The equation seems quite hard to solve symbolically. You can try numerical methods, starting from random initial points:
eqns2 = eqns /.
Subscript[\[Alpha], i_] :> ToExpression["a" <> ToString[i]];
sol = FindRoot[eqns2,
{{a1, RandomInteger[{5, 20}] Pi/180},
{a2, RandomInteger[{5, 20}] Pi/180},
{a3, RandomInteger[{5, 20}] Pi/180},
{a4, RandomInteger[{5, 20}] Pi/180}}]
eqns2[[All, 1]] /. sol
After some non convergent attempts I found a reasonable candidate solution:
{a1 -> 0.184056, a2 -> 0.280866, a3 -> 0.539386, a4 -> 0.573635}
for which the equations are satisfied to machine precision and the Jacobian is nonsingular.