I have the command:
Solve[{Cos[η] == Sqrt[3/6],
Sin[α] Cos[η] == Sqrt [1/6],
Sin[α] Sin[η] == Sqrt [2/6]}, {η, α}]
I get the output
{}.
What's the issue?
As said by @Mr.Wizard, you don't have a solution because it is a overdetermined system of equations, i.e., there are more equations than unknowns. If you look at @Mr.Wizard 's plot,
it is possible to note that the three equations has no common intersection. But if you elliminate one of the equations you can find two points of solution.
NSolve[{Sin[α] Cos[η] == Sqrt[1/6.],
Sin[α] Sin[η] == Sqrt[2/6.]}, {η, α}]
NSolve[{Cos[η] == Sqrt[3/6],
Sin[α] Sin[η] == Sqrt[2/6.]}, {η, α}]
NSolve[{Cos[η] == Sqrt[3/6],
Sin[α] Cos[η] == Sqrt[1/6.]}, {η, α}]
(*{{η -> -2.186276035465284`, α -> -0.7853981633974483`}, {\
η -> 0.9553166181245092`, α -> 0.7853981633974483`}}
{{η -> -0.7853981633974483`, α -> -0.9553166181245092`}, \
{η -> 0.7853981633974483`, α -> 0.9553166181245092`}}
{{η -> -0.7853981633974483`, α ->
0.6154797086703874`}, {η -> 0.7853981633974483`, α ->
0.6154797086703874`}}*)
ContourPlot[{Cos[η] == Sqrt[3/6], Sin[α] Cos[η] == Sqrt[1/6], Sin[α] Sin[η] == Sqrt[2/6]}, {η, -Pi, Pi}, {α, -Pi, Pi}]
$\endgroup$