WEBVTT
Kind: captions
Language: en
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Okay, so if we just ask where sine of x is a half, that would be a minimum competence question and you are suppose to know where sine of x is a half
00:00:19.540 --> 00:00:23.480
Okay what if we do sine 3x equals a half?
00:00:23.780 --> 00:00:27.100
and we tell you that x is between zero and pi
00:00:27.700 --> 00:00:29.780
Well half as we say that
00:00:30.620 --> 00:00:33.180
so if it's between zero and pi
00:00:33.800 --> 00:00:36.280
well you just have to play with it a bit
00:00:36.280 --> 00:00:38.060
you say to yourself, well first of all
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3x has to be pi over 2. Lets do it in degrees though.
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If you first do it in degrees, okay?
00:00:45.860 --> 00:00:47.380
Then convert in the end
00:00:47.700 --> 00:00:49.460
so you say to yourself well
00:00:50.220 --> 00:00:54.860
where is sine equal to half, whats equal to have in the first quadrant?
00:00:55.580 --> 00:00:56.300
30 degrees
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then you go over here to the second quadrant
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and its also
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equal to a half because
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right we know what they say about the albany students
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so its true in the second quadrant
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and we can keep going around
00:01:13.600 --> 00:01:17.680
so instead of staying true to 30 degrees, well put 150 degrees
00:01:17.680 --> 00:01:22.400
and its going to be true to more places in a matter of fact its true there's an infinite number of them
00:01:23.700 --> 00:01:25.220
so if we add 360 degrees
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so let's just test this out for now
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so if we divide
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so this is where 3x is equal so if I divide everything by 3
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you get 10 degrees
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50 degrees
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130 degrees
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So those are all less then a 180 degrees because pi is a 180 degrees right?
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so all of those work
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Are there more answers? well lets see
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I have all the way round here and that was 390. We keep going
00:01:51.980 --> 00:01:55.060
Thats 510 degrees, how did I get these numbers?
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I add 360
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I add 360
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Divide that by 3 and I gonna get 170 degrees
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Thats still less then 180 right?
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The next one I come up with is gonna more than 180
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Because I'm already at 170
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So now I don't need degrees, I just convert it into radians. How do you convert 10 degrees into radians?
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Have any ideas?
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Multiply it by pi/180
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Take 50 and multiply it by pi/180
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Take 130 times pi/180
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and 170 times pi/180
00:02:32.960 --> 00:02:34.160
you can stop there
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thats a hundred perfect answer
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You dont have to reduce those if you don't want to
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so one of the ways you can make your life easier is to do the degrees first and switch it into radians at the end
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Well you have to find all angles until you get to 180 degrees
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So you just going until you get 180 and stop
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Its not gonna be that many of them. We're not gonna have 20 of them
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20 is a lot
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however you shouldnt assume there is only 1
00:03:09.980 --> 00:03:12.960
you should assume -- there's 3x
00:03:13.100 --> 00:03:14.820
the clue is 3x
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360 degrees
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Normally if its 360 degrees, its two answers
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because 3, 2 times 3 is 6x
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So for 180 degrees its not necessary half of that
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Can i do another one of these? yeah!
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how do I know where to start off? Where does sine equal a half?
00:03:35.740 --> 00:03:40.060
Sine is the a half at 30 degrees, thats how I know to start there, right?
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okay, we understand what i did? So you say to yourself where does sine equal 1/2
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30 degrees
00:03:47.200 --> 00:03:47.860
150
00:03:48.620 --> 00:03:51.820
and keep going okay? and then when you divide by 3
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then you keep going until you run out of numbers less than 180 degrees
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Go ahead -- if its worth it, i'll do it
00:04:05.360 --> 00:04:06.720
sure lets do that one
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2pi? now im just gonna -- okay
00:04:15.280 --> 00:04:17.040
we have cosine of 3x
00:04:18.100 --> 00:04:19.780
equals square root of 3/2
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x is between, this is one of the one from my practice stuff right?
