Checking solutions of two-variable linear inequalities example

WEBVTT
Kind: captions
Language: en

00:00:00.590 --> 00:00:04.640
Determine whether the
ordered pairs 3 comma 5 and 1

00:00:04.640 --> 00:00:09.740
comma negative 7 are solutions
to the inequality 5x minus 3y

00:00:09.740 --> 00:00:14.040
is greater than or equal to 25.

00:00:14.040 --> 00:00:17.060
So again, let me just try
each of these ordered pairs.

00:00:17.060 --> 00:00:19.290
We could try what happens
when x is equal to 3

00:00:19.290 --> 00:00:21.720
and y is equal to 5
in this inequality

00:00:21.720 --> 00:00:23.314
and see if it satisfies it.

00:00:23.314 --> 00:00:25.230
And then we could try
it for 1 and negative 7.

00:00:25.230 --> 00:00:27.220
So let's do that first.

00:00:27.220 --> 00:00:29.750
Let's do it first for 3 and 5.

00:00:29.750 --> 00:00:31.970
So when x is 3, y is 5.

00:00:31.970 --> 00:00:34.410
Let's see if this
actually gets satisfied.

00:00:34.410 --> 00:00:39.176
So we get 5 times 3.

00:00:39.176 --> 00:00:40.050
Let me color code it.

00:00:40.050 --> 00:00:45.070
So this is 5-- I didn't want to
do it in that color-- 5 times

00:00:45.070 --> 00:00:52.200
3 minus 3 times 5.

00:00:56.380 --> 00:01:02.640
Let's see if this is
greater than or equal to 25.

00:01:02.640 --> 00:01:05.900
So 5 times 3 is 15.

00:01:05.900 --> 00:01:09.350
And then from that, we're
going to subtract 15,

00:01:09.350 --> 00:01:12.915
and let's see if that is
greater than or equal to 25.

00:01:12.915 --> 00:01:15.150
Put that question mark
there because we don't know.

00:01:15.150 --> 00:01:17.700
And 15 minus 15, that is 0.

00:01:17.700 --> 00:01:22.810
So we get the expression 0 is
greater than or equal to 25.

00:01:22.810 --> 00:01:23.970
This is not true.

00:01:23.970 --> 00:01:26.640
0 is less than 25.

00:01:26.640 --> 00:01:28.580
So this is not true.

00:01:28.580 --> 00:01:30.130
This is not true.

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So this ordered pair is not
a solution to the inequality.

00:01:35.030 --> 00:01:39.450
So this is not a solution.

00:01:39.450 --> 00:01:41.810
You put in x is
3, y is 5, you get

00:01:41.810 --> 00:01:43.840
0 is greater than
or equal to 25,

00:01:43.840 --> 00:01:46.360
which is absolutely not true.

00:01:46.360 --> 00:01:49.390
Now let's try it with
1 and negative 7.

00:01:49.390 --> 00:02:00.140
So we have 5 times 1
minus 3 times negative 7

00:02:00.140 --> 00:02:03.120
needs to be greater
than or equal to 25.

00:02:03.120 --> 00:02:07.120
5 times 1 is 5, and then
minus 3 times negative 7

00:02:07.120 --> 00:02:08.229
is negative 21.

00:02:08.229 --> 00:02:11.295
So it becomes minus
negative 21 is

00:02:11.295 --> 00:02:13.610
to be greater than
or equal to 25.

00:02:13.610 --> 00:02:17.300
This is the same thing as
5 plus 21-- subtracting

00:02:17.300 --> 00:02:20.170
a negative same thing as
adding the positive-- is

00:02:20.170 --> 00:02:22.520
greater than or equal to 25.

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And 5 plus 21 is 26 is indeed
greater than or equal to 25.

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So this works out.

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So this is a solution.

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And just to see if
we can visualize

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this a little bit better, I'm
going to graph this inequality.

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I'm not going to show you
exactly how I do it this time,

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but I'm going to show you
where these points lie relative

00:02:43.280 --> 00:02:44.055
to this solution.

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So we have 5-- let me
do this is a new color.

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So we have 5x-- that's
not a new color.

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Having trouble
switching colors today.

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We have 5x minus 3y is
greater than or equal to 25.

00:03:04.090 --> 00:03:05.590
Let me write this
inequality in kind

00:03:05.590 --> 00:03:06.927
of our slope-intercept form.

00:03:06.927 --> 00:03:08.260
So this would be the same thing.

00:03:08.260 --> 00:03:10.830
If we subtract 5x
from both sides,

00:03:10.830 --> 00:03:14.020
we get negative
3y is greater than

00:03:14.020 --> 00:03:18.510
or equal to negative 5x plus 25.

00:03:18.510 --> 00:03:20.420
I just subtracted
5x from both sides.

00:03:20.420 --> 00:03:23.850
So that gets eliminated, and you
have a negative 5x over here.

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Now let's divide both
sides of this equation,

00:03:26.210 --> 00:03:29.182
or I should say this
inequality, by negative 3.

00:03:29.182 --> 00:03:31.140
And when you divide both
sides of an inequality

00:03:31.140 --> 00:03:33.730
by a negative number, multiply
or divide by a negative number,

00:03:33.730 --> 00:03:35.580
it swaps the inequality.

00:03:35.580 --> 00:03:37.360
So if you divide both
sides by negative 3,

00:03:37.360 --> 00:03:42.920
you get y is less than or
equal to negative 5 divided

00:03:42.920 --> 00:03:46.560
by negative 3 is 5 over 3x.

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And then 25 divided by
negative 3 is minus 25/3.

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So this is now the
expression or the inequality,

00:03:57.860 --> 00:04:03.850
y is less than or equal
to 5/3 x minus 25/3.

00:04:03.850 --> 00:04:05.930
So if I wanted to
graph this-- I'll

00:04:05.930 --> 00:04:08.224
try to draw a relatively
rough graph here,

00:04:08.224 --> 00:04:10.140
but really just so that
we can visualize this.

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So our y-intercept
is negative 25/3.

00:04:17.740 --> 00:04:20.380
That's the same thing
as negative 8 and 1/3.

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So that's 1, 2,
3, 4, 5, 6, 7, 8,

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and a little bit more than 8.

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So our y-intercept is
negative 8 and 1/3 like that.

00:04:30.120 --> 00:04:32.860
And it has a slope of 5/3.

00:04:32.860 --> 00:04:36.320
So that means for every 3 it
goes to the right, it rises 5.

00:04:36.320 --> 00:04:41.120
So it goes 1, 2, 3, it rises 5.

00:04:41.120 --> 00:04:43.515
So the line is going to
look something like this.

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I'm drawing a very
rough version of it.

00:04:45.590 --> 00:04:49.950
So the line will look something
like that, this line over here.

00:04:49.950 --> 00:04:55.480
That's if it was a y is
equal to 5/3 x minus 25/3.

00:04:55.480 --> 00:04:56.810
But here we have an inequality.

00:04:56.810 --> 00:04:59.740
It's y is less than or equal to.

