Positional Astronomy:
The relation between ecliptic and equatorial coordinates

To convert between the ecliptic and equatorial coordinates of a point X, we draw the spherical triangle PKX, where P is the North Celestial Pole, and K is the north pole of the ecliptic.

Apply the cosine rule to the corner at K:

Alternatively, apply same rule to the other corner at P, and get:

Now try applying the sine rule to the same triangle,

Grouping these three relations together, we have:

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