~ being sine would be fantastic. That said, my impression from the J wiki is that the o. circle function, rather than being a poor solution to a difficult problem, has the same motivation as !: Foreign; it’s meant to let sin, cos, etc exist in the standard library you import by default, without having to reserve the words sin, cos, etc.
I think so, but I haven’t absorbed enough J/APL philosophy to understand why it would be such a crime to have functions available with “lengthy” names like “sin.”
J/APL apologists tend to get poetic about how eventually things do become readable, but I find it hard to imagine that is true for !:. It looks like exactly what people complain about, readability-wise.