00:04:28.000 --> 00:04:30.880
So lets try it for 2 minutes, do it in degrees
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so ask yourself, wheres cosine, uh
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where is cosine equal to radical 3/2 in degrees
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Try to find all the places
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the clue is
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thats 3x so 3 times 2, that 6 so 6 answers
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alright i think that long enough
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so cosine of 3x, so wheres cosine equal radical 3/2
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well lets see
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cosine is positive in the first quadrant
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and again down here on in the fourth quadrant
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cosine of 30 degrees is the square root of 3/2
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and now youre going to go all the way to the fourth quadrant
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so thats 330 degrees
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so i have 3x equals 30 degrees
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or 330 degrees
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so if i was doing graphs
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okay ill show it to you have to graph another way in a second
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so now lets go around this circle the full time, 360 degrees and and then go another 30
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and you go all the way around to here and thats 690 degrees
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we all have paper to work on
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theres lots of blank space
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and lets divide by 3 and see how we do, thats 10 degrees
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110 degrees
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130 degrees
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230 degrees
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so far i have all these possibilities, have to go all the way to 360
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i mean we might spot a pattern but
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lets add another 300 degrees to each of these. so i get 750
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and 1050
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I divide by 3 and i get 250
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and 350
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thats what you get
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how do i know im done? because the next one is going to be more than 360 degrees
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and now i want to convert these into radians
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I multiply by pi and divided by 180
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so 10pi/180
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or you can make it pi/18
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okay and now i get 110pi/180
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also known as 11pi/18 and so on
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isnt that a nice fast way to do these?
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now why is that happening?
00:07:02.820 --> 00:07:04.980
so what does cosine3x look like?
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cosine3x looks just like cosinex
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except I squished it
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by a factor of 3
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so thats 2pi
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so thats 2pi/3
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and 4pi/3
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so when i want to find what equal to the square root of 3/2. i have 1,2,,4,5,6 solutions. Thats the theory behind it
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the rest of the theory you are thinking will it show on the midterm
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but thats why you get 6 answers, because you cross the graph 6 times
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okay so lets do something else
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some other type of question. notice i covered that back up, i dont want you spending so much time thinking about it
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do a graphing problem, sure!
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im gonna make one up
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you could but im gonna make one up
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lets just start off with something like that
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alright lets do this one
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so i have some changes coming here
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do you get your test back, oh thats a good question. You get your midterms back in recitation
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you dont go to recitation you dont get your midterm back
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i end up with a box at the end of the semester, you know what i do..
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so i have two things going on here. ive go this 4, i got the 2 and i got pi/6
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so this tells me, move my curve up two
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and this tells me stretch the curve by 4
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so you can stretch it by 4 or move it up 2
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thats really what happens
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you start at
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you start going up 4 and down 4 and then you move it up2 and now you are here okay?
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so you go up 4 from here and down 4 from here
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this tells me move pi/6 to the right
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i havent shrunk the curve at all
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so the length of the curve is gonna be 2pi+6
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its perfectly fine leaving it like that
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you can turn that into 13pi/6
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why not? i dont care. if you do degrees and then you convert back at the end
00:10:03.840 --> 00:10:06.540
you just take the 30 and add 60 to it
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alright so your sine curve
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its gonna look like that okay. it should cross in the middle, dotted line
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all you gotta do
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thats half way, which would be
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7pi/6
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you want to find half way between the 2 numbers
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take a 1pi/6 and the 13, so you take the 1 and the 13
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add them together. that makes 14. divide it by 2. you get 7
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anytime you want to find the half of 2 numbers, you add them and dived them by 2
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thats all you had to do, thats the whole curve
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isnt it easier
00:10:54.120 --> 00:10:57.800
well im either gonna graph first. so i dont want to start avariation on that yet
00:11:19.340 --> 00:11:21.100
alright so how do we do this
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well first
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very important, we have to pull the 3 out
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okay so you have to do that
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and notice when you take the 3 out of the x
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you are taking the 3 out of pi. and that makes pi/3. and how do i know thats right
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3 times pi/3, will give you pi
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dont over think these things
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now we can do the same kind of thing we did before
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were still shifted up 2
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still go up 4
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and down 4
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we start pi/3 to the right
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and you add 2pi, to get to the end of the curve
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so pi/3
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plus 2pi
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is 7pi./3
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so it right on 7pi/3
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here comes the fun part
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okay lets say the middle is 4pi/3
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now what about this 3? oaky
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take your values at the x axis and dived them by 3
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so this now becomes pi/9
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4pi/9, 7pi/9
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thats how the factor you pressed the curve by 3
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all the numbers get divided by 3
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that is your horizontal compression
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do you want me to make it harder? cause i can make it harder i thought this was pretty clear
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no the 3 does not cancel out the 2/3
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either way
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we just need to be multiply by x because whats happening is that is in parenthesis
00:13:40.360 --> 00:13:48.280
when youre told, you have a 3 multiply by x you c=are compressing it by 3. We did this when we talked about transformation
00:13:48.460 --> 00:13:54.760
because you need, you take pi/9 and multiply it by 3 and you get pi/3
00:13:55.320 --> 00:14:00.120
so thats why you can divide a number by 3. because this multiplies in that
00:14:03.180 --> 00:14:05.580
good so lets do a piecewise function
00:14:07.080 --> 00:14:08.720
it was 40 divided by 4
00:14:18.420 --> 00:14:23.700
you would like a word prolblem? sure could we do the ones that were in the packet?
