p=sin²(θ/2)=½-½cos(θ) (using trig identity).
dp/dθ=½sin(θ).
For small changes ∆p and ∆θ, ∆p/∆θ=dp/dθ.
Therefore ∆p=0.069=½sin(θ)∆θ, ∆θ=0.138/sin(θ).
When p=0.16, cos(θ)=1-2sin²(θ/2)=1-2p=0.68.
sin(θ)=√(1-cos²(θ))=√(1-0.4624)=0.7332 approx.
∆θ=0.138/0.7332=0.1882.
So the margin of error in θ is 0.1882 radians (10.7838º).
When p=0.16, sin²(θ/2)=0.16, sin(θ/2)=0.04, θ=2arcsin(0.04)=0.0800 (4.5849º)
θ=0.08±0.1882 or 4.5849±10.7838 degrees [-0.1082,0.2682] or [-6.1989º,15.3687º].