Rachel: hi jason again
 
me: hi
 
Rachel: thanks for your program! it helped a lot
 
me: You're welcome
 
Rachel: do you have a little bit of time to help me with math?
  
test tomorrow, i dont get it
 
me: Maybe a few minutes, but not much time
 
Rachel: yay thianks
 
me: What are you having trouble with?
 
Rachel: solving trig functions algebraically
  
with no calculator
 
me: you mean like http://www.sosmath.com/trig/Trig5/trig5/trig5.html ?
 
Rachel: i don't think so...
  
like 2sin2x= rad3
 
me: If you know the your 45-45-90 and 30-60-90 triangles, you can solve sin x = \sqrt{3} / 2
 
Rachel: what's the 45 45 90 one again?
 
me: it's an isosceles right triangle. just use Pythagorean theorem
 
Rachel: rad 2, rad 2, 2?
 
me: sure
 
usually expressed as x:x:x\sqrt{2}
 
Rachel: x/sqrt2?
 
me: why?
  
*what?
 
Rachel: no
 
that's what i was asking
 
me: no, \sqrt is the square-root command in LaTeX
 
Rachel: i know
  
why is the third side x/sqrt2?
 
me: x\sqrt{2} = x * \sqrt{2}
 
Rachel: ohh ok
  
ohh i see
  
and then the other triangle is x, xrad3, 2x?
 
me: yes
 
Rachel: ok, with the example u gave
  
sinx = cos30
  
?
 
me: so what's x, then?
 
Rachel: rad3/2….
 
me: no, that's sin x
 
Rachel: how do i find it then?
 
me: draw the triangle, find an angle in the triangle whose sine is \sqrt{3}/2
 
Rachel: what do u mean?
 
me: how did you get sin x = cos 30?
 
Rachel: ohh
  
sin 60
 
me: ok, so now solve 2sin2x= rad3
 
Rachel: why 2x?
  
why is it 2sin2x?
 
me: because that's what you asked me to start off with :-P
 
Rachel: oh ok
 
i'm still confused...
  
idk how
 
me: can you solve 2sin y = rad 2?
  
sorry,
2sin y = rad 3
 
Rachel: sin60…but i don't think that's the right answer
 
me: so what's y?
  
or are you saying that y = sin 60?
 
Rachel: no
  
i'm not saying that
  
but idk how to find what y is
 
me: can you tell me what sin y is?
 
Rachel: rad3/2
 
rad(3)/2
 
me: can you give me a number such that the sine of that number is rad3/2?
 
Rachel: 6-
  
60
 
me: ok, so y = 60 is one solution to the equation 2sin y = rad3
 
Rachel: why?
 
if siny = sin 60, does that mean y=60?
 
me: (another solution is 60 + 360, or 60 + 360 * 2, or 60 - 360)
  
It means that y = 60 is one solution.
  
(the full solution set is y = 60 + 360 k for all integers k)
 
Rachel: right
 
me: so give me a solution to 2 sin(2x) = \sqrt{3} / 2
  
er 2 sin(2x) = \sqrt{3}
 
Rachel: is x=30 one answer?
  
x= 30 + 360k?
 
me: x = 30 is one answer
 
you missed some answers in your general solution, though
  
(why did you divide only the 60 by 2, and not all of y?)
 
Rachel: huh?
  
would it be x = 30 + 180 k?
 
me: how'd you get 30?
 
Rachel: yeah divided by 2
 
except it says solve in terms of pi, so would it be x= (π/6) + 2πk?
 
me: which equation were you solving?
 
Rachel: 2sin2x= √3
 
me: why 2πk?
 
Rachel: period of sin?
 
me: solve 2sin y = rad3, in radians
 
Rachel: or would it only be πk because divided by 2
 
me: figure it out :-P
 
Rachel: y=π/3 + somethingπk
  
idk how to figure it out
 
me: what can you add to y which leaves sin y unchanged?
 
Rachel: 0?
 
me: (what's the smallest positive nonzero number?)
 
Rachel: oh
  
:p
  
.00000000000000000000001
  
jk
  
idk
 
me: Nope, that changes sin y
 
it changes it by about .00000000000000000000001
 
Rachel: 45?
  
lol
 
me: 45 radians?
 
Rachel: π?
 
me: go get graph paper, and plot sin x from x = 0 to x = 4 π. Then draw the point corresponding to (π/3, sin(π/3)), and then think about my question again
 
Rachel: oh!
 
me: and make your graph accurate
 
Rachel: it's pi
 
me: don't answer me now, go graph it
 
Rachel: i graphed it in my head
 
me: graph it on paper, carefully
 
and label 0, π/3, 2π/3, π, 4π/3, 5π/3, 2π, 7π/3, 8π/3, 3π, 10π/3, 11π/3, 4π
 
Rachel: i dont getthis part
  
Then draw the point corresponding to (π/3, sin(π/3))
 
me: draw the point (0, 0)
 
Rachel: ok
 
me: draw the point (π/2, 1)
 
Rachel: didnt draw π/2...
 
me: guess
 
it's right in the middle of 0 and π
  
:-P
 
Rachel: ok
  
lol
  
then what
 
me: now draw the point (π, sin(π))
 
then draw the point (π/3, sin(π/3))
 
Rachel: sinπ?
 
me: what about it?
 
Rachel: how do i draw π, sinπ?
 
me: what's sin(π)?
 
Rachel: 0
  
wait no
 
me: yes
 
Rachel: yes
 
me: can you draw (π, sin(π)) now?
 
Rachel: yes
 
me: why couldn't you before?
 
Rachel: i didnt realize/connect that sinπ=0
 
me: what did you think sinπ was?
 
