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Note that solve is usually used to solve algebraic equations. (The examples of the doc string of solve also contain some trigonometric equations, but mixing them with more variables... I also try it in some specific non-algebraic cases, but do not hope to be successful...)

To get the system solved i am using two variables $s=\sin\theta$, $c=\cos\theta$ instead of the one variable $\theta$. This moves the system into the algebraich world. The code

# i am using s, c for sin(theta), cos(theta)...
Fs, Na, Nb, s, c = var('Fs Na Nb s c')

eq1 = 0 ==  Fs + 0.16*Na           - 10.0*s
eq2 = 0 ==            Na           - 10.0*c
eq3 = 0 == -Fs           + 0.26*Nb -  6.0*s
eq4 = 0 ==                      Nb -  6.0*c
eq5 = 1 == s^2 + c^2

solns = solve( [eq1, eq2, eq3, eq4, eq5]
               , Fs, Na, Nb, s, c
               , solution_dict=True)

for sol in solns:
    for key in sol:
        print "%2s = %s" % ( key, sol[key].n(30) )
    print

then delivers

 c = 0.98104950
 s = 0.19375728
Nb = 5.8862970
Na = 9.8104950
Fs = 0.36789356

 c = -0.98104950
 s = -0.19375728
Nb = -5.8862970
Na = -9.8104950
Fs = -0.36789356

(Getting $\theta$ now should be one more code line.)