My point is that your arbitrary sketch fails at clarifying the question. Theta theta what is theta there is no theta in your non-clarifying arbitrary sketch. Also, did I mention we have no clue what theta is supposed to be?
So if you can't point out what theta is supposed to be, then I suppose the generic advice to do that substitution is about it. Well, that and maybe look at what the taylor expansion of sqrt(1+x) is for x<<1, but who knows if that is relevant.
without knowing what theta is , we are just grasping at straws.
you can try a couple of options, and let us know how it goes (I don't feel like trying them out myself).
i. sin(theta) = D / (R + H), i.e. theta is angle AOB
ii. sin(theta) = R / (R + H), i.e. theta is angle OAB
iii. theta is angle BOC, and provides information about the refraction capability of earth. so it would be required to calculate the overall distance from A to C through B.
try to work through each, and who knows, maybe one of them may match with what you have initially provided.
Or at least Rihanna didn't dispute this when I showed it in a previous thread (this is the third thread asking exactly the same question). Then again, maybe Rihanna doesn't know what theta is either.
Disclaimer: The question relates to a flying thingie that's further away, so curvature of the earth needs to be taken into account, and the simple equation shown here can't be used.
Or at least Rihanna didn't dispute this when I showed it in a previous thread (this is the third thread asking exactly the same question). Then again, maybe Rihanna doesn't know what theta is either.