Le jeudi 4 décembre 2014 15:08:38 UTC+1, SiL588 . a écrit :
>
> Hello,
> i have to solve a simple two equations system, where
>
> eq0=v==V_f*cos(psi_d)
> eq1=l*phi==V_f*sin(psi_d)
>
> but when i do
>
> solve ([eq0,eq1],psi_d)
>
> the output is : [ ].
> What's wrong?
> Thank you very much.
>
Solving either eq0 or eq1 gives you a solution :
sage: s0=solve(eq0, psi_d) ; s0
[psi_d == arccos(v/V_f)]
sage: s1=solve(eq1, psi_d) ; s1
[psi_d == arcsin(l*phi/V_f)]
Your problem is that you have a hidden condition : apart the obvious ones (
abs(v/V_v)<=1 and abs(l*phi/V_f)<=1), the two solutions thus obtained must
be compatible :
sage: s0[0].rhs()==s1[0].rhs()
arccos(v/V_f) == arcsin(l*phi/V_f)
which might be betted understood using the well-known equality
sin^2x+cos^2x=1 :
sage: map(lambda u,v:u+v, map(lambda t:(cos(t).trig_expand())^2,
solve(eq0,psi_d)),
map(lambda t:(sin(t).trig_expand())^2,
solve(eq1,psi_d)))
[cos(psi_d)^2 + sin(psi_d)^2 == l^2*phi^2/V_f^2 + v^2/V_f^2]
leading you to the condition l^2*phi^2/V_f^2 + v^2/V_f^2==1. Beware : since
we squared the previous results, this eqiality might have spurious
solutions...
HTH,
--
Emmanuel Charpentier
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