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So for any x, the y's that
satisfy it are the y's

00:05:04.490 --> 00:05:09.550
that equals 5/3 x minus 25/3--
that would be on the line,

00:05:09.550 --> 00:05:10.970
so it would be
that point there--

00:05:10.970 --> 00:05:12.820
and all the y's less than it.

00:05:12.820 --> 00:05:16.360
So the solution is this
whole area right over here.

00:05:16.360 --> 00:05:20.160
Since it's less than or equal
to, we can include the line.

00:05:20.160 --> 00:05:22.360
The equal to allows us
to include the line,

00:05:22.360 --> 00:05:25.260
and the less than tells us we're
going to go below the line.

00:05:25.260 --> 00:05:26.931
And we can verify
that by looking

00:05:26.931 --> 00:05:28.180
at these two points over here.

00:05:28.180 --> 00:05:31.870
We saw that 3 comma 5 is
not part of the solution.

00:05:31.870 --> 00:05:35.010
So 3 comma 5 is 1, 2--
it's right about there

00:05:35.010 --> 00:05:35.760
and then up 5.

00:05:35.760 --> 00:05:39.252
So 3 comma 5 is
right above here.

00:05:39.252 --> 00:05:40.710
It's in this region
above the line,

00:05:40.710 --> 00:05:42.590
and notice not part
of the solution.

00:05:42.590 --> 00:05:48.610
And then 1 comma negative 7 is
going to be right over here.

00:05:48.610 --> 00:05:50.020
It's almost on the line.

00:05:50.020 --> 00:05:54.630
So 1 comma negative 7 is
going to be right over there.

00:05:54.630 --> 00:05:58.016
But it is, at least,
within this solution area.

00:05:58.016 --> 00:06:00.140
So hopefully that gives
you a little bit more sense

00:06:00.140 --> 00:06:01.514
of how to visualize
these things.

00:06:01.514 --> 00:06:05.200
And we'll cover this in more
detail in future videos.


Parallel Line Equation

WEBVTT
Kind: captions
Language: en

00:00:00.650 --> 00:00:03.100
Write the equation of
a line that is parallel

00:00:03.100 --> 00:00:07.260
to line 2x minus
4y is equal to 8

00:00:07.260 --> 00:00:10.550
and goes through the point 3, 0.

00:00:10.550 --> 00:00:16.440
So because this line is parallel
to this right over here,

00:00:16.440 --> 00:00:20.080
that just means that it
has the exact same slope.

00:00:20.080 --> 00:00:21.980
So it's essentially
saying, find the equation

00:00:21.980 --> 00:00:24.650
of a line that has the
same slope as this line

00:00:24.650 --> 00:00:26.361
and goes through this point.

00:00:26.361 --> 00:00:28.610
So first, let's figure out
what the slope of this line

00:00:28.610 --> 00:00:30.310
is, and the easiest
way to do that is

00:00:30.310 --> 00:00:34.040
to put this in kind of
slope, y-intercept form.

00:00:34.040 --> 00:00:38.670
So we have 2x minus
4y is equal to 8.

00:00:38.670 --> 00:00:40.280
We want to solve
for y, we want this

00:00:40.280 --> 00:00:43.020
to be in the form y
is equal to mx plus b.

00:00:43.020 --> 00:00:46.480
So let's add-- even
better, let's subtract 2x

00:00:46.480 --> 00:00:51.230
from both sides, so we can
isolate the y's on the left.

00:00:51.230 --> 00:00:52.510
So that cancels out.

00:00:52.510 --> 00:00:58.150
And we have negative 4y is
equal to negative 2x plus 8,

00:00:58.150 --> 00:01:00.120
and I'm writing the
negative 2x first

00:01:00.120 --> 00:01:03.520
just because that gets us
closer to the mx plus b form.

00:01:03.520 --> 00:01:06.730
Now, we could divide both sides
of this equation by negative 4

00:01:06.730 --> 00:01:08.790
so we can isolate the y.

00:01:08.790 --> 00:01:11.920
So we divide everything
by negative 4,

00:01:11.920 --> 00:01:15.540
and the left hand side, we
just have a y is equal to

00:01:15.540 --> 00:01:18.710
and negative 2 divided
by negative 4 is 1/2.

00:01:18.710 --> 00:01:24.180
So y is equal to 1/2x, and
then 8 divided by negative 4

00:01:24.180 --> 00:01:25.750
is negative 2.

00:01:25.750 --> 00:01:27.350
So we could write
plus negative 2

00:01:27.350 --> 00:01:29.790
or we could just write minus 2.

00:01:29.790 --> 00:01:33.600
So we just rewrote
this line over here,

00:01:33.600 --> 00:01:36.100
but because we wrote
it in mx plus b form,

00:01:36.100 --> 00:01:37.950
we now know its slope.

00:01:37.950 --> 00:01:41.530
The slope of this line is 1/2.

00:01:41.530 --> 00:01:44.480
That right there is the slope.

00:01:44.480 --> 00:01:46.460
So the equation of the
line that we care about

00:01:46.460 --> 00:01:48.430
is going to have the same slope.

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It's going to have the
form y is equal to 1/2

00:01:51.130 --> 00:01:54.880
x plus some y-intercept.

00:01:54.880 --> 00:01:57.060
It's not exactly
this line. b is going

00:01:57.060 --> 00:01:58.870
to be different than
this negative 2 here,

00:01:58.870 --> 00:02:00.520
but b is going to
have the same slope.

00:02:00.520 --> 00:02:04.430
They're going to intersect the
y-axis in a different place,

00:02:04.430 --> 00:02:06.780
but they're going to be
parallel to each other.

00:02:06.780 --> 00:02:09.400
So now, we need to figure
out how to figure out b.

00:02:09.400 --> 00:02:12.970
And to do that we have this
information right over here.

00:02:12.970 --> 00:02:17.930
On this line, the one we care
about, when x is 3, y is 0.

00:02:17.930 --> 00:02:22.950
When x is 3, y is equal
to 0, or we could say 0

00:02:22.950 --> 00:02:28.610
is equal to 1/2 times 3 plus b.

00:02:28.610 --> 00:02:32.130
We just have to find a b
that sets up this equation so

00:02:32.130 --> 00:02:34.190
that when x is equal
to 3, y is equal to 0.

00:02:34.190 --> 00:02:39.880
So we get 0 is
equal to 3/2 plus b.

00:02:39.880 --> 00:02:51.450
Subtract 3/2 from both sides,
and we get negative 3/2

00:02:51.450 --> 00:02:54.230
is equal to b, or b is
equal to negative 3/2.

00:02:54.230 --> 00:02:56.490
So we have the
equation for our line

00:02:56.490 --> 00:02:59.510
that is now parallel to this
equation, and goes to the point

00:02:59.510 --> 00:03:00.800
3 comma 0.

00:03:00.800 --> 00:03:04.610
It is y is equal to-- it has
the same slope as this guy

00:03:04.610 --> 00:03:10.680
up here, 1/2x, and now we know
its y-intercept, minus 3/2.

00:03:14.200 --> 00:03:16.352
And we are done.

