Does a player have to announce both of the triggers on a card such as Jeskai Ascendancy, or can they just say “trigger”, acknowledging it once?

The rules for Missed Triggers focus on when you have to acknowledge your triggers (short version: when they matter). We do not require you to announce them when they trigger, it's just a “best practice”, for clear communication.

I recommend some alone time with section 2.1 of the IPG. :)

d:^D