How to find the precise and correct range of a specific angle in a triangle given certain conditions?

Question

In $\triangle$ABC, let angles A, B, and C be the three interior angles of the triangle, and given that $\frac{\sin B}{\sin A} = 2\sqrt{3} \sin C$, then the range of values for $B + \frac{\pi}{6}$ is _ , and the range of values for $\frac{\sin C}{\sin A} + \frac{\sin A}{\sin C}$ is _.

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This method does not determine the correct range.

FunctionRange[{B + \[Pi]/6, Sin[B]/Sin[A] == 2 Sqrt[3] Sin[C1], 
  A + B + C1 == \[Pi], a/Sin[A] == b/Sin[B] == c/Sin[C1], 
  a^2 + c^2 - b^2 == 2 a  c  Cos[B]}, {a, b, c, A, B, C1}, t]

get the result is:

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Since A, B, and C1 are the interior angles of a triangle, each angle ranges within the open interval (0, π). Adding this condition does not yield a more precise or correct answer.

FunctionRange[{B + \[Pi]/6, Sin[B]/Sin[A] == 2 Sqrt[3] Sin[C1], 
  A + B + C1 == \[Pi], a/Sin[A] == b/Sin[B] == c/Sin[C1], 
  a^2 + c^2 - b^2 == 2 a  c  Cos[B], 0 < A < \[Pi], 0 < B < \[Pi], 
  0 < C1 < \[Pi]}, {a, b, c, A, B, C1}, t]

This code get the result.

How can I obtain this final precise result as follows:

Answer 1

Use Reduce after the evaluation.

s = FunctionRange[{B + \[Pi]/6, Sin[B]/Sin[A] == 2  Sqrt[3]  Sin[C1], 
    A + B + C1 == \[Pi], a/Sin[A] == b/Sin[B] == c/Sin[C1], 
    a^2 + c^2 - b^2 == 2  a   c   Cos[B]}, {a, b, c, A, B, C1}, t];

Reduce[{s, 0 <= t <= \[Pi]}, t]

(C[1] == 0 && t == (5 \[Pi])/6) || (C[2] == 0 && \[Pi]/6 < t < (5 \[Pi])/6)

Answer 2

It is sufficient to consider only the three angles of the triangle!

ContourPlot and constraint \[Gamma] == Pi - \[Alpha] - \[Beta] show the range of possible values

ContourPlot[
 Sin[\[Beta]] == 
  2 Sqrt[3] Sin[\[Alpha]] Sin[Pi - \[Alpha] - \[Beta]], {\[Alpha], 0, 
  Pi}, {\[Beta], 0, Pi} , GridLines -> {None, { 4 Pi/6 }}, 
 FrameLabel -> {\[Alpha], \[Beta]}]

range of \[Beta] : 0<\[Beta]<2Pi/3

Reduceconfirms this result

Reduce[Sin[\[Beta]] == 
   2 Sqrt[3] Sin[\[Alpha]] Sin[Pi - \[Alpha] - \[Beta]] && 
  0 < \[Alpha] < Pi && 0 < \[Beta] < Pi, {\[Alpha], \[Beta]}, Reals]//FullSimplify