00:03:16.352 --> 00:03:17.810
And actually, if
you're interested,

00:03:17.810 --> 00:03:20.230
you could graph these two and
see that they're parallel.

00:03:20.230 --> 00:03:21.730
So if you were to
graph these-- I'll

00:03:21.730 --> 00:03:24.810
do just a very rough
graph really fast.

00:03:24.810 --> 00:03:26.900
So this is my y-axis.

00:03:26.900 --> 00:03:28.890
This is my x-axis.

00:03:28.890 --> 00:03:33.242
This line up here,
I'll do in blue,

00:03:33.242 --> 00:03:35.450
which is the same thing as
this line right over here.

00:03:35.450 --> 00:03:37.560
I just algebraically
manipulated it.

00:03:37.560 --> 00:03:39.240
It has a y-intercept
at negative 2.

00:03:39.240 --> 00:03:40.010
So negative 2.

00:03:40.010 --> 00:03:44.560
So 1, 2, at negative 2,
and has a slope of 1/2.

00:03:44.560 --> 00:03:47.150
So that means that for
every one it goes over,

00:03:47.150 --> 00:03:49.540
it only goes 1/2 up, or
for every 2 it goes over,

00:03:49.540 --> 00:03:51.050
it only goes 1 up.

00:03:51.050 --> 00:03:52.990
So it's going to look
something like this.

00:03:52.990 --> 00:03:55.110
I'll try my best to draw it.

00:03:55.110 --> 00:03:57.010
It's going to look
something like that.

00:03:57.010 --> 00:04:02.185
And so this line over here, its
y-intercept is at negative 3/2,

00:04:02.185 --> 00:04:04.560
and negative 3/2 is the same
thing as negative 1 and 1/2,

00:04:04.560 --> 00:04:06.060
so it's going to
be right over here.

00:04:06.060 --> 00:04:07.730
So it's going to
be right above it.

00:04:07.730 --> 00:04:09.670
Right over there and it's going
to have the exact same slope.

00:04:09.670 --> 00:04:11.220
So it's going to
be right above it.

00:04:11.220 --> 00:04:12.910
It's going to be parallel.

00:04:12.910 --> 00:04:14.510
Right above it, just like that.

00:04:14.510 --> 00:04:16.779
I'll do my best
to draw parallel,

00:04:16.779 --> 00:04:18.070
and so it would look like that.

00:04:18.070 --> 00:04:20.329
They're parallel, but
they're different lines

00:04:20.329 --> 00:04:22.900
because they have
different y-intercepts.


Negative numbers introduction

WEBVTT
Kind: captions
Language: en

00:00:00.510 --> 00:00:03.180
In this video, I want
to familiarize ourselves

00:00:03.180 --> 00:00:05.250
with negative numbers,
and also learn

00:00:05.250 --> 00:00:07.440
a bit of how do we
add and subtract them.

00:00:07.440 --> 00:00:09.130
And when you first
encounter them,

00:00:09.130 --> 00:00:11.570
they look like this deep
and mysterious thing.

00:00:11.570 --> 00:00:13.080
When we first
count things, we're

00:00:13.080 --> 00:00:14.620
counting positive numbers.

00:00:14.620 --> 00:00:17.080
What does a negative
number even mean?

00:00:17.080 --> 00:00:19.280
But when we think
about it, you probably

00:00:19.280 --> 00:00:22.770
have encountered negative
numbers in your everyday life.

00:00:22.770 --> 00:00:24.487
And let me just give
you a few examples.

00:00:24.487 --> 00:00:26.070
So before I actually
give the example,

00:00:26.070 --> 00:00:27.611
the general idea is
a negative number

00:00:27.611 --> 00:00:29.445
is any number less than the 0.

00:00:34.580 --> 00:00:36.850
And if that sounds strange
and abstract to you,

00:00:36.850 --> 00:00:39.350
let's just think about it in a
couple of different contexts.

00:00:42.546 --> 00:00:45.110
If we're measuring the
temperature-- and it

00:00:45.110 --> 00:00:46.950
could be in Celsius
or Fahrenheit.

00:00:46.950 --> 00:00:49.790
Let's just assume we're
measuring it in Celsius.

00:00:49.790 --> 00:00:51.490
And so, let me
draw a little scale

00:00:51.490 --> 00:00:53.270
that we can measure
the temperature on.

00:00:53.270 --> 00:00:57.000
And so, let's say this
is 0 degrees Celsius.

00:00:57.000 --> 00:01:00.570
That is 1 degree Celsius,
2 degrees Celsius,

00:01:00.570 --> 00:01:02.990
3 degrees Celsius.

00:01:02.990 --> 00:01:05.730
Now let's say that it's
a pretty chilly day,

00:01:05.730 --> 00:01:10.120
and it's currently
3 degrees Celsius.

00:01:10.120 --> 00:01:11.760
And someone who
predicts the future

00:01:11.760 --> 00:01:13.780
tells you that it
is going to get

00:01:13.780 --> 00:01:16.980
4 degrees colder the next day.

00:01:16.980 --> 00:01:18.360
So how cold will it be?

00:01:18.360 --> 00:01:21.320
How can you represent
that coldness?

00:01:21.320 --> 00:01:23.680
Well, if it only
got 1 degree colder,

00:01:23.680 --> 00:01:24.809
we would be at 2 degrees.

00:01:24.809 --> 00:01:26.600
But we know we have to
go 4 degrees colder.

00:01:26.600 --> 00:01:31.650
If we got 2 degrees colder,
we would be at 1 degree.

00:01:31.650 --> 00:01:34.846
If we got 3 degrees
colder, we would be at 0.

00:01:34.846 --> 00:01:36.270
But 3 isn't enough.

00:01:36.270 --> 00:01:38.300
We have to get 4 degrees colder.

00:01:38.300 --> 00:01:41.190
So we have to actually
go one more below the 0.

00:01:43.940 --> 00:01:50.510
And that one below 0,
we call that negative 1.

00:01:50.510 --> 00:01:52.760
And so, you can kind of
see that the number line,

00:01:52.760 --> 00:01:54.990
as you go to the right
of 0 on the number line,

00:01:54.990 --> 00:01:56.792
it increases in positive values.

00:01:56.792 --> 00:01:58.500
As you go to the left
of the number line,

00:01:58.500 --> 00:01:59.930
you're going to have negative 1.

00:01:59.930 --> 00:02:01.690
Then you're going
to have negative 2.

00:02:01.690 --> 00:02:03.931
And you're going
to have negative 3.

00:02:03.931 --> 00:02:05.430
And you're going
to have-- depending

00:02:05.430 --> 00:02:07.510
on how you think
about it-- you're

00:02:07.510 --> 00:02:09.759
going to have larger
negative numbers.

00:02:09.759 --> 00:02:12.340
But I want to make it
very clear-- negative 3

00:02:12.340 --> 00:02:16.180
is less than negative 1.

00:02:16.180 --> 00:02:19.150
There is less heat in
the air then negative 1.

00:02:19.150 --> 00:02:20.980
It is colder.

