what is the function of the trigger coupling in an oscilloscope?
Let's say you have a scope with an "intrinsic" sensitivity of 10 volts full scale. That is, with no amplification it shows a range of 0 to 10 volts. Consider a 0.1 volt AC signal riding on a 100 volt baseline. How would you display it? Using DC coupling, by scaling down to 10 volts (DC) you could show it, but the AC component would be reduced by a factor of 10 as well. This would have you trying to see a .01 volt signal on a 10 volt signal, or approximately 0.1% of the screen. You would find that very difficult to see. Instead, if you use AC coupling, you can then eliminate the DC, then add a gain of 100 to the result, and see the 0.1 volts occupying the entire vertical range of the screen.

Now let's think about triggering. With the triggering run from the same input signal as the display, it is reasonable to think of the triggering circuit as being in essentially the same position as the display. So it should come as no surprise that the trigger will have a hard time reliably picking up a 10 mV level riding on 10 volts. Instead, by using AC coupling and gain, the trigger level can be precisely adjusted to give a stable display.