How can I check if we can observe a certain star (given its equatorial coordinates) from a certain location on Earth (given its DMS coordinates)? I am given the equatorial coordinates of a star (e.g. Sirius has RA: 6h45m8.9s and Dec = -16°42'52.1"). Now my question is: can we observe this star from a specific location on Earth (e.g. from Brussels 50°51'1.62" N 4°20'55.61" E)? If yes, how can I justify this using the coordinates of the star and location on Earth?
You can check whether the star is anti-circumpolar i.e. if it never rises above the horizon (if its upper culmination altitude is less than zero). If it is, you will never see it from that location. If it is not, at some point during the year you will be able to see it in conditions with minimal light pollution if its upper culmination altitude is around zero (and, of course, the conditions do not have to be as good if the upper culmination altitude is greater, so you may want to increase the upper culmination bound by a few degrees to make sure you can actually see the star).
A way to justify being able to observe a star whose upper culmination altitude is big enough is to state that the Sun goes along the ecliptic across a fixed celestial sphere during the year, and at some point must be on the other side of the celestial sphere compared to the star. That day, when the Sun sets, the star will be visible.
Sirius, in particular, can be seen from Brussels tonight at around 2 AM: