First, I'm really sorry for this somewhat vague and possibly just silly question. I also apologize if the following context runs a bit long. But please trust me that I'm asking with total sincerity and that my end goal is to find a starting point to grasp the one area of basic high school math that has always been just out of reach.
To begin: I always hated math as a young child (multiplication tables, carry-the-one, borrow-from-left, etc), and it was only later when math actually started to get interesting that I realized I didn't hate math, I hated rote memorization of tables and blind-faith rules/functions/tricks that were easy to forget under pressure.
Somewhere in middle school (pre-algebra, pre-geometry), things started to click. I don't want to suggest I was a math genius by any stretch, but I found that if I was keeping up conceptually with one module, the intro into the next had a nice "Oh, yes, of course! That is the next logical step!" feeling, so that I got to be the obnoxious kid who rolled his eyes at anyone struggling with a specific concept, thinking that it was so obvious that if we already know that the volume of a cylinder is the area of the circle on top times the height, why wouldn't a cone be one-third of that?
As I continued through high school, things got trickier, but overall either things made sense, or eventually made sense if I carefully retraced my steps to see where I got lost, and occasionally, things made no damn sense until there was an awesome pop in my head, like realizing that matrices weren't really crazy and magical, they were just a way of lining up all of the variables in such a way that they could be easily dealt with all at once.
Then we got to trigonometry functions. I basically bowed out at this point, took some honorably low B and C grades, knowing I just wasn't getting something, and never took any formal math classes ever again.
Two things I've learned since then:
You aren't really borrowing or carrying any one's, it's just a handy way of treating the top number as 10 + itself.
Unless you are taking math classes in college meant for math majors, you always use a calculator to get the actual numbers when figuring sine, cosine, and tangent.
That second part is crucial to my question. 20 years after hitting this math wall, I find out that it's not just easier to use a calculator for these functions, it's pretty much required (unless you've got your grandfather's slide rule, but this is basically the same idea, look it up).
So my questions are:
Are the trigonometric functions inherently something you just accept and learn and find a place in your brain for so that further concepts derive intuitively from that starting point, or do these concepts derive in a clear and somewhat intuitive (or at least straightforward) way from lower-level concepts that I've managed to not quite fit together on every 3-5 year skimming I attempt on the topic?
I'm sure, since there are those who can actually provide proofs or calculate the functions without a calculator, that these functions were not just "found" or "dreamt of" or were the ramblings of one insane genius who could only provide the ratios, not the reasoning. So I get that they are arrived at from lower-level math. So what I really want to know is if my struggle to understand it the same way I had come to understand every other math concept is basically where I'm going wrong, like my long overdue discovery that calculators were essential to the process.
(really sub 2) - If it really is just "hard" or "learned" or "work", I can accept that. Learning Latin verbs was hard and I knew I was getting low marks because I wasn't doing the work. But if there is some natural progression, and anyone can take a decent guess at some specific connection/concept that is most likely the "missing piece" (either because it usually is, or because this all sounds too familiar, or because you've got a knack for figuring out what's wrong just from senseless ramblings), I welcome any feedback or suggestions.
Note that a big part of my initial apology (and reason for wording things as I have) is that I'm not looking for a tutor or a drawn-out lesson in trig (not here, at least), only some validation that either I am missing something that should make this easier than I'm making it sound, or that it actually is hard and my mistake is expecting every time to spot that thing I was missing.
Thanks, as always.