00:02:20.980 --> 00:02:24.300
There's less temperature there.

00:02:24.300 --> 00:02:27.180
So let me just
make it very clear.

00:02:27.180 --> 00:02:39.622
Negative 100 is much
smaller than negative 1.

00:02:39.622 --> 00:02:41.580
You might look at 100,
and you might look at 1,

00:02:41.580 --> 00:02:43.121
and your gut reaction
is to say, wow,

00:02:43.121 --> 00:02:44.860
100 is a much larger
number than 1.

00:02:44.860 --> 00:02:46.570
But when you think
about it, negative 100

00:02:46.570 --> 00:02:49.510
means there's a
lack of something.

00:02:49.510 --> 00:02:52.260
If it's negative 100 degrees,
there's a lack of heat.

00:02:52.260 --> 00:02:55.580
So there's much less heat here
then if we had negative 1.

00:02:55.580 --> 00:02:57.320
Let me give you another example.

00:02:57.320 --> 00:03:01.840
Let's say in my bank
account today, I have $10.

00:03:01.840 --> 00:03:03.090
Let me do this in a new color.

00:03:03.090 --> 00:03:11.130
So let's say today I have $10.

00:03:11.130 --> 00:03:13.120
Now let's say I go
out there-- because I

00:03:13.120 --> 00:03:21.035
feel good about my $10-- and
let's say I go and spend $30.

00:03:21.035 --> 00:03:22.660
And for the sake of
argument, let's say

00:03:22.660 --> 00:03:24.630
I have a very flexible
bank, one that

00:03:24.630 --> 00:03:26.930
lets me spend more
money than I have--

00:03:26.930 --> 00:03:28.360
and these actually exist.

00:03:28.360 --> 00:03:30.160
So I spend $30.

00:03:30.160 --> 00:03:32.850
So what's my bank account
going to look like?

00:03:32.850 --> 00:03:35.220
So let me draw a
number line here.

00:03:35.220 --> 00:03:37.660
And you might already have
an intuitive response.

00:03:37.660 --> 00:03:41.679
I will owe the bank some money.

00:03:41.679 --> 00:03:42.970
So let me write this over here.

00:03:42.970 --> 00:03:47.070
So tomorrow, what
is my bank account?

00:03:47.070 --> 00:03:49.740
So you might immediately
say, look, if I have $10,

00:03:49.740 --> 00:03:52.840
and I spend $30,
there's $20 that

00:03:52.840 --> 00:03:55.180
had to come from
someplace, and that $20 is

00:03:55.180 --> 00:03:56.260
coming from the bank.

00:03:56.260 --> 00:03:58.970
So I'm going to
owe the bank $20.

00:03:58.970 --> 00:04:02.530
And so, in my bank account,
to sure how much I have,

00:04:02.530 --> 00:04:07.240
I could say $10 minus $30
is actually negative $20.

00:04:07.240 --> 00:04:13.240
So in my bank account tomorrow,
I'm going to have negative $20.

00:04:13.240 --> 00:04:18.115
So if I say I have negative $20,
that means that I owe the bank.

00:04:18.115 --> 00:04:18.990
I don't even have it.

00:04:18.990 --> 00:04:21.314
Not only do I have
nothing, I owe something.

00:04:21.314 --> 00:04:22.230
It's going in reverse.

00:04:22.230 --> 00:04:25.660
Here, I have something to spend.

00:04:25.660 --> 00:04:28.710
If my $10 is in my bank, that
means the bank owes me $10.

00:04:28.710 --> 00:04:31.730
I have $10 that I
can use to go spend.

00:04:31.730 --> 00:04:33.490
Now all of a sudden,
I owe the bank.

00:04:33.490 --> 00:04:35.156
I've gone in the
other direction.

00:04:35.156 --> 00:04:36.530
And if we use a
number line here,

00:04:36.530 --> 00:04:38.488
it should hopefully make
a little bit of sense.

00:04:38.488 --> 00:04:39.400
So that is 0.

00:04:39.400 --> 00:04:42.920
I'm starting off with $10.

00:04:42.920 --> 00:04:47.630
And spending $30 means I'm
moving 30 spaces to the left.

00:04:47.630 --> 00:04:51.140
So if I move 10 spaces to the
left-- if I only spend $10,

00:04:51.140 --> 00:04:52.600
I'll be back at 0.

00:04:52.600 --> 00:04:56.570
If I spend another $10,
I'll be at negative $10.

00:04:56.570 --> 00:04:58.920
If I spend another
$10 after that,

00:04:58.920 --> 00:05:01.950
I will be at negative $20.

00:05:01.950 --> 00:05:04.510
So each of these distances--
I spent $10, I'd be at 0.

00:05:04.510 --> 00:05:06.720
Another $10, I'd
be at negative $10.

00:05:06.720 --> 00:05:09.470
Another $10, I would
be at negative $20.

00:05:09.470 --> 00:05:12.505
And this whole distance right
here is how much I spent.

00:05:15.050 --> 00:05:17.090
I spent $30.

00:05:17.090 --> 00:05:20.119
So the general idea, when
you spend or if you subtract

00:05:20.119 --> 00:05:21.660
or if we're getting
colder, you would

00:05:21.660 --> 00:05:23.160
move to the left
of the number line.

00:05:23.160 --> 00:05:24.610
The numbers would get smaller.

00:05:24.610 --> 00:05:26.837
And we now know they
can get smaller than 0.

00:05:26.837 --> 00:05:28.420
They can go to
negative 1, negative 2,

00:05:28.420 --> 00:05:31.630
they can even go to
negative 1.5, negative 1.6.

00:05:31.630 --> 00:05:34.590
You're getting more and more
negative the more you lose.

00:05:34.590 --> 00:05:36.830
If you're adding-- if I
go and get my paycheck,

00:05:36.830 --> 00:05:38.840
I will move to the right
of the number line.

00:05:41.414 --> 00:05:43.080
Now with that out of
the way, let's just

00:05:43.080 --> 00:05:45.010
do a couple of more
pure math problems.

00:05:45.010 --> 00:05:47.270
And just think
about what it means

00:05:47.270 --> 00:05:56.330
if we were to say 3 minus 4.

00:05:56.330 --> 00:05:58.277
So once again, this is
exactly the situation

00:05:58.277 --> 00:05:59.860
that we did here
with the temperature.

00:05:59.860 --> 00:06:01.829
We're starting at 3,
and we're subtracting 4.

00:06:01.829 --> 00:06:03.370
So we're going to
move 4 to the left.

00:06:03.370 --> 00:06:05.770
We go 1, 2, 3, 4.

00:06:05.770 --> 00:06:09.870
So that gets us to negative 1.

00:06:09.870 --> 00:06:11.370
And when you're
starting to do this,

00:06:11.370 --> 00:06:13.680
you really understand what
a negative number means.

00:06:13.680 --> 00:06:16.074
I really encourage you to
visualize the number line,

00:06:16.074 --> 00:06:17.490
and really move
along it depending

00:06:17.490 --> 00:06:19.550
on whether you're
adding and subtracting.

