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Solve doesn't give an answer with system of equations


I have to solve the following system of equations:


Solve[{k*Sech[d]*Cos[b]/Sqrt[A^2 + B^2] == 1, 
k*Sech[d]*Sin[b]/Sqrt[A^2 + B^2] == 0, A*k*Tanh[d]/(A^2 + B^2)+c1 == 1,

B*k*Tanh[d]/(A^2 + B^2)+c2 == 0, k*Sech[k + d]*Cos[a + b]/Sqrt[A^2 + B^2] == 1, 
k*Sech[k + d]*Sin[a + b]/Sqrt[A^2 + B^2] == 0.1, 
A*k*Tanh[k + d]/(A^2 + B^2)+c1 == 1.1, 
B*k*Tanh[k + d]/(A^2 + B^2)+c2 == 0}, {k, d, a, b, A, B, c1, c2}]

but Solve will just compute forever and give no result. I've tried also with Reduce and NSolve, but with no luck. I can use approximated answers, so I've tried to use the Taylor series of the second order of the functions Sech Tanh Cos and Sin, but still Solve couldn't give me an answer.


I there something else I could try?


Thank you.



Answer



Notice that most of your transcendal functions are neatly separated into pairs of equations.



This will get you most of the way to your solution


sys={k*Sech[d]*Cos[b]/Sqrt[A^2+B^2]==1, k*Sech[d]*Sin[b]/Sqrt[A^2+B^2]==0, 
  A*k*Tanh[d]/(A^2+B^2)+c1==1, B*k*Tanh[d]/(A^2+B^2)+c2==0, 
  k*Sech[k+d]*Cos[a+b]/Sqrt[A^2+B^2]==1, k*Sech[k+d]*Sin[a+b]/Sqrt[A^2+B^2]==1/10, 
  A*k*Tanh[k+d]/(A^2+B^2)+c1==11/10, B*k*Tanh[k+d]/(A^2+B^2)+c2==0};
sol=Eliminate[sys, {Sech[d], Tanh[d], Sech[k+d], Tanh[k+d]}]

which instantly tells you


(* B==0 && c2==0 && Cos[a+b]==10 Sin[a+b] && Sin[b]==0 && A!=0 *)


Follow that with


sys /. ToRules[sol] /. True -> Sequence[]

which instantly tells you


(* {(k Cos[b] Sech[d])/Sqrt[A^2] == 1,
    (10 k Sech[d+k] Sin[a+b])/Sqrt[A^2]==1,
       (k Sech[d+k] Sin[a+b])/Sqrt[A^2]==1/10,
     c1+(k Tanh[d])/A==1,
     c1+(k Tanh[d+k])/A==11/10}*)


Notice one of those equations is obviously redundant and can be eliminated. Then notice c1 can then be eliminated giving even one fewer equation. Then notice that information from the first step is telling you that b is a multiple of Pi and Cos[b] is either +1 or -1.


If you make use of all this information then you might be able to use Reduce to get to the final solution.


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