For a spin one-half system Sakurai leaves it to the reader to verify that knowledge of the three ensemble averages [(Subscript[S, x])], [(Subscript[S, y])],[(Subscript[S, z])] is sufficient to reconstruct the density operator (this is also problem 3.10).  Showing that is algebraically messy, and well suited to do in Mathematica.

$Assumptions = θ ∈Reals && ϕ ∈ Reals  && α ∈Reals && β ∈ Reals ;
plus := {{Cos[θ/2]}, {Sin[θ/2] E^(I ϕ)}} ;
minus := {{Sin[θ/2] E^(-I ϕ)}, {-Cos[θ/2] }} ;

(* spin matrix *)
sigma = ( Join[plus // Transpose, minus // Transpose] // Transpose) /. θ -> 2 θ ;

(* verify the eigenvalues *)
(*sigma . plus  -plus// Simplify // MatrixForm
sigma . minus  +minus// Simplify // MatrixForm*)

sigmaX = {{0, 1}, {1, 0}} ;
sigmaY = {{0, -I}, {I, 0}} ;
sigmaZ = {{1, 0}, {0, -1}} ;


trace[a_] := ConjugateTranspose[plus] . a . plus + ConjugateTranspose[minus] . a . minus ;
rho = α plus . ConjugateTranspose[plus] + β minus . ConjugateTranspose[minus] ;

tx = trace[rho . sigmaX] // FullSimplify ;
ty = trace[rho . sigmaY] // FullSimplify  ;
tz = trace[rho . sigmaZ] // FullSimplify  ;

Grid[{
{"|+> = ",plus},
{"|-> = ",minus},
{"σ = ",sigma},
{"(Subscript[σ, x]) = ",sigmaX},
{"(Subscript[σ, y]) = ",sigmaY},
{"(Subscript[σ, z]) = ",sigmaZ},
{"ρ = ",rho},
{"ρ (Subscript[σ, x]) = ",rho.sigmaX//Simplify},
{"ρ (Subscript[σ, y]) = ",rho.sigmaY//Simplify},
{"ρ (Subscript[σ, z]) = ",rho.sigmaZ//Simplify},
{"tr(ρ (Subscript[σ, x])) = ", tx },
{"tr(ρ (Subscript[σ, y])) = ", ty },
{"tr(ρ (Subscript[σ, z])) = ", tz }
}] //TraditionalForm





(*Solve[{
x == tx [[1,1]],
z== tz [[1,1]]
}, {α,β}] // FullSimplify // TraditionalForm*)

Since α - β = 2 α - 1, we have

( 2 α - 1 ) (OverHat[n]) = (tr(ρ (Subscript[σ, x])), tr(ρ (Subscript[σ, y])), tr(ρ (Subscript[σ, z]))),

or

α = ((1 + (Sqrt[(tr^2)(ρ (Subscript[σ, x])) + (tr^2)(ρ (Subscript[σ, y]))  + (tr^2)(ρ (Subscript[σ, z]))]))/2)
β = ((1 - (Sqrt[(tr^2)(ρ (Subscript[σ, x])) + (tr^2)(ρ (Subscript[σ, y]))  + (tr^2)(ρ (Subscript[σ, z]))]))/2)