00:06:19.550 --> 00:06:20.760
Now let's do a couple more.

00:06:20.760 --> 00:06:27.400
Let's say I have 2 minus 8.

00:06:27.400 --> 00:06:30.240
And we'll think about more ways
to do this in future videos.

00:06:30.240 --> 00:06:33.550
But once again, you just
want to do the number line.

00:06:33.550 --> 00:06:35.920
You have a 0 here.

00:06:35.920 --> 00:06:37.870
Let me draw the
spacing a little bit.

00:06:37.870 --> 00:06:39.600
So we have 0 here.

00:06:39.600 --> 00:06:41.830
We're at 1, 2.

00:06:41.830 --> 00:06:43.960
If we're subtracting
8, that means

00:06:43.960 --> 00:06:46.860
we're going to
move 8 to the left.

00:06:46.860 --> 00:06:50.630
So we're going to go 1
to the left, 2 the left.

00:06:50.630 --> 00:06:52.890
So we've gone 2 to
the left to get to 0.

00:06:52.890 --> 00:06:54.830
We have to move how
many more to the left?

00:06:54.830 --> 00:06:56.600
Well, we've already
moved 2 the left.

00:06:56.600 --> 00:06:59.280
To get to 8, we have to
move 6 more to the left.

00:06:59.280 --> 00:07:06.650
So we're going to have to
move 1, 2, 3, 4, 5, 6 more

00:07:06.650 --> 00:07:07.260
to the left.

00:07:07.260 --> 00:07:08.800
Well, where is that
going to put us?

00:07:08.800 --> 00:07:09.800
Well, we were at 0.

00:07:09.800 --> 00:07:15.590
This is a negative 1, negative
2, negative 3, negative 4,

00:07:15.590 --> 00:07:18.560
negative 5, negative 6.

00:07:18.560 --> 00:07:24.760
So 2 minus 8 is negative 6.

00:07:24.760 --> 00:07:26.440
2 minus 2 would be 0.

00:07:26.440 --> 00:07:29.440
When you're subtracting 8,
you're subtracting another 6.

00:07:29.440 --> 00:07:30.830
So we'd go up to negative 6.

00:07:30.830 --> 00:07:33.120
We go 6 below 0.

00:07:33.120 --> 00:07:34.245
Let me do one more example.

00:07:34.245 --> 00:07:36.578
And this one will be a little
less conventional for you,

00:07:36.578 --> 00:07:38.040
but hopefully it
will make sense.

00:07:40.760 --> 00:07:43.490
And I'll do this in a new color.

00:07:43.490 --> 00:07:47.570
Let me take negative 4 minus 2.

00:07:47.570 --> 00:07:49.199
So we're starting at
a negative number,

00:07:49.199 --> 00:07:50.740
and then we're
subtracting from that.

00:07:50.740 --> 00:07:54.030
And if this seems confusing,
just remember the number line.

00:07:56.770 --> 00:07:59.050
So this is 0 right here.

00:07:59.050 --> 00:08:05.420
This is negative 1, negative
2, negative 3, negative 4.

00:08:05.420 --> 00:08:06.800
So that's where we're starting.

00:08:06.800 --> 00:08:09.750
Now we're going to
subtract 2 from negative 4.

00:08:09.750 --> 00:08:11.570
So we're going to
move 2 to the left.

00:08:11.570 --> 00:08:14.940
So if we subtract 1, we
will be at negative 5.

00:08:14.940 --> 00:08:19.820
If we subtract another 1, we
are going to be at negative 6.

00:08:19.820 --> 00:08:21.610
We are going to
be at negative 6.

00:08:21.610 --> 00:08:23.072
So this is negative 6.

00:08:23.072 --> 00:08:24.530
Let's do another
interesting thing.

00:08:24.530 --> 00:08:28.490
Let's start at negative 3.

00:08:28.490 --> 00:08:30.230
Let's say we have negative 3.

00:08:30.230 --> 00:08:34.280
Instead of subtracting something
from that, let's add 2 to it.

00:08:34.280 --> 00:08:36.230
So where would this put
us on the number line?

00:08:36.230 --> 00:08:39.620
So we're starting at negative
3, and we're adding 2,

00:08:39.620 --> 00:08:41.809
so we're going to
move to the right.

00:08:41.809 --> 00:08:44.460
So you add 1, you
become negative 2.

00:08:44.460 --> 00:08:46.680
But if you add another
1, which we have to do,

00:08:46.680 --> 00:08:47.710
you become negative 1.

00:08:47.710 --> 00:08:49.260
You move 2 to the right.

00:08:49.260 --> 00:08:54.422
So negative 3 plus
negative 2 is negative 1.

00:08:54.422 --> 00:08:56.130
And you can see for
yourself but this all

00:08:56.130 --> 00:08:59.870
fits our traditional notion
of adding and subtracting.

00:08:59.870 --> 00:09:03.910
If we start at negative
1, and we subtract 2,

00:09:03.910 --> 00:09:06.540
we should get negative 3--
kind of reverse of this thing

00:09:06.540 --> 00:09:07.250
up here.

00:09:07.250 --> 00:09:09.349
Negative 3 plus 2 gets us there.

00:09:09.349 --> 00:09:10.890
Then if we start
then, we subtract 2,

00:09:10.890 --> 00:09:13.306
we should get back to negative
3, and we see that happens.

00:09:13.306 --> 00:09:17.720
If you start at negative
1 right over here,

00:09:17.720 --> 00:09:20.110
and you subtract 2--
you move 2 to the left--

00:09:20.110 --> 00:09:21.469
you get back to negative 3.

00:09:21.469 --> 00:09:23.010
So hopefully, this
starts to give you

00:09:23.010 --> 00:09:28.012
a sense of what it means to
deal with, or add and subtract,

00:09:28.012 --> 00:09:28.720
negative numbers.

00:09:28.720 --> 00:09:30.560
But we're going to give
a lot more examples

00:09:30.560 --> 00:09:31.090
in the next video.

00:09:31.090 --> 00:09:32.590
And we're actually
going to see what

00:09:32.590 --> 00:09:34.936
it means to subtract
a negative number.


Basic trigonometry II

WEBVTT
Kind: captions
Language: en

00:00:00.800 --> 00:00:03.017
Let's just do a ton of more examples,

00:00:03.017 --> 00:00:07.036
just so we make sure that we're getting this trig function thing down well.

00:00:07.036 --> 00:00:11.447
So let's construct ourselves some right triangles.

00:00:11.447 --> 00:00:13.668
Let's construct ourselves some right triangles,

00:00:13.668 --> 00:00:15.186
and I want to be very clear.

00:00:15.186 --> 00:00:18.042
The way I've defined it so far, this will only work in right triangles.

00:00:18.042 --> 00:00:23.475
So if you're trying to find the trig functions of angles that aren't part of right triangles,

00:00:23.475 --> 00:00:25.704
we're going to see that we're going to have to construct right triangles,

00:00:25.704 --> 00:00:27.867
but let's just focus on the right triangles for now.

