How did you get that information? How did you, in a non-magical way, go from information about an angle to information about a ratio?
If students don't know this information, then perhaps they are studying applications. So, what applications are students taught in typical trigonometric texts? Periodic behavior perhaps? Like sound? Only perhaps a brief blurb in the text that application is even possible. Perhaps they look at something about an incline plane. It is unlikely that they will touch projectiles.
It appears that trigonometry is there to give students some sense of mild comfort for future work in physics or engineering. This makes me think, "Why not statistics instead?"
> How did you, in a non-magical way, go from information about an angle to information about a ratio?
By having a right triangle?
The rest of your post seems to show that you want trig to be about periodic behavior, when it really is about triangles. That's what trigonometry means - measuring triangles.
Yes, trig has applications to periodic behavior, projectiles, differential equations, inclined planes, and all kinds of other stuff. But the point of a trig class is not to teach the applications. The point is to teach the tools, and maybe touch on the applications.
The problem is this. Using compass and ruler constructions there is a set of angles you can construct, and you can calculate sin and cos for those angles. You can even write the values for those out explicitly. However no part of this construction sheds light on how to find sin and cos for angles that you don't know how to construct. Or even gives good intuition that no matter how you do it, you can define it in a way that makes sense for all angles.
In fact we draw a picture, people look at it, and their intuition tells them that things will work out. Very few students will notice the logical gaps.
But to close the logical gaps, you need to start with Calculus first, and then derive trig formulas from that.
(Yes, I'm aware of the history here. Euclid presented trig reasonably rigorously a very long time before Calculus. Newton invented Calculus in the 1600s, and then used it as a heuristic to figure out answers that he then rederived using trig in The Principia. Leibniz reinvented Calculus in part based on inspiration from Newton's work. None of this was made formally correct until the late 1800s.)
(I have no opinion on pedagogical arguments about which is best to present first. I believe that we present trig first as a holdover from a curriculum where The Elements was the standard textbook until very recently.)
Well... you can use the half-angle formulas and the angle addition formulas to calculate sin and cos for angles that are arbitrarily close to the ones that you want. Add to that the idea that sin and cos must be continuous (I consider that intuitively obvious from a unit circle, but I don't know how to make that argument rigorous), and you can start to interpolate. You can in fact use these methods to calculate sin and cos for an arbitrary angle to any desired degree of precision... if you have the patience. It will be shorter to use the series derived from calculus, I'll admit.
I do not believe that there is an argument for continuity without starting with Calculus. Certainly starting from ruler and compass constructions it is not obvious.
That said, if you have enough Calculus to define how to measure the arclength of a segment of the circle, you can quickly prove that sin and cos in radians exist, have a nice power series, and so on.
It is like x^y with x positive. We can manually define it every rational y. But the easiest way to get a rigorous and straightforward definition is to prove the algebraic properties of the integral of 1/x, use that to define the logarithm, define its inverse function to be the exponential, prove its algebraic properties, then define x^y as e^(y*log(x)). And it all just works.
It's easy to show more or less directly (by comparing arclength to straight-line length), and certainly without calculus, that the absolute difference between sin(x + delta) and sin(x) is at most |delta|.
Replying really late, just in case anybody reads this. (I went on vacation, and this occurred to me then.)
If you don't have calculus, you don't have anything like a delta-epsilon proof of continuity. But without calculus, you also don't know that you need it. So you just assume (correctly) that you can interpolate, and it works just like you expect, and life goes on.
As young students, we are taught that these functions are ratios of sides on right angles. It is fairly intuitive for them that provided they can draw a right triangle, they just need to measure the sides. And it is easy to accept that someone computed sines and cosines for a lot of angles and put it in a table.
My point being, that we didn't need (in the past) calculus to compute trigonometric functions, and I don't see how it is a burden for students to be introduced to those functions without the Calculus definition.
And the reality is, that the definition of sine as a ratio of the catheti and hypotenuse is a rigorous definition of the function. Strictly, this sine is different from the sine of calculus. The first, the sine from Euclidean geometry, assigns a real to pair of rays, while the calculus sine, is function from the real numbers to the reals. And it does take some work to link them formally.
