA^3*SC + SA*SB^3 + SA*SC^3 + SB^3*SC + SB*SC^3) + 24*(SA^2*SB*SC + SA*SB^2*SC + SA*SB*SC^2) - 16*(SA^2*SB^2 + SA^2*SC^2 + SB^2*SC^2) - 4*(SA^3 + SB^3 + SC^3) + 4*(SA^2*SB + SA^2*SC + SA*SB^2 + SA*SC^2 + SB^2*SC + SB*SC^2) ) + ( 16*SA^2*SB^2*SC^2 + 8*(SA^3*SB*SC + SA*SB^3*SC + SA*SB*SC^3) - 16*(SA^2*SB^2*SC + SA^2*SB*SC^2 + SA*SB^2*SC^2) + (SA^4 + SB^4 + SC^4) - 4*(SA^3*SB + SA^3*SC + SA*SB^3 + SA*SC^3 + SB^3*SC + SB*SC^3) + 6*(SA^2*SB^2 + SA^2*SC^2 + SB^2*SC^2) + 4*(SA^2*SB*SC + SA*SB^2*SC + SA*SB*SC^2) ) ; Intriguingly, tidying the above up for publication produces a reduction in numerical rounding error by an order of magnitude! WFL