I'm trying to figure out the error in the opening angle for a cone created with kinematic neutron imaging. The angle is defined as:
$$\theta = \sin^{-1}\sqrt{\frac{Ep}{E}}$$
And I want to find the error in this angle. I don't know how to propagate error through an inverse sine function so I made a substitution where:
$$
u = \sin^2(\theta) \\
u = \frac{Ep}{E}
$$
My work is attached, but my delta-theta at the end doesn't have units of radians or degrees. Where did I go wrong? Here is a LaTex file showing my entire process. 
Your problem being $$\theta = \sin^{-1}\left(\sqrt{\frac{Ep}{E}}\right)$$ let us start with $$\theta = \sin^{-1}(X) \qquad \text{where} \qquad X=\sqrt{\frac{Ep}{E}}$$ Using derivatives $$\Delta \theta=\frac{\Delta X}{\sqrt{1-X^2}}$$ Now $$X=\sqrt{\frac{Ep}{E}}\implies \log(X)=\frac 12 \log(Ep)-\frac 12 \log(E)$$ Using again derivatives, then $$\frac{\Delta X} X=\frac 12 \frac{\Delta Ep}{Ep}+\frac 12 \frac{\Delta E}{E}\implies \Delta X=??? \implies \Delta \theta=???$$