# Error propagation in cone width for kinematic neutron imaging

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: $$\theta = \sin^{-1}\left(\sqrt{\frac{Ep}{E}}\right)\,.$$ 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: $$u = \sin^2(\theta)\,,\qquad u = \frac{Ep}{E}\,.$$ My work is attached, but my delta-theta at the end doesn't have units of radians or degrees. Where did I go wrong?

I.e. if your new variable $$q$$ is related to the old variables $$x$$, $$y$$, and $$z$$ by $$q = f(x,y,z)$$, then
$$dq = \frac{\partial f}{\partial x}dx +\frac{\partial f}{\partial y}dy+ \frac{\partial f}{\partial z}dz.$$
I think the key for you will be the differential of $$\arcsin(x)$$: $$d \arcsin(x) = \frac{dx}{\sqrt{1-x^2}}.$$ and that of $$\sqrt{E_p/E}$$: $$d\sqrt{\frac{E_p}{E}} = \frac{dE_p}{2\sqrt{E_p E}} -\frac{1}{2}\sqrt{\frac{E_p}{E}} \frac{dE}{E}$$
Try using those to expand out the differential of $$\theta=\arcsin\left(\sqrt{\frac{E_p}{E}}\right)$$.