00:00:27.867 --> 00:00:31.344
So let's say that I have a triangle,

00:00:31.344 --> 00:00:33.897
where let's say this length down here is seven,

00:00:33.897 --> 00:00:37.757
and let's say the length of this side up here,

00:00:37.757 --> 00:00:39.452
let's say that that is four.

00:00:39.452 --> 00:00:42.516
Let's figure out what the hypotenuse over here is going to be.

00:00:42.516 --> 00:00:45.720
So we know -let's call the hypotenuse, "h"-

00:00:45.720 --> 00:00:52.200
we know that h squared is going to be equal to seven squared plus four squared,

00:00:52.200 --> 00:00:55.194
we know that from the Pythagorean theorem,

00:00:55.194 --> 00:00:57.469
that the hypotenuse squared is equal to

00:00:57.469 --> 00:01:01.974
the square of each of the sum of the squares of the other two sides.

00:01:01.974 --> 00:01:04.533
h squared is equal to seven squared plus four squared.

00:01:04.533 --> 00:01:09.776
So this is equal to forty-nine plus sixteen,

00:01:09.776 --> 00:01:11.800
forty-nine plus sixteen,

00:01:11.800 --> 00:01:18.553
forty nine plus ten is fifty-nine, plus six is sixty-five.

00:01:18.553 --> 00:01:21.107
It is sixty five. So this h squared,

00:01:21.107 --> 00:01:25.705
let me write: h squared -that's different shade of yellow-

00:01:25.705 --> 00:01:28.818
so we have h squared is equal to sixty-five.

00:01:28.818 --> 00:01:33.533
Did I do that right? Forty nine plus ten is fifty nine, plus another six is sixty-five,

00:01:33.533 --> 00:01:37.600
or we could say that h is equal to, if we take the square root of both sides,

00:01:37.600 --> 00:01:39.200
square root

00:01:39.200 --> 00:01:42.933
square root of sixty five. And we really can't simplify
this at all.

00:01:42.933 --> 00:01:44.699
This is thirteen.

00:01:44.699 --> 00:01:47.463
This is the same thing as thirteen times five,

00:01:47.463 --> 00:01:50.388
both of those are not perfect squares and

00:01:50.388 --> 00:01:51.804
they're both prime so you can't simplify this any more.

00:01:51.804 --> 00:01:55.467
So this is equal to the square root of sixty five.

00:01:55.467 --> 00:02:02.114
Now let's find the trig, let's find the trig functions for this angle up here.

00:02:02.114 --> 00:02:05.457
Let's call that angle up there theta.

00:02:05.457 --> 00:02:06.533
So whenever you do it

00:02:06.533 --> 00:02:09.467
you always want to write down - at least for me it works out to write down -

00:02:09.467 --> 00:02:11.714
"soh cah toa".

00:02:11.714 --> 00:02:13.120
soh...

00:02:13.120 --> 00:02:16.464
...soh cah toa. I have these vague memories

00:02:16.464 --> 00:02:18.786
of my trigonometry teacher.

00:02:18.786 --> 00:02:21.293
Maybe I've read it in some book. I don't know - you know, some... about

00:02:21.293 --> 00:02:23.867
some type of indian princess named "soh cah toa" or whatever,

00:02:23.867 --> 00:02:26.123
but it's a very useful mnemonic,

00:02:26.123 --> 00:02:27.564
so we can apply "soh cah toa".

00:02:27.564 --> 00:02:31.046
Let's find, let's say we want to find the cosine.

00:02:31.046 --> 00:02:34.436
We want to find the cosine of our angle.

00:02:34.436 --> 00:02:37.965
We wanna find the cosine of our angle, you say: "soh cah toa!"

00:02:37.965 --> 00:02:40.800
So the "cah". "Cah" tells us what to do with cosine,

00:02:40.800 --> 00:02:43.027
the "cah" part tells us

00:02:43.027 --> 00:02:46.371
that cosine is adjacent over hypotenuse.

00:02:46.371 --> 00:02:51.433
Cosine is equal to adjacent over hypotenuse.

00:02:51.433 --> 00:02:55.798
So let's look over here to theta; what side is adjacent?

00:02:55.798 --> 00:02:57.702
Well we know that the hypotenuse,

00:02:57.702 --> 00:03:00.767
we know that that hypotenuse is this side over here.

00:03:00.767 --> 00:03:04.761
So it can't be that side. The only other side that's kind of adjacent to it that

00:03:04.761 --> 00:03:07.133
isn't the hypotenuse, is this four.

00:03:07.133 --> 00:03:10.473
So the adjacent side over here, that side is,

00:03:10.473 --> 00:03:14.374
it's literally right next to the angle,

00:03:14.374 --> 00:03:15.754
it's one of the sides that kind of forms the angle

00:03:15.754 --> 00:03:17.133
it's four over the hypotenuse.

00:03:17.133 --> 00:03:21.108
The hypotenuse we already know is square root
of sixty-five.

00:03:21.108 --> 00:03:25.380
so it's four over the square root of sixty-five.

00:03:25.380 --> 00:03:29.142
And sometimes people will want you to rationalize the denominator which means

00:03:29.142 --> 00:03:32.625
they don't like to have an irrational number in the denominator,

00:03:32.625 --> 00:03:35.227
like the square root of sixty five,

00:03:35.227 --> 00:03:39.359
and if they - if you wanna rewrite this without a irrational number in the denominator,

00:03:39.359 --> 00:03:41.634
you can multiply the numerator and the denominator

00:03:41.634 --> 00:03:43.306
by the square root of sixty-five.

00:03:43.306 --> 00:03:45.094
This clearly will not change the number,

00:03:45.094 --> 00:03:48.122
because we're multiplying it by something over itself,

00:03:48.122 --> 00:03:49.111
so we're multiplying the number by one.

00:03:49.111 --> 00:03:52.780
That won't change the number, but at least it gets rid of the irrational number in the denominator.

00:03:52.780 --> 00:03:54.127
So the numerator becomes

00:03:54.127 --> 00:03:57.800
four times the square root of sixty-five,

00:03:57.800 --> 00:04:03.461
and the denominator, square root of 65 times square root of 65, is just going to be 65.

00:04:03.461 --> 00:04:07.130
We didn't get rid of the irrational number, it's still there, but it's now in the numerator.

00:04:07.130 --> 00:04:09.777
Now let's do the other trig functions

00:04:09.777 --> 00:04:12.401
or at least the other core trig functions.

00:04:12.401 --> 00:04:14.399
We'll learn in the future that there's actually a ton of them

00:04:14.399 --> 00:04:15.443
but they're all derived from these.

00:04:15.443 --> 00:04:19.733
so let's think about what the sine of theta is. Once again
go to "soh cah toa".

00:04:19.733 --> 00:04:25.474
The "soh" tells what to do with sine. Sine is opposite over hypotenuse.

00:04:25.474 --> 00:04:29.200
Sine is equal to opposite over hypotenuse.

00:04:29.200 --> 00:04:31.372
Sine is opposite over hypotenuse.

00:04:31.372 --> 00:04:34.390
So for this angle what side is opposite?

