I agree with everything you say. That having been said, learning SOHCAHTOA is at least a little different from, say, learning about staff notation. What SOHCAHTOA is used to memorize is not, in itself, something for which there is any deeper underlying concept to understand*: it’s just a matter of familiarity with what terminology denotes what. There’s no good reason that “sine” couldn’t denote what we currently refer to as cosine or tangent or cosecant or what have you, and so forth; it’s all just the arbitrary caprice of linguistic history. And thus, there is no other way to learn this than by memorization of the standard, since all there is to this is… memorization of the standard.
So, yeah, remembering how to apply the mantra SOHCAHTOA and understanding trigonometry are largely orthogonal; one shouldn’t conflate knowledge of terminological conventions with knowledge of substantial mathematical facts. But if one does want to learn the terminological conventions, it’s not as though there’s any more insightful, “Ah, that’s what’s really going on” way to come to know them than by rote.
My only complaint, then, is that SOHCAHTOA is subpar, even for those purposes, being such an arbitrary, made-up word, rather than something independently memorable and resistant to corruption (such as “HOMES”). Even if one did learn it with the “sew” pronunciation, there’s no great guarantee that one would remember it that way, as evidenced both by Google and by my students on occasion.
*: Yeah, alright, you could make the slight observation that co-f(x) always denotes f(the complementary angle to x). But this is largely opaque even to mathematicians, only just barely of any use in generally remembering what denotes what, and again just a curiosity of terminology, rather than a significant trigonometric fact.
Actually, as far as musical notation goes, apart from “Each space is separated by a semitone from the neighboring ledger line, in linear order”, what deeper underlying concept is there to grasp?
