Tan is not invertible as a whole. ArcTan is the inverse of the restriction to the interval between -Pi/2 to Pi/2. Your angle 10Pi/11 is outside that interval. The angle between -Pi/2 and Pi/2 which has the same Tan as 10Pi/11 is precisely -Pi/11. Have a look:

With[{\[Theta] = 10 Pi/11}, 
 Graphics[{Circle[], PointSize[Large], 
   Point[{Cos[\[Theta]], Sin[\[Theta]]}], Line[{{1, -1}, {1, 1}}], 
   Line[{{Cos[\[Theta]], Sin[\[Theta]]}, 
     2 {Cos[\[Theta] - Pi], Sin[\[Theta] - Pi]}}]}, Axes -> True]]