@oAlt That's actually fair. In Korea, you're considered "1" when you're born and gain a year every New Year's, regardless of your birthday. It is indeed possible he can be in 6th grade and thirteen, at least from where he is.
so - you technically don't need to know how the proofs work to use the formulas, but it's useful to get a general understanding of how they work because (1) you can re-derive them if you forget the formulas, and (2) it helps you with similar things
i assume you're asking for help understanding the proof they give
by tilting my head a little to the left (to be precise, angle β to the left), and then dropping a vertical line from the top point down onto the middle line
your method works too! just showing you that there's another way to see it, and it might be clearer visually. i know thinking visually helps me a lot more than equations
so, how that you know gamma is secretly just alpha, can you find the lengths of Q and R?
I have to confess I've always hated that way of proving the addition and subtraction formulas. It only works for angles in particular ranges, and it perpetuates the WRONG idea that trigonometry is about triangles. (It isn't, it's about circles. (cos t, sin t) are the coordinates of a point moving anticlockwise in a circle of radius 1 about the origin with velocity 1.)
no one teaches complex numbers before trigonometry unless you have a dad who was a math professor and wrote a thesis that involved complex equations. so me.
(though I never cared much for proofs - I just memorized this stuff)
So, what happens when you take the point (x,y) in the plane and rotate it anticlockwise through an angle t? Answer: you get the matrix (cos t -sin t; sin t cos t) times (x; y) where ";" means "new line" so (x; y) is a column vector. (Or you can write it out in formulae for old and new x, if you don't know about matrices.)
So now do that for the angles alpha, beta, alpha+beta and note that rotating through alpha+beta has to be the same thing as rotating through alpha and then through beta.
And now setting the relevant things to equal one another immediately gives you the formulae for cos(A+B) and sin(A+B).
Using A and -B instead of A and B gives you the difference formulae too.
No complicated diagrams with possibly-unexpected construction lines; you just do the obvious thing at each step and it works out :-).
Yeah, obviously this is no good if you haven't already covered rotation matrices. But I think the best path to proving the cos(A+B) etc. formulas goes via understanding rotation matrices first. Or maybe complex numbers, which can do something similar a bit more neatly.
Though I think the best way to get there neatly with complex numbers also requires a bit of calculus, which again people don't generally teach before trigonometry though I think they kinda should.
(More precisely: they should, for people who are eventually going to learn both and have an interest in pure mathematics. Of course that's far from being everyone.)
I don't have an answer to that all worked out. But it would definitely be more than one curriculum, doing things in a different order depending on where you're headed and what stuff you're eventually going to learn.
E.g., I think it can be true that if you're going to learn either A or B but not both then you should learn A, and also that if you are going to learn both then you should learn B firts.
In practice, that will probably be decided for them by the education system, on the basis of things like asking them whether they want to do subject X next year, or looking at how well they do in exams. That's not optimal but it's suboptimal in the same way as most things in education are suboptimal.
Not sure. (There's a tradeoff between making decisions that are better for each child and introducing extra complexity to the system, which may reduce the resources available for actual teaching and cause other problems.)
Again, I haven't sat down and tried to devise a complete mathematics curriculum, and doubtless if I did I'd discover lots of difficulties I've never even thought of.
The sort of situation I have in mind with "A and/or B" is this: A is some practically-useful thing that you can learn how to do practically-useful things with by following fairly simple procedures. B is something related that's more general, more abstract, more accurate, etc., but also more difficult. If you learn B first then you will understand A better and not just be following rote procedures. But not everyone needs to know about B, and maybe not everyone is capable of understanding it.
(I don't believe that in general you should learn more abstract/general things first. Often the generalized abstractions are much easier to make sense of once you've seen concrete examples. The best process if you have plenty of time and motivation is probably to start concrete, then look at the bigger picture, then return to the concrete level with better understanding.)
it is! and that's a historical mistake - it's much more about circles than triangles
@GarethMcCaughan yes, i agree that the proof there is not ideal - i prefer the complex number (or rotation matrix) derivation too. but i wasn't sure if north knew about complex numbers, and he did ask about the proof in his book (so i tried to find something close to whatever his book did)
@Deusovi For the avoidance of doubt, I wasn't at all suggesting that you were wrong to explain that proof -- as you say, it's likely the one North has been shown.
