This is a question of action sequence. I don't think the rules specify anything here, so I'll throw it out for general discussion.
Let's say my Archon gets womped by an enemy and my Arete is reduced to zero -- I'll die unless I sacrifice a Muse to restore my Arete to its permanent rating. Which of course I will do. Now, here's the question:
Before I sacrifice the Muse, I burn off all but 1 point of it and use those points as normal. Then I sacrifice the now 1-point Muse and restore my Arete. Is this legal?
When does burning a Muse fall in the order of actions in relation to sacrificing a Muse to live?
In the game that Matt runs on Skype, we're encouraged to keep a few low-value muses around for burning this way.
Personally, I just try to keep out of situations where I'm going to lose Arete/Hubris