Very interesting discussion! I did a less theoretical approach a while ago to calculate the same thing: I just go to timeanddate.com, find the city I'm interested in, go to its sun page (example for New York https://www.timeanddate.com/sun/usa/new-york), and find the table that shows the difference in daylight minutes per day. No math or programming needed, just copy-paste and some buttons in Excel.
Not as satisfying as a derivation here, but a quicker way to get the answer.
EDIT: I did a spot check for Rovaniemi, Finland. This city is far north enough that the sun is up all day (66.5 degrees). But the graph on this page seems to be a little bit off: it requires an even higher latitude for that to show up.
Not as satisfying as a derivation here, but a quicker way to get the answer.
EDIT: I did a spot check for Rovaniemi, Finland. This city is far north enough that the sun is up all day (66.5 degrees). But the graph on this page seems to be a little bit off: it requires an even higher latitude for that to show up.