I don't think this system has a solution (over the reals). If we define
f[θ_, ψ_] := 2 Cos[θ] - 1.5 Sin[θ] - 0.08 Cos[ψ]^2 + 0.08 Sin[ψ]^2 - 0.05 Cos[θ/2]/θ
g[θ_, ψ_] := 2 Sin[θ] + 1.5 Cos[θ] - 0.16 Cos[ψ] Sin[ψ] - Sin[θ/2]
then we'd like to know where
f[θ, ψ] == g[θ, ψ] == 0
If we plot all three functions, it appears they don't simultaneously intersect.
Plot3D[{f[x, y], g[x, y], 0}, {x, -10, 10}, {y, -10, 10}]