Solving a system of 3 equations containing trigonometric functions [duplicate]

Question

I want to solve the following three equations for variables $\alpha$, $\beta$ and $\gamma$:

system = 
  {1/√((Sin[β/2])^2 + (Cos[β/2])^2 (Sin[(γ + α)/2])^2) Sin[β/2] Sin[(γ - α)/2] - 0.8819 == 0, 
   1/√((Sin[β/2])^2 + (Cos[β/2])^2 (Sin[(γ + α)/2])^2) Sin[β/2] Cos[(γ - α)/2] - 0.37947 == 0, 
   1/√((Sin[β/2])^2 + (Cos[β/2])^2 (Sin[(γ + α)/2])^2) Cos[β/2] Sin[(γ + α)/2] - 0.277 == 0};

How can this system be solved quickly?

Answer 1

Try

NMinimize[{1, {system  , 0 < \[Alpha] < 4 Pi, 0 < \[Beta] < 4 Pi,0 < \[Gamma] < 4 Pi}}, {\[Alpha], \[Beta], \[Gamma]}]
(*{1., {\[Alpha] -> 10.5561, \[Beta] -> 3.86852, \[Gamma] -> 0.309435}}*)