It was asking about velocity at a given point, (i.e. instantaneous rate of change) but I'm not sure how I was supposed to solve that without you know, calculus or just guessing
well the slope is changing over time >.>; and I would use that fact plus the one bobble just said (which is true for a sin (x) graph... but this is a bit taller than a normal sin :) )
if this isn't obvious to you, that suggests there's something suboptimal in your understanding of functions somewhere around here that may cause trouble in the future. I don't mean that there's some fact you don't know, just that it seems like in some way you may not be thinking about things right. I am aware that this is a super-unhelpful thing to say.
("this" = "tan 2x = sin 2x / cos 2x, given what you know about the relationship between sin, cos, tan")
that's the thing that should feel obvious and if it doesn't it may be worth staring at it for a while and thinking to see whether it becomes more obvious
sometimes you do need to think of it as meaning a specific value. If someone writes down x^3-x^2+x-1=0 and says "solve for x", they're asking you what particular values of x make the thing true.
It's an unfortunate fact that the underlying idea of algebra -- you can give names to things -- turns out to have two quite fundamentally different uses (1. expressing that something is always true by using a variable to stand for "anything"; 2. a notation for when you want to say something about particular things without yet saying exactly what they are).
I suspect a lot of people are at least a bit confused by this.
I agree - it's one reason that I think that they should be given separate letters when being set up... then you can think of them separately until that becomes second nature :/
It would be an interesting experiment to try presenting this stuff in schools with explicit quantifiers everywhere -- quantifiers are things like the ∀ for "for all", and there's another one with an upside-down "E" meaning "there exists" -- and see whether it ends up being less confusing or more confusing.
It happens that for an A that's the same as reflecting about a horizontal line, and for an E it's the same as reflecting about a vertical one, but I'm pretty sure the right way to think of it is as a half-turn in both cases.
I'd personally find it more helpful if we used different variables to mean different things, but inevitably will probably become simplified because people are lazy
like my proofs for example
The variables magically disappear when they ask me to simplify a trig function
(mostly because the variables themselves weren't being manipulated at this point, so I could just drop the "x" or the "u" or whatever. Bad habit though)
One remark about that "gradient of a sine curve where it crosses its midline" thing earlier. This is the same as asking about the gradient of the graph of sin t at t=0, of course. And, provided you express angles in radians which everyone always should, the key fact is that sin t ~= t when t is small. And that's "obvious" because sin t is the y-coordinate of a point on the unit circle that's moved anticlockwise through a distance t, and when t is small [... continues]
... that bit of the unit circle looks a lot like the straight line x=1, and moving a distance t along that line obviously takes you to (1,t) whose y-coord is just t.
Ha. Feel free to ignore it if it was way too confusing.
Radians are better than degrees because (1) the definition is super-simple and has no arbitrary magic numbers in it and (2) lots of facts about trig functions and angles and whatnot are simpler when expressed that way.
Definition of cos and sin: if you start at (1,0) and walk a distance t around the unit circle (anticlockwise, by convention), then you end up at (cos t, sin t). Note that I never used the word "triangle"; trigonometry is about circles, not triangles.
Everyone knows pi, not everyone knows tau, so use pi.
It might have been better if we'd gone with tau instead of pi originally, but we didn't and the world is not going to change.
Some things come out nicer in terms of pi, some things come out nicer in terms of tau, some things would actually be nicer if the thing we had a special symbol for were pi/2.
Here's how I feel about degrees, by the way. Imagine that we'd never invented exponents, but we have a function called sqr that does something squaring-like. The idea is that sqr operates on distances, and its output is an area converted into a distance, so what it does depends on your units, and it's conventional to use units called squrbles with the property that sqr(180 squrbles) = 180 squrbles. [... continues]
And then any time you or I would just square something, in this hypothetical nightmare-world you're supposed to express whatever-it-is in squrbles, and since with everything in squrbles sqr(x) = x^2/180 you end up with a lot of gratuitous factors of 180 everywhere in what would otherwise be nice simple mathematical facts.
And your calculator has a button that does sqr, but by default it assumes you're working in squrbles, so when you enter 1 and hit sqr you get 0.00555555... or whatever it is.
And then one day someone says "wait, why don't we adopt units where sqr(x) just equals x times x?" but everyone says no, we're used to squrbles.
I think the reason we have degrees is that 360 is a nice round number that approximately equals the number of days in a year, and the orbit of the earth around the sun is roughly circular.
Anyway, I claim that although there isn't a formula for (say) sin or cos that's as simple as x*x, they are functions that behave much more nicely when you work in radians, so much so that we shouldn't think of angles as having units at all; they're just numbers, and the functions we use them with are defined "in radians".
e.g. sin x = x/1 - x^3/(1x2x3) + x^5/(1x2x3x4x5) - x^7/(1x2x3x4x5x6x7) + ... -- the series goes on for ever, but for any value of x the terms eventually get small enough fast enough that it all adds up to a definite value.
Measurements are always best described from a base identity. Angles are "how far around the most basic circle do I have to move to push this line's intersection point to the other line's intersection point?" Degrees are that, as are radians. It's just that degrees are "how many 1/360ths of that circle do we pass?" and radians are literally "how far did we travel"
incidentally, the thing about years and circles and whatnot isn't conspiracy theory, it's a perfectly serious suggestion. Blame it on the ancient Babylonians.
I got stuck on "what the heck is cos^2(2x)" until I remembered our little conversation about the "x can be anything"
I'm still not sure how my teacher simplified it down. @GarethMcCaughan And what did you mean by cos^4 (x)? That seems to be the route my teacher took, but I can't read her handwriting
@Sciborg Perhaps you can understand why I'm confused? :P
Amusingly, on the very first line she already had that common factor -- it's the "cos^2 x" at the start -- but she expanded it all out, and then decided to undo it a couple of lines later.
