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]]