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The most precise and quite intuitive formula I could find online is here: https://www.celestialprogramming.com/snippets/angleBetweenEclipticAndHorizon.html

$$\cos I = \cos ϵ \sin ϕ − \sin ϵ \cos ϕ \sin θ$$

ϵ is the obliquity of the ecliptic (e.g. 23.4°), ϕ is latitude, θ is the Local Sidereal Time.

Even though it makes sense that it depends on the sidereal time, latitude and inclination angle, I can't find a demonstration for this formula, so I'm wondering how can I demonstrate this formula?

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    $\begingroup$ @astridlovespie Welcome to Stack Exchange! By "demonstrate" do you mean to plot it, or explain it, or derive it, or... We don't normally say "demonstrate a formula". Thanks! $\endgroup$
    – uhoh
    Commented Jun 24, 2023 at 1:54
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    $\begingroup$ See en.wikipedia.org/wiki/Spherical_law_of_cosines $\endgroup$
    – PM 2Ring
    Commented Jun 24, 2023 at 4:54
  • $\begingroup$ Local sidereal time is an angle. 0 radians is when the plane of your local meridian passes the vernal equinox pi radians is 12 hours later. $\endgroup$
    – stretch
    Commented Jun 24, 2023 at 22:24
  • $\begingroup$ @ Greg Miller he may just be looking for the derivation. No one seems to know what he means by demonstration. His question is about your formula on a different site, $\endgroup$
    – stretch
    Commented Jul 1, 2023 at 23:52

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