OMG that shirt! I have taught from that text. Not my favorite - a classic, but somewhat dry. But the shirt is fun!
I've definitely read book that used their convention, even if they've backed off from it.
arcsin(x) was a multi-valued function defined on [-1, 1] and Arcsin(x) was the restriction of that to a specific range. IIRC [-pi/2, pi/2], just like they say.
I think this was my highschool trig book. Which is holy, and whose conventions you shall not disparage. Amazon. I think I'm remembering the convention being associated with the right book.
This is pretty analogous to the (afaik quite common) convention in complex analysis for the logarithm, log being multivalued and Log being the standard branch.
Man that was a good text book. I should see if I can't find that and double check; it'd be fun to reread parts of it.