00:14:23.700 --> 00:14:25.940
the packet problems? you do yes
00:14:35.760 --> 00:14:37.040
then ill make one up
00:14:38.360 --> 00:14:40.200
not easy to make up actually
00:14:53.700 --> 00:15:00.240
alright do you want to do senor, or do we wnat to do like light that does the electricity bill
00:15:03.160 --> 00:15:10.200
do we want rodrego or do we want or do we what the electricity one?
00:15:12.920 --> 00:15:21.280
we will do rodirgo, alright he is driving from lancing to columbus
00:15:21.540 --> 00:15:24.980
lancing is a street in michigan, colombus is in ohio
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just in case you were wondering
00:15:37.020 --> 00:15:42.380
rodrigo drives at 50 miles per hour for 2 hours
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my god speed up
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we have to figure out how far Rodrigo goes
00:15:48.580 --> 00:15:52.180
so those of you who love physics, remember your 7th grade
00:15:56.400 --> 00:15:57.360
word problems
00:16:00.900 --> 00:16:03.060
rate times time equals distance
00:16:03.060 --> 00:16:05.800
or if you read it the other direction, distance equals time times rate
00:16:06.920 --> 00:16:07.940
youll get drt
00:16:11.380 --> 00:16:18.520
so to figure out how far rodrigo goes, you take how fast hes going and figure out how long he goes for. 50 mph is 50 miles,
00:16:18.520 --> 00:16:20.140
fifty miles times 2 hours is 100 miless
00:16:20.900 --> 00:16:21.460
and so on
00:16:21.680 --> 00:16:24.880
so for the first phase of the journey
00:16:29.400 --> 00:16:35.240
you want to find out where rodrigo is, you just multiply 50 by how long he has been driving
00:16:35.760 --> 00:16:37.760
and you keep this up for 2 hours
00:16:42.480 --> 00:16:44.540
but then in 2 hours, something happens
00:16:46.740 --> 00:16:48.100
it shifts locations
00:16:48.940 --> 00:16:54.220
how do you have 2 different equal signs. dont worry about it for now, ill explain
00:16:54.220 --> 00:17:00.940
there are 2 hours he magical stomps on the brakes, so he is only goign 30 mph
00:17:05.240 --> 00:17:07.720
now hes on 25a at 5 oclock
00:17:08.600 --> 00:17:11.320
he would beat long island record anytime
00:17:11.940 --> 00:17:19.900
now hes only driving 30 mph but be careful, how far has rodrigo gone after 2 hours? hes gone a 100 miles
00:17:20.280 --> 00:17:23.100
so he starts a 100 mile mark
00:17:24.860 --> 00:17:25.900
and now 30 mph
00:17:26.400 --> 00:17:32.080
its not t hours its t minus 2 hours, because you want to take away those first 2 hours
00:17:33.300 --> 00:17:38.380
because at 2 and a half hours you dont wan 2 1/2 you want a half here
00:17:39.220 --> 00:17:40.880
plus the original 100
00:17:41.300 --> 00:17:42.420
and thats at dawn
00:17:45.800 --> 00:17:46.840
for 2 more hours
00:17:47.960 --> 00:17:49.880
how i do know for 2 more hours?
00:17:50.400 --> 00:17:51.840
he has to get 160 miles
00:17:51.840 --> 00:17:54.620
hes already gone a 100 miles so he has 60 left to go
00:17:56.000 --> 00:17:58.800
30 miles an hour to go 60 miles takes 2 hours
00:17:59.880 --> 00:18:02.480
now i couldve put an equal sign here
00:18:02.840 --> 00:18:04.440
or here it doesnt matter
00:18:05.140 --> 00:18:10.260
its sort of customary to put the equal sign that one but it doesnt matter
00:18:10.520 --> 00:18:12.280
cause he instantly shifts
00:18:12.960 --> 00:18:16.860
or stops on the break. now he stops at preise for a while
00:18:18.300 --> 00:18:24.220
so now we have a 160 miles, so if you want to know where is he you would say hes gone a 160 miles
00:18:25.320 --> 00:18:29.420
and that last for an hour. an hour
00:18:30.440 --> 00:18:32.920
i dont know, I guess shes worth an hour
00:18:33.160 --> 00:18:37.460
I dont know how attractive she is she have a good conversation blah blah blah
00:18:39.680 --> 00:18:44.080
think they are having a good meal, maybe its a happy meal from mcdonalds
00:18:46.440 --> 00:18:48.320
now he says oh boy am i late
00:18:49.260 --> 00:18:55.500
so he has to get through the remain distance to columbus and he is now is live at 65 mph
00:18:58.660 --> 00:19:01.540
just so you know the speed limit is the limit
00:19:01.760 --> 00:19:04.920
so you good under the speed limit, its not a suggestion
00:19:05.700 --> 00:19:06.260
its a law
00:19:07.760 --> 00:19:12.800
well i think its a suggest its like hey how about 65, you go sure 65 sounds good
00:19:13.360 --> 00:19:16.640
and for that you have to take away the first 5 hours
00:19:21.200 --> 00:19:25.900
and you can keep that up until he gets there
00:19:26.080 --> 00:19:30.140
so he has to go those last 65 miles, im sorry 90 miles
00:19:32.800 --> 00:19:37.020
thats 65 mph plus another 5 miles, you can leave it like that.