Rachel: idk
 
me: well figure out what you thought it was. Try to recreate what your thought processes were when I told you "plot (π, sin(π))" and you said "I don't know how" or "I don't know where that is"
 
Rachel: sin(π/3) = √3)/2?
  
i just had no clue before
 
wait why then draw the point (π/3, sin(π/3))
 
me: That's not the question I asked you. I want to know what you were thinking when I told you draw (π, sin(π)).
  
(we'll get to (π/3, sin(π/3)) soon)
 
Rachel: i wasn't thinking anything
  
i had no clue what it was
  
i wasn't thinking it was something else
 
me: That's the wrong way to approach math. Have you ever done stream-of-consciousness writing? (/do you know what it is?)
 
Rachel: yes, we did that today
 
but it's not on the test
  
no
  
idk
  
english
  
we did it in english today
 
me: Yes, that's what I'm talking about.
 
Rachel: ok, well i understand now that it''s not good to think that way, but i need to be able to think tomorrow on my test, so please dont waste both of our time trying to figure out why i wasnt thinking
 
me: I'm not trying to figure that out. I'm trying to teach you to react to math by thinking (in a useful way), instinctually.
 
Rachel: ok, good idea, just not right now please
 
me: Every time you ask me a (new) math question in the future, I'm going to have you do that for a minute or two on every concept related to that question. (I only want sentences related to the concepts, so keep going with stream-of-consciousness until you've actually thought about the concept for 2 minutes and have things to say about it. I won't care if they're things like "to plot things, I have to draw lines on paper.")
 
Rachel: ok
 
me: anyway, I told you to draw (π/3, sin(π/3)) for the following reason
  
did you notice that all the points I've mentioned are on the sine curve?
 
Rachel: yes, especially considering you only mentioned 3 points
 
me: and, btw, go draw (π/3, sin(π/3)))
 
Rachel: well 4
  
but idk how to draw that
 
sinπ/3 = √3)/2
  
which isnt on my y axis
 
me: but it is on your sine curve
 
Rachel: therefore, since i have no brain, i cant do it
  
lol
 
me: so it's in the same place on the y-axis that the sine curve is at that point
 
Rachel: alright
  
did it
 
me: now find the next point on the sine curve with the same y-value as that point
  
and tell me what that point is
 
Rachel: 2π/3, whatever the y value of the other point was
 
sinπ/3
 
me: ok, and what's the one after that?
 
Rachel: √3/2
 
me: yes, sinπ/3 = √3/2
What's the next point?
 
Rachel: 7π/3
 
me: and the next one?
 
Rachel: 3π
 
me: you drew your picture wrong. sin 3π = 0
 
Rachel: no i drew my picture correctly
  
i read it wrong
 
8π
  
/3
 
me: ok. so how far to the right do you have to go to be guaranteed of finding another point like this?
  
(how far do you have to go to be guaranteed of finding a point at exactly the spot you pick?)
 
Rachel: 5π/3
  
k
  
5πk/3
 
me: at exactly the spot you pick?
 
Rachel: what?
  
0?
 
me: what if you go 5π/3 to the right of π/3?
 
Rachel: 6π/3
  
=
  
2π
 
me: ok, so what are all the solutions to sin(x) = \sqrt{3} / 2?
 
Rachel: it was sin2x
  
= √3/2
 
me: I know. Solve this one first.
 
Rachel: oh
  
ok
  
um
 
x= π/3 + 2πk
 
me: anything else?
 
Rachel: k= all integers?
 
me: what about 2π/3?
 
Rachel: what about it?
 
me: is it a valid solution?
 
Rachel: um
 
yes
 
me: is it of the form π/3 + 2πk?
 
Rachel: ohhh right! i think ir remember that now!
  
2 equations!
 
me: ... you shouldn't need to remember it :-P
 
Rachel: x = 2π/3 + 2πk or x = π/3 + 2πk
 
me: ok. so now what are the solutions to sin(2x)
= √3/2 ?
 
Rachel: x = π/6 + πk or x = π/3 + πk
 
me: good
 
Rachel: yay! i understand that now!
 
me: good
 
Rachel: but how do i do that for tan?
  
tan confuses me
 
me: why does it confuse you?
 
Rachel: actually i think i can do it
  
um
  
next sub-topic
  
well question about sub-topic
  
what precisely does it mean "modeling with trig functions"?
 
me: If you move a laser pointer back and forth for Toby to look at, and then plot the position of that as a function of time, it'll look almost like a sine curve. Noticing that it looks almost like a sine curve, and figuring out what sine curve it looks most similar to (what a, k, c, d are for a sin(k x + c) + d), that's and example of "modeling with trig functions"
 
Rachel: ohh ok thanks
  
next
  
affect of A, B, C, and D on circle and trig function
 
i know how it affects trig funvctions, but not sure about the circle
 
me: I'm not sure what A, B, C, and D you're talking about
 
Rachel: like x=asinB(x-c) +d
 
me: x = a sin( B( x - c ) ) + d ?
 
that's not a circle, nor a graph
 
Rachel: that's the parametric equation, right?
 
me: no. a parametric equation is like x = f(t), not x = f(x)
 
Rachel: right
  
um
 
me: and you also need a y = g(t)
 
Rachel: it's the equation of the trig function, no?
 
me: I don't know what you're talking about
 
Rachel: are you familiar with the equation?
 
me: with x on both sides, no
 
Rachel: oh
  
ok didnt mean that then
 
me: have you looked at that section in your textbook?
 
Rachel: meant y= asinb(x-c) +d
  
and not yet, i figured i wanted to ask you while you were offering the help
 
me: I don't know what you're asking me to do. If you have a specific question, ask. Otherwise, go look at your textbook/notes and let me get back to work
 
Rachel: ok, will u still be here though for me to ask about specific questions?
 
me: maybe
 
Rachel: alright, well thanks so much for your help!
  
wait