61–70 of 89 posts
Earlier quoted context omitted.
> An integral is the inner product of a function and a suitably-dimensioned unit. Does that mean that ∫f(x)dx is f.(dx, dx, …) = f(x_0)·dx + f(x_1)·dx + f(x_2)·dx + … for all x in the domain?
Integral[f(x)g(x)dx] defines an inner product on a space of functions (glossing over exactly what functions) on an interval. So Integral[f(x) dx] is the inner product (f, 1), where "1" is the constant function g(x) = 1 on the interval.
Earlier quoted context omitted.
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 a…
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…
But even a disembodied being of pure reason might eventually discover continuity via logic->topology.
Earlier quoted context omitted.
I've seen an impl somewhere where the (int => rat) bit from the Isabelle/HOL design was (stream rat). That's about the best I have at the moment.
BTW, in what sense is that coinductive? For what functor is it a final coalgebra?
Earlier quoted context omitted.
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 a…
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…
Earlier quoted context omitted.
> 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 kind…
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…
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...
Earlier quoted context omitted.
BTW, in what sense is that coinductive? For what functor is it a final coalgebra?
Streams are coinductive, thus reals are the type of final coalgebras of (rat * -) which satisfy the Cauchy condition. So something like (Sigma (mu (rat * -)) isCauchy).
Earlier quoted context omitted.
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 co…
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…
Earlier quoted context omitted.
Integral[f(x)g(x)dx] defines an inner product on a space of functions (glossing over exactly what functions) on an interval. So Integral[f(x) dx] is the inner product (f, 1), where "1" is the constant function g(x) = 1 on the interval.
So ∫f(x)g(x)dx = (f.g)·dx?
Earlier quoted context omitted.
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'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|.
Of course that assumes that arclength is well-defined. The standard approach to which is, of course, Calculus.
Earlier quoted context omitted.
> 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 kind…
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 wa…
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.