00:19:39.980 --> 00:19:42.140
that is an equal sign and hes done
00:19:45.740 --> 00:19:50.320
the key to a lot of these problems you have to think about what is going on in each space
00:19:50.800 --> 00:19:56.520
tell yourself of story, you want to break up every different interval
00:19:58.500 --> 00:20:05.020
how did i get the 90/65, alright so you got 160 miles he has to go 250 miles and he 90 miles left to go
00:20:05.560 --> 00:20:09.160
and remember rate times time equals distance. so 65
00:20:10.680 --> 00:20:14.720
t equals 90 miles so the time it is going to take is 90/65
00:20:15.600 --> 00:20:18.860
plus 90/65 passed the 5 hours
00:20:18.860 --> 00:20:22.700
you are going to do this in physics but its going to be much harder
00:20:23.640 --> 00:20:24.140
yes?
00:20:32.860 --> 00:20:39.960
yes i put the 5 hours and added 90/65. thats the remaining time
00:20:56.160 --> 00:20:57.760
add the 5 hours so 3 times
00:20:59.100 --> 00:21:04.640
so in four hours, he stop at preise for an hour, is 5 hours and now he drives from ther
00:21:05.700 --> 00:21:10.780
alright i can try and make up a word problem but its hard to do on the spot
00:21:12.060 --> 00:21:13.980
can we do the electrcity one?
00:21:49.040 --> 00:21:54.160
the electricity is sold for residential customers for a following rate plan
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for the first 250kmph, 1 hour you use of energy
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the first 250kwatt hours
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that a customers use cost 24 cents per kwatt hour
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if we want to find the cost
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look at this wave and you say to yourself the structure the 3 sets of things
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for h starts out at 0 till it gets to 250
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illl move it down a bit
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then from 250 to the next 500 so thats up to 750
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thats the 3 times. the 3 structures
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so the first piece
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is 24 cents
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times the number of hours
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and the after 250 kwatt hours
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how much money does that cost, well youre gonna have to do a little multiplication and youll get 60 bucks
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so for the second piece hes already spent 60 dollars
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and its gonna be 26 times hours after 250
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so far so good?
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wheres this come from?
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so 60 dollars is 24 cents times the first 250 kwatt hours
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so you just do 24 times 250 and you work the decimal place
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thats 60 dollars then its 26 cents
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times the number of hours after the 250 kwat hours
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you can just add it you dont actually have to figure out what 60 is but be careful. youll have lost of scrap paper
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you should be able to do it
00:24:26.240 --> 00:24:30.400
i dont thin the numbers are particular hard that we came up with
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and now what happens
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so how much is the second time it cost you
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well its gonna be 500 kwatt hours
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at 26 cents a kwatt hour
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thats a 130 dollars
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so now its a 130 bucks
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a 190 dollars because you spent 60 and then you spend another 130 and get a 190 dollars
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plus 28 cents
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times everything minus 750
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how come it never equal 750
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now it can equal too
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you know theres one in every crowd
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pointing out all my flaws this is terrible goo thing were not married
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alright next one
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I can try and make one up but there hard to make up on the spot
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well how about a inverse? are we ready to do some? lets practice one
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there you go find f inverse of x
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alright lets do the inverse of this
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so what do we do?
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we write this as y
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and then you switch x and y
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and now we just have to isolate y
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we can cross multiply
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and get 7x, 4y cubed minus 1
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add 1 to both sides
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and you get 7x plus 1
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equals 4y cubed
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dived by 4
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and take the cubed root
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that is your inverse function
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how do we know its an inverse?
