Jurij,
I think {?p, 180, 185} should probably be {?p, 180 °, 185 °} unless you intended radians.
root = Table[
FindRoot[{450 == Lp Cos[?p] + L1 Cos[?1] + L2 Cos[?2] + L3 Cos[?3],
530 == Lp Sin[?p] + L1 Sin[?1] + L2 Sin[?2] + L3 Sin[?3]},
{{?2, 90 °}, {?3, 10 °}}], {?p, 180 °, 185 °}]
(* {{?2 -> 1.47161, ?3 -> 0.0806623}} *)
The next part of your question is not clear to me. Evaluating the code
{xp, yp} = {Lp Cos[?p], Lp Sin[?p]};
{x1, y1} = {xp + L1 Cos[?1], yp + L1 Sin[?1]};
{x2, y2} = {x1 + L2 Cos[?2], y1 + L2 Sin[?2]};
{x3, y3} = {x2 + L3 Cos[?3], y2 + L3 Sin[?3]};
Gives
{{xp, yp}, {x1, y1}, {x2, y2} {x3, y3}}
(*
{{180 Cos[?p], 180 Sin[?p]},
{75 Sqrt[3] + 180 Cos[?p], 75 + 180 Sin[?p]},
{(75 Sqrt[3] + 420 Cos[?2] + 180 Cos[?p]) (75 Sqrt[3] + 420 Cos[?2] + 460 Cos[?3] + 180 Cos[?p]),
(75 + 420 Sin[?2] + 180 Sin[?p]) (75 + 420 Sin[?2] + 460 Sin[?3] + 180 Sin[?p])}}
*)
What do you mean by "each of the combinations of Theta2 and Theta3"?