00:04:34.390 --> 00:04:38.430
We just go opposite it, what it opens into, it's opposite the seven

00:04:38.430 --> 00:04:41.200
so the opposite side is the seven.

00:04:41.200 --> 00:04:44.468
This is, right here - that is the opposite side

00:04:44.468 --> 00:04:47.800
and then the hypotenuse, it's opposite over hypotenuse.

00:04:47.800 --> 00:04:51.109
The hypotenuse is the square root of sixty-five.

00:04:51.109 --> 00:04:52.966
Square root of sixty-five.

00:04:52.966 --> 00:04:55.133
and once again if we wanted to rationalize this,

00:04:55.133 --> 00:04:59.933
we could multiply times the square root of 65 over the square root of 65

00:04:59.933 --> 00:05:04.298
and the the numerator, we will get seven square root of 65

00:05:04.298 --> 00:05:07.966
and in the denominator we will get just sixty-five again.

00:05:07.966 --> 00:05:10.474
Now let's do tangent!

00:05:10.474 --> 00:05:12.796
Let us do tangent.

00:05:12.796 --> 00:05:14.793
So if i ask you the tangent

00:05:14.793 --> 00:05:17.394
of - the tangent of theta

00:05:17.394 --> 00:05:20.784
once again go back to "soh cah toa".

00:05:20.784 --> 00:05:23.106
The toa part tells us what to do with tangent

00:05:23.106 --> 00:05:24.800
it tells us...

00:05:24.800 --> 00:05:27.053
it tells us that tangent

00:05:27.053 --> 00:05:29.867
is equal to opposite over adjacent

00:05:29.867 --> 00:05:33.137
is equal to opposite over

00:05:33.137 --> 00:05:35.867
opposite over adjacent

00:05:35.867 --> 00:05:38.709
So for this angle, what is opposite? We've already figured it out.

00:05:38.709 --> 00:05:41.124
it's seven. It opens into the seven.

00:05:41.124 --> 00:05:42.533
It is opposite the seven.

00:05:42.533 --> 00:05:46.372
So it's seven over what side is adjacent.

00:05:46.372 --> 00:05:48.200
well this four is adjacent.

00:05:48.200 --> 00:05:51.295
This four is adjacent. So the adjacent side is four.

00:05:51.295 --> 00:05:54.330
so it's seven over four,

00:05:54.330 --> 00:05:56.133
and we're done.

00:05:56.133 --> 00:05:59.375
We figured out all of the trig ratios for theta. let's do another one.

00:05:59.375 --> 00:06:00.416
Let's do another one.

00:06:00.416 --> 00:06:02.719
i'll make it a little bit concrete 'cause right now we've been saying,

00:06:02.719 --> 00:06:06.434
"oh, what's tangent of x, tangent of theta." let's make it a little bit more concrete.

00:06:06.434 --> 00:06:08.431
Let's say...

00:06:08.431 --> 00:06:10.799
let's say, let me draw another right triangle,

00:06:10.799 --> 00:06:13.772
that's another right triangle here.

00:06:13.772 --> 00:06:17.533
Everything we're dealing with, these are going to be right triangles.

00:06:17.533 --> 00:06:21.109
let's say the hypotenuse has length four,

00:06:21.109 --> 00:06:26.357
let's say that this side over here has length two,

00:06:26.357 --> 00:06:31.790
and let's say that this length over here is going to be two times the square root of three.

00:06:31.790 --> 00:06:33.462
We can verify that this works.

00:06:33.462 --> 00:06:36.467
If you have this side squared, so you have - let me write it down -

00:06:36.467 --> 00:06:38.803
two times the square root of three squared

00:06:38.803 --> 00:06:42.471
plus two squared, is equal to what?

00:06:42.471 --> 00:06:46.467
this is two. There's going to be four times three.

00:06:46.467 --> 00:06:49.763
four times three plus four,

00:06:49.763 --> 00:06:53.478
and this is going to be equal to twelve plus four is equal to sixteen

00:06:53.478 --> 00:06:57.800
and sixteen is indeed four squared. So this does equal four squared,

00:06:57.800 --> 00:07:01.790
it does equal four squared. It satisfies the pythagorean theorem

00:07:01.790 --> 00:07:06.133
and if you remember some of your work from 30 60 90 triangles

00:07:06.133 --> 00:07:07.781
that you might have learned in geometry,

00:07:07.781 --> 00:07:11.450
you might recognize that this is a 30 60 90 triangle.

00:07:11.450 --> 00:07:13.133
This right here is our right angle,

00:07:13.133 --> 00:07:15.867
- i should have drawn it from the get go to show that this is a right triangle -

00:07:15.867 --> 00:07:20.366
this angle right over here is our thirty degree angle

00:07:20.366 --> 00:07:23.385
and then this angle up here, this angle up here is

00:07:23.385 --> 00:07:26.125
a sixty degree angle,

00:07:26.125 --> 00:07:27.797
and it's a thirty sixteen ninety because

00:07:27.797 --> 00:07:31.791
the side opposite the thirty degrees is half the hypotenuse

00:07:31.791 --> 00:07:36.800
and then the side opposite the 60 degrees is a squared of 3 times the other side

00:07:36.800 --> 00:07:38.432
that's not the hypotenuse.

00:07:38.432 --> 00:07:40.159
So that said, we're not gonna ...

00:07:40.159 --> 00:07:43.415
this isn't supposed to be a review of 30 60 90 triangles although i just did it.

00:07:43.415 --> 00:07:46.933
Let's actually find the trig ratios for the different angles.

00:07:46.933 --> 00:07:51.295
So if i were to ask you or if anyone were to ask you, what is...

00:07:51.295 --> 00:07:54.639
what is the sine of thirty degrees?

00:07:54.639 --> 00:07:58.447
and remember 30 degrees is one of the angles in this triangle but it would apply

00:07:58.447 --> 00:08:01.698
whenever you have a 30 degree angle and you're dealing with the right triangle.

00:08:01.698 --> 00:08:05.135
We'll have broader definitions in the future but if you say sine of thirty degrees,

00:08:05.135 --> 00:08:09.035
hey, this angle right over here is thirty degrees so i can use this right triangle,

00:08:09.035 --> 00:08:12.133
and we just have to remember "soh cah toa"

00:08:12.133 --> 00:08:17.116
We rewrite it. soh, cah, toa.

00:08:17.116 --> 00:08:22.782
"sine tells us" (correction). soh tells us what to do with sine. sine is opposite over hypotenuse.

00:08:22.782 --> 00:08:26.358
sine of thirty degrees is the opposite side,

00:08:26.358 --> 00:08:30.723
that is the opposite side which is two over the hypotenuse.

00:08:30.723 --> 00:08:32.395
The hypotenuse here is four.

00:08:32.395 --> 00:08:35.646
it is two fourths which is the same thing as one-half.

00:08:35.646 --> 00:08:40.800
sine of thirty degrees you'll see is always going to be equal to one-half.

00:08:40.800 --> 00:08:44.144
now what is the cosine?

00:08:44.144 --> 00:08:46.867
What is the cosine of thirty degrees?

