# Solving trigonometric equations

I need to solve these three trigonometric equations, and figure out the a, b and c. But, this takes too long time, and I am not sure the answer is right. Thus, How can I progress my code to speed it up and get the right answers.

f[Ax_, Ay_] := (a - Ay)/(Ax + Ay + a);

eq1 = f[81590621330, 46002067383];
eq2 = f[91677087285, 56769009856];
eq3 = f[107042438235, 73485959705];

NSolve[{eq1 == (Cos[b Degree])^2 - (Sin[b Degree])^2*(Sin[c Degree + 5 Degree])^2,
eq2 == (Cos[b Degree])^2 - (Sin[b Degree])^2*(Sin[c Degree + 10 Degree])^2,
eq3 == (Cos[b Degree])^2 - (Sin[b Degree])^2*(Sin[c Degree + 15 Degree])^2}, {a, b, c}];


Seems, this equations have no solution.

Build functions, bring all equations to one side, which should be zero. Mimimum of square of the three functions should be zero for a solution of a,b,c values. This is not zero here..

alleqs = {eq1 == (Cos[
b Degree])^2 - (Sin[b Degree])^2*(Sin[c Degree + 5 Degree])^2,
eq2 == (Cos[
b Degree])^2 - (Sin[b Degree])^2*(Sin[
c Degree + 10 Degree])^2,
eq3 == (Cos[
b Degree])^2 - (Sin[b Degree])^2*(Sin[
c Degree + 15 Degree])^2};

allfunc = alleqs[[All, 1]] - alleqs[[All, 2]]

NMinimize[allfunc.allfunc, {a, b, c}]

(*   {0.0000268218, {a -> -11.1683, b -> 122.798, c -> -111.169}}   *)


Test with ContourPlot whether this is a global minimum.