Originally my question is rooted in neutrino physics where neutrinos are produced at a height h above the earth (which has radius r). Here one would like to know the distance L from the source to a specific point on the earth's surface as a function of zenith angle $\phi$, r and h. I have drawn this in the picture below to make it more comprehensible:

distance to the circumference at a given zenith angle

Now I have found through some research that the answer should be

$L=-rcos(\phi)+\sqrt{r^2 cos^2(\phi)+h^2+2rh}$

But I cant get the trigonometry right to derive this solution.


By cosine formula, we have


Solving, we have



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