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:
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You aren't really borrowing or carrying any one's, it's just a handy way of treating the top number as 10 + itself.
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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:
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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?
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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.
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(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.
Best Answer
Why would someone learn the values of trigonometric functions? When I was child, we usually didn't use calculator in the elementary school but we used a table. As I was nine years old I asked my mother how did the editor fill this tables. She told me, that he might have constructed triangles and measured their sides. Of course she was wrong but her idea wasn't totally bad. The trigonometry is an old discipline, older than most the techniques of computing the values of trigonometric functions. The base concept of trigonometry is the similarity. Thales has known that he can compute the height of a pyramid from the length of its shadow using a stick. There was a lot of problems in astronomy and "ancient engineering" which needed the concrete values of trigonometric functions. When someone needed badly a value of a trigonometric function, in the worst case one could measure it. On the other hand we can deduce some special value of trig functions: we know $\sin \frac{\pi}{4}$, $\sin \frac{\pi}{3}$, $\sin \frac{\pi}{6}$, $\cos \frac{\pi}{4}$, $\cos \frac{\pi}{3}$, $\cos \frac{\pi}{6}$ and the trigonometric addition formulae are quite old things too, so one could compute the value of $\sin k\frac{\pi}{2^n}$ which is enough to approximation but gives us a very uncomfortable method. But this isn't the way in which the value of sinus is given in our tables or in way is it computed by a calculator. In the XVII century Newton or someone else noticed that sinus and cosinus satisfy the following differential equation $$\ddot{y}=-y. $$ If sinus and cosinus would be a polynomial -- in that time people believed every function is like polynomial -- then this would give us a recursion of their coefficients. Let be $$\sin t = \sum\limits_{k=0}^na_kt^k$$ then $$\ddot\sin t=\sum\limits_{k=0}^n -k(k-1)a_kt^{k-2} $$ Our differential equation and the fact that a polynomial determines its coefficients show us $$a_{k-2}=-k(k-1)a_k$$ Because $\sin 0=0$ and $\dot\sin 0= \cos 0= 1$ we've got a recursion for it's coefficients. It sounds great, but there is a problem with it. If $a_0$ or $a_1$ isn't 0 then the recursion never stops that is the degree of sin must be infinity. Well, this is the origin of the notion of power series... (I'm really sorry for my very unpolished English)