# 4th degree nonlinear simultaneous equations with 3 unknowns

What method should be used to solve the following nonlinear simultaneous equations for $z_1, z_2, z_3$

$$a_2z_1^2 + a_1z_2^2 - (z_1z_2 \tan(t))^2 - z_1z_2 2b_1/\cos(t)^2 = b_1/\cos(t)^2 - a_1a_2$$

$$a_3z_2^2 + a_2z_3^2 - (z_2z_3 \tan(t))^2 - z_2z_3 2b_2/\cos(t)^2 = b_2/\cos(t)^2 - a_2a_3$$

$$a_1z_3^2 + a_3z_1^2 - (z_3z_1 \tan(t))^2 - z_3z_1 2b_3/\cos(t)^2 = b_3/\cos(t)^2 - a_3a_1$$

where $a_1,a_2,a_3,b_1,b_2,b_3,t$ are known.

Any help would be appreciated.

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Please use Latex formatting meta.math.stackexchange.com/questions/1773/… –  leonbloy Aug 6 '12 at 18:16
What about $t$ ? –  leonbloy Aug 6 '12 at 18:18
I'm not really sure how this should be tagged, but this isn't a nonlinear-optimization SFAICT... anyway, Ali, your unknowns are algebraically related, so Gröbner bases might be a good idea. –  Ｊ. Ｍ. Aug 7 '12 at 0:16
@J.M. could you work out the Gröbner basis idea? And what is SFAICT? –  draks ... Aug 7 '12 at 12:26
I invented a tag, algebraic-systems. Any better ideas? –  Gerry Myerson Aug 8 '12 at 1:36