Yes, just give an initial point close to your solution:

eqns3 = eqns /. 
   Subscript[\[Alpha], i_] :> 
    ToExpression["a" <> ToString[i]]*Pi/180;
FindRoot[eqns3, {{a1, 10}, {a2, 16}, {a3, 31}, {a4, 33}}]

and you will get

{a1 -> 10.5456, a2 -> 16.0925, a3 -> 30.9046, a4 -> 32.8669}

Hi Gianluca, how you doing? i hope you are fine with good health. Could you please help me out to find the solution of this system of equations. i have attached the mathematica file as well.

Attachments:

If you do it this way

Select[Solve[{1 - Cos[3 x Degree] + Cos[3 y Degree] == 0, 
     1 - Cos[5 x Degree] + Cos[5 y Degree] == 0}, {x, y}] // N // 
  Simplify, FreeQ[#, Complex] &]

you will find your solution about 5th from the end. It's complicated because there are infinitely many solutions to parameterize. Another way is to give bounds for x and y:

N@Reduce[{1 - Cos[3 x Degree] + Cos[3 y Degree] == 0, 
   1 - Cos[5 x Degree] + Cos[5 y Degree] == 0, 0 < x < 180, 
   0 < y < 180}, {x, y}]