1. How to calculate when a star, a planet, the moon, or the sun will rise and set*
*If you know its right ascension and declination and your latitude and longitude

2. Notation
a = right ascension
d = declination
f = latitude
a=altitude

3. We make use of spherical trig
The standard formula for the altitude of an object is: sin(a) = sin(d)sin(f) + cos(d) cos(f) cos(H)
If a = 0° (the object is on horizon, either rising or setting), then this equation becomes: cos(H) = - tan(f) tan(d)
This gives the semi-diurnal arc H: the time between the object crossing the horizon, and crossing the meridian.
Knowing the Right Ascension of the object, and its semi-diurnal arc, we can find the Local Sidereal Time of meridian transit, and hence calculate its rising and setting times.
One should put in a correction for atmospheric refraction, but we will neglect it for the moment

4. East Lansing f = +42.7 At this time of year, when does Arcturus rise?
Arcturus a = 14h 14 min +19D19’
cos(H) = - tan(f) tan(d) = -(0.923)(0.351) = H =109 degrees
In time units H = 109/15 = hours. Arcturus will rise 4.7 hours before it reaches the meridian and set 7.3 hours after it passes the meridian. That is, it will rise 7.3 hours before LST = 14H 14M and set 7.3 hours later than that time.

5. So when is LST = 14H14M on Jan. 20?
LST at local midnight on Jan. 21 is about
4 x 2 = 8 hours
So LST = 14H14M will come at about 6:13am local time or
6:14 +0:37 EST = 6:50 am. And we predict that Arcturus will rise at 7.3 hours before that or at about 11:30pm.
We’ve neglected refraction and some other small effects.

6. Let’s calculate when the sun sets
On Jan. 20 the sun is at a = 20H08M d=-20D10’
cos(H) = - tan(f) tan(d) = -(0.923)(-0.367) = H = 70.2
70.2/15 = 4.7 hours. The sun would set at about 4.7 hours after local noon. Or at about 4H42min + 37 min = 5:19 pm EST
The actual setting time for that date is 5:36pm
How could we make our calculation more accurate?

7. Corrections
Refraction: varies with your location and even the weather but is about 34 arcmin at the horizon
For the sun to set, the top must disappear. Thus even without refraction the center of the sun can be 15 arcmin below the horizon before the top sets:
So in the formula sin(a) = sin(d)sin(f) + cos(d) cos(f) cos(H)
For a use = -49’ instead of 0. Then we get
= (-0.345)(0.678) + (0.939)(0.735)cos H so H = 71.4
or 4 hours 46 min. When we add the 37 minutes that brings our setting time to 5:23 pm , closer but still early
What have we forgotten?

8. The Equation of Time

9. Taking account of the equation of time
We find that on Jan. 20 the equation of time is about -12 minutes. That is, apparent solar time is about 12 minutes earlier than mean solar time. When we add 12 minutes to our answer we get
5:23 + 0:12 =5:35pm
Pretty close to the standard value

10. Celebrations all around!