How do I solve vector loop equations

edit

Well...

sage: var('x tb');
sage: Ra=1/2; Rb=3; ta=pi/2;
sage: Sys = [Ra*cos(ta)+Rb*cos(tb)==x, Ra*sin(ta)+Rb*sin(tb)==0] ; Sys
[3*cos(tb) == x, 3*sin(tb) + 1/2 == 0]

Sage's default solver (i. e. Maxima's) isn't especially good at solving trigonometric systems :

sage: solve(Sys, [x, tb])
[3*cos(tb) == x, 3*sin(tb) + 1/2 == 0]
sage: solve(Sys, [x, tb], to_poly_solve=True)
[3*cos(tb) == x, 3*sin(tb) + 1/2 == 0]
sage: solve(Sys, [x, tb], to_poly_solve="force")
[3*cos(tb) == x, 3*sin(tb) + 1/2 == 0]

but Sympy's can solve this one :

sage: solve(Sys, [x, tb], algorithm="sympy")
[{tb: pi + arcsin(1/6), x: -1/2*sqrt(35)}, {tb: -arcsin(1/6), x: 1/2*sqrt(35)}]

(and, FWIW, so can the gratis Wolfram engine :

sage: mathematica.Solve(Sys, [x, tb])
{{x -> -Sqrt[35]/2, tb -> ConditionalExpression[-Pi + ArcTan[1/Sqrt[35]] + 
     2*Pi*C[1], Element[C[1], Integers]]}, 
 {x -> Sqrt[35]/2, tb -> ConditionalExpression[-ArcTan[1/Sqrt[35]] + 
     2*Pi*C[1], Element[C[1], Integers]]}}

but that's hardly surprising...).

More interesting is the general case (i. e. non-constant Ra, Rb, ta) :

sage: var("Ra, Rb, ta");
sage: Sys = [Ra*cos(ta)+Rb*cos(tb)==x, Ra*sin(ta)+Rb*sin(tb)==0]

Maxima can't solve this (no surprise...) :

sage: solve(Sys, [x, tb])
[Ra*cos(ta) + Rb*cos(tb) == x, Ra*sin(ta) + Rb*sin(tb) == 0]
sage: solve(Sys, [x, tb], to_poly_solve="force")
[Ra*cos(ta) + Rb*cos(tb) == x, Ra*sin(ta) + Rb*sin(tb) == 0]

but Sympy (and Mathematica) can :

sage: solve(Sys, [x, tb], algorithm="sympy")
[{tb: pi + arcsin(Ra*sin(ta)/Rb),
  x: Ra*cos(ta) - sqrt(-Ra^2*sin(ta)^2/Rb^2 + 1)*Rb},
 {tb: -arcsin(Ra*sin(ta)/Rb),
  x: Ra*cos(ta) + sqrt(-Ra^2*sin(ta)^2/Rb^2 + 1)*Rb}]
sage: mathematica.Solve(Sys, [x, tb])
{{x -> Ra*Cos[ta] - Sqrt[Rb^2 - Ra^2*Sin[ta]^2], 
  tb -> ConditionalExpression[ArcTan[-(Sqrt[Rb^2 - Ra^2*Sin[ta]^2]/Rb), 
      -((Ra*Sin[ta])/Rb)] + 2*Pi*C[1], Element[C[1], Integers]]}, 
 {x -> Ra*Cos[ta] + Sqrt[Rb^2 - Ra^2*Sin[ta]^2], 
  tb -> ConditionalExpression[ArcTan[Sqrt[Rb^2 - Ra^2*Sin[ta]^2]/Rb, 
      -((Ra*Sin[ta])/Rb)] + 2*Pi*C[1], Element[C[1], Integers]]}}

EDIT : Coaxing Maxima's default solver to give a solution.

You have a system of two equations with three unknowns ; one of the has to become a parametter of the solution(s). Choosing ta as a parameter allows us to write :

sage: S1 = Sys[1].solve(tb) ; S1
[tb == -arcsin(Ra*sin(ta)/Rb)]
sage: Sol = S1+Sys[0].subs(S1).solve(x) ; Sol
[tb == -arcsin(Ra*sin(ta)/Rb),
 x == Ra*cos(ta) + sqrt(-Ra^2*sin(ta)^2/Rb^2 + 1)*Rb]

The (almost) equivalence of this answer and those given by Sympy ant the Wolfram engine is left as an (interesting !) exercise to the reader...

HTH,

Abbreviated @Emmanuel answer

Ra=0.5; 
Rb=3.0; 
ta=pi/2; 
var('x tb'); 
sol = solve([Ra*cos(ta)+Rb*cos(tb)==x, 
             Ra*sin(ta)+Rb*sin(tb)==0.0],x,tb,
             algorithm='sympy') 
sol

[{tb: -0.167448079219689, x: 2.95803989154981},
 {tb: 3.30904073280948, x: -2.95803989154981}]