What kind of pedagogical or pragmatic relevance do you see trig as a building block for? I would answer that question by saying that it most likely comes up again either in physics or engineering contexts, or in a standardized exam like MCAT. And only in the sense of familiarity with the unit circle and trig functions.
What other foundation or learning pathway do you see trig serving as? Somebody else mentioned that trig serves use by teaching students that calculus has rich applications. So then I question, what kind of applications are students learning in trig? And if students are to learn rich examples of calculus applications, then why not statistics, which is also relevant to the bio / social sciences? Also, couldn't we mash trig inside calculus?
It seems to me that if I know trig, I can use it to solve the set of geometric problems to which it is applicable, without having been taught any special applications of trig. That's valuable in itself.
Then I take physics, and I find a whole bunch of other applications. I take calculus, and I find a bunch more uses. I take mechanics, and I find a bunch more. But it is not the job of trig to teach me those applications (though hints would be useful). It's not trig's job to teach me physics - that's a job for physics. But I need trig as a foundation.
I'm not sure that I answered your question, though...
But that's my point: it's not really about triangles; that's just a rather mundane application. Trig is really about complex exponentials that solve f'' + f = 0.
It is about triangles. The idea that sine is the solution of this ODE for f(0)=0 and f'(0)=1 is quite modern.
I would say a course on trigonometry usually covers (my experience):
trigonometric functions
exact value of them for the angles 30, 45, 60, 90 ... degrees
Trigonometric formulas for the sum and difference of angles. A formuka for the double and the half angle.
Law of sine and law of cosine
Lots of relations derived from the Pythagoras theorem (sin^2+cos^=1)
how to solve trigonometric equations
With all this, you are equipped to completely determine a triangle, knowing some of its and the length of some its sides. As as application, I was taught, how to measure heights and distances provided you can measure angles.
Thus, without trigonometry, it would be fairly hard to take a course on analytic geometry.
Now, how would the course be enhanced by introducing sine as the solution of an ODE?
I think I am missing something because, I am unable to see why it is huge burden to introduce sine and cosine without their rigorous definition. At which age, are students taught trigonometry? And what does a course on trigonometry covers? What would you think they would be able to do without it?
When we were introduced the sine and the cosine function, we were already familiar with Thales theorem, so therefore we could show that this ratio was a constant.
I am quite sure historically as well sine and cosine predate the more formal construction of those functions, be it as a series, solution of an ODE or inverse of arc sin (and this defined as an integral)...
I think I wasn't clear in saying that I believe the current pedagogical value of trigonometry is in giving students a brief familiarity with the trig functions when they see it again in the context of physics or engineering. Or standardized testing. I think those are the likely scenarios where students are going to be seeing relevance in trigonometry.
What other foundation or learning pathway do you see trigonometry serving as? Somebody else mentioned that it gives students a sense of applications, so they know that calculus is not for nothing. So then I question: what applications? And I pose, how about statistics?
It may be because of your math education but I was introduced to trigonometry in junior high in China. I had, and my classmates had, no trouble understanding them as functions of angles coming from ratios. It is the glue that binds circles and triangles and squares and ... . By high school analytic geometry greatly expanded their scope and use. This is the problem I saw in American high school when I moved to US: shallow introduction to mathematical topics made them vapid and jejune. Ancient Greeks were enthralled by trigonometry, ancient Egyptians built Pyramids and ancient Chinese built great dams with trigonometry. Calling it practically useless and pedagogically only useful as a prep for calculus is going too far.
My introduction was through model rocketry. Somewhere along the line, Al-Biruni's method for finding the radius of the earth was brought up. That was in the early '70s, in what would be "junior high" in the US (elementary school in my part of Canada). It was always about right triangles, not periodic functions, at the beginning, which makes a whole lot of sense - trigonometry was both useful and used for a whole lot of years before calculus was invented. And like logarithms in the pre-scientific-calculator days, there was a point where one turned to tables for practical reasons without thinking of the table values as "magicical" - we were taught how to calculate intermediate values to the limits of practicality. Is there any practical sense (a sense that would be useful for people who would be entering the trades track) in being able to calculate much more accurately than you can measure angles?