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ask yourself what you are doing x
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first thing you is you cube it
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then you multiply by 4, then you subtract 1, last you divide by 7
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so first you multiply by 7, second you add 1 and third divide by 4 and last you get the cubed root
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so you reverse the order and you reverse the operations lol
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thats how you know you are doing it correctly
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you can also check, make up a number and plug it in and see what happens
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this comes out 0. sorry not true, well 3/7 so we plug in 3/7 here and you get 1
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3/7 times 7 is 3. plus 1 is 4 divided by 4 is 1 cubed 1 is 1.
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so you can always check but putting in a number and getting an answer and putting an answer in the other one
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how do you know if the inverse does not exist?
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if you have an even power of x, youre going to havea problem okay?
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the only need of a problem is when you have x squared
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then when you do the inverse you have to plus or minus the square root of x
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how do you get 2 values
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that would be bad
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these are cubed roots there wont be an issue
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lets do another one, lets do a slightly harder one
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lets find the inverse
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what do we do? we switch x and y
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okay what are our steps?
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first you cross multiply
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and we get
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3xy minus 7x
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equals 2y plus 11
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now we are going to group, factor and divide
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first we group, everything with a y goes on one side, everything without a y goes on the other side
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its 3xy minus 2y, 7x plus 11
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now we are gonna factor out the y
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y, 7x plus 11/3x minus 2
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and that is that is the inverse
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okay some people will have the 3xy on this side and the 11 on that side
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so thats why when i wrote the answers up, i gave both possibilities there both correct
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you might of had minus 7x minus 11, everything over 2minus 3x same thing
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either form is fine. you either this or this
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either it doesnt matter
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lets practice domain
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find the domains of both of those
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i dont know if you would need to use notation but you might, i dont remember if we require it or not
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my advice, if youre not sure then first write the answer without interval notations so you can just demonstrate what you are doing even if you screw up with the interval
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make sure its clear that you know what you are doing
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okay whats the domain of the square root of 3 minus 5x
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you cant take a square root of a negative number
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3 minus 5x
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has to be greater than or equal to 0
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move the 5x to the other side
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divide by 3
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the domain you can write either x is greater then or equal to 3/5
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or you can right it in interval notation like that
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so far so good?
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you can not take the square root of a negative number so the domain inside has to be greater then or equal to zero
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okay what about the second one?
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you can take the cube root one out of 1
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so you can either right down all real numbers, all reals, you can use that fun r symbol
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or you can right infinity and minus infinity
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i can take a cubed because itd be 3/5
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you take the square root of 0, thats 0
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alright so you can include that for square root
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i didnt i moved the x to the other side
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alright lets make it slightly harder
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how bout that
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so we look at this, its just like before we got a square root
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so 2x minus 1 needs to be greater then or equal to 0 0
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we have 2x minus 1 greater or equal to 0
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2x is greater then one
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x is greater then or equal to a half
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the denominator can not be zero
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so
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x minus 10 can not be 0. so x cannot equal 10
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so that could be your answer
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or if you want to do interval notation, you start at a half go to 10
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and then you stop at 10 and start again
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without including 10, you can do it that way
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the inverse is fine with me
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we do not specify interval notation, so you should not feel obligated to use interval notation
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but you should be doing a lot of webassign interval notation, you should be able to do this by now
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okay i will do a couple more. then i will kick you guys out and go and sleep
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how bout this one?
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sinx equals 8/9, and this tells me im in the second quadrant
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so i draw a triangle in the second quadrant
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i put in an x
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and i do sine
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now i need to figure out the tangent
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tangent requires opposite and adjacent and i only have the opposite and i need the adjacent
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so i am going to need to figure out the missing side
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so i use the pythagorean theorem
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8 squared plus 2 squared equals 9 squared
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do a little arithmetic
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and ill get b square root of 17
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because 8 plus 9 equals 17
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funny how that works
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works everytime
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plus or minus square root of 17, works everytime
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whats 10 squared, its 100, whats 11 squared
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121. whats 10 plus 11, 21 works everytime
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okay so i want to find tangent, tanx
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well is it 8/the square root of 17? not is negative 8/square root of 17
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why is it negative? were in the second quadrant
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do a composite function
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you want to do f(gx)
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you take gx and you put it inside f
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so fog of f, if you like to call ot that
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square root of2/x cubed minus 1
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you dont have to do anything to that, you can just leave it as 2/x cubed minus 1
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if i want to do g(fx)
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I take fx and put it inside g
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so g(fx)
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is i take the square root of x minus one and put it in ther
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heres what i want you to do. I want you to relax, memorize what you need to memorize good luck on the exam and see you next week