00:08:46.867 --> 00:08:50.135
Once again go back to "soh cah toa".

00:08:50.135 --> 00:08:52.643
The cah tells us what to do with cosine.

00:08:52.643 --> 00:08:56.033
Cosine is adjacent over hypotenuse.

00:08:56.033 --> 00:08:59.051
So for looking at the thirty degree angle it's the adjacent.

00:08:59.051 --> 00:09:01.791
This, right over here is adjacent. it's right next to it.

00:09:01.791 --> 00:09:05.467
it's not the hypotenuse. it's the adjacent over the hypotenuse.

00:09:05.467 --> 00:09:09.129
so it's two square roots of three

00:09:09.129 --> 00:09:13.633
adjacent over...over the hypotenuse, over four.

00:09:13.633 --> 00:09:16.977
or if we simplify that, we divide the numerator and the denominator by two

00:09:16.977 --> 00:09:20.646
it's the square root of three over two.

00:09:20.646 --> 00:09:22.782
Finally, let's do the tangent.

00:09:22.782 --> 00:09:27.800
The tangent of thirty degrees,

00:09:27.800 --> 00:09:30.305
we go back to "soh cah toa".

00:09:30.305 --> 00:09:31.699
soh cah toa

00:09:31.699 --> 00:09:34.800
toa tells us what to do with tangent. It's opposite over adjacent

00:09:34.800 --> 00:09:38.804
you go to the 30 degree angle because that's what we care about, tangent of 30.

00:09:38.804 --> 00:09:42.101
tangent of thirty. Opposite is two,

00:09:42.101 --> 00:09:46.200
opposite is two and the adjacent is two square roots of three.

00:09:46.200 --> 00:09:48.045
It's right next to it. It's adjacent to it.

00:09:48.045 --> 00:09:49.439
adjacent means next to.

00:09:49.439 --> 00:09:52.039
so two square roots of three

00:09:52.039 --> 00:09:54.454
so this is equal to... the twos cancel out

00:09:54.454 --> 00:09:56.776
one over the square root of three

00:09:56.776 --> 00:10:00.723
or we could multiply the numerator and the denominator by the square root of three.

00:10:00.723 --> 00:10:05.367
So we have square root of three over square root of three

00:10:05.367 --> 00:10:08.804
and so this is going to be equal to the numerator square root of three and then

00:10:08.804 --> 00:10:12.473
the denominator right over here is just going to be three.

00:10:12.473 --> 00:10:15.800
So that we've rationalized a square root of three over three.

00:10:15.800 --> 00:10:17.442
Fair enough.

00:10:17.442 --> 00:10:20.693
Now lets use the same triangle to figure out the trig ratios for the sixty degrees,

00:10:20.693 --> 00:10:22.457
since we've already drawn it.

00:10:22.457 --> 00:10:28.328
so what is... what is the sine of the sixty degrees?

00:10:28.328 --> 00:10:30.166
and i think you're hopefully getting the hang of it now.

00:10:30.166 --> 00:10:34.253
Sine is opposite over adjacent. soh from the "soh cah toa".

00:10:34.253 --> 00:10:36.668
for the sixty degree angle what side is opposite?

00:10:36.668 --> 00:10:39.315
what opens out into the two square roots of three,

00:10:39.315 --> 00:10:42.566
so the opposite side is two square roots of three,

00:10:42.566 --> 00:10:45.306
and from the sixty degree angle the adj-oh sorry

00:10:45.306 --> 00:10:47.999
its the opposite over hypotenuse, don't want to confuse you.

00:10:47.999 --> 00:10:50.507
so it is opposite over hypotenuse

00:10:50.507 --> 00:10:54.315
so it's two square roots of three over four. four is the hypotenuse.

00:10:54.315 --> 00:10:59.981
so it is equal to, this simplifies to square root of three over two.

00:10:59.981 --> 00:11:05.507
What is the cosine of sixty degrees? cosine of sixty degrees.

00:11:05.507 --> 00:11:10.244
so remember "soh cah toa". cosine is adjacent over hypotenuse.

00:11:10.244 --> 00:11:13.667
adjacent is the two sides, right next to the sixty degree angle.

00:11:13.667 --> 00:11:17.907
So it's two over the hypotenuse which is four.

00:11:17.907 --> 00:11:20.972
So this is equal to one-half

00:11:20.972 --> 00:11:24.176
and then finally, what is the tangent?

00:11:24.176 --> 00:11:27.984
what is the tangent of sixty degrees?

00:11:27.984 --> 00:11:32.349
Well tangent, "soh cah toa". Tangent is opposite over adjacent

00:11:32.349 --> 00:11:34.671
opposite the sixty degrees

00:11:34.671 --> 00:11:36.400
is two square roots of three

00:11:36.400 --> 00:11:38.000
two square roots of three

00:11:38.000 --> 00:11:39.919
and adjacent to that

00:11:39.919 --> 00:11:42.733
adjacent to that is two.

00:11:42.733 --> 00:11:44.800
Adjacent to sixty degrees is two.

00:11:44.800 --> 00:11:48.650
So its opposite over adjacent, two square roots of three over two

00:11:48.650 --> 00:11:52.644
which is just equal to the square root of three.

00:11:52.644 --> 00:11:54.641
And I just wanted to -look how these are related-

00:11:54.641 --> 00:11:57.984
the sine of thirty degrees is the same as the cosine of sixty degrees.

00:11:57.984 --> 00:12:01.333
The cosine of 30 degrees is the same thing as the sine of 60 degrees

00:12:01.333 --> 00:12:03.966
and then these guys are the inverse of each other

00:12:03.966 --> 00:12:05.635
and i think if you think a little bit about this triangle

00:12:05.635 --> 00:12:07.105
it will start to make sense why.

00:12:07.105 --> 00:12:08.461
we'll keep extending
this and

00:12:08.461 --> 99:59:59.999
give you a lot more practice in the next few videos.


Basic trigonometry

WEBVTT
Kind: captions
Language: en

00:00:00.550 --> 00:00:05.322
In this video, I want to give
you the basics of trigonometry.

00:00:05.322 --> 00:00:08.334
And it sounds like a
very complicated topic,

00:00:08.334 --> 00:00:10.000
but you're going to
see that it's really

00:00:10.000 --> 00:00:14.980
just the study of the ratios
of sides of triangles.

00:00:14.980 --> 00:00:18.550
The "trig" part of trigonometry
literally means triangle.

00:00:18.550 --> 00:00:21.860
And the "metry" part
literally means measure.

00:00:21.860 --> 00:00:23.810
So let me just give
you some examples here.

00:00:23.810 --> 00:00:26.350
And I think it'll make
everything pretty clear.

00:00:26.350 --> 00:00:27.970
So let me draw some
right triangles.

00:00:27.970 --> 00:00:29.580
Let me just draw
one right triangle.

00:00:29.580 --> 00:00:31.325
So this is a right triangle.

00:00:31.325 --> 00:00:33.460
And when I say it's
a right triangle,

00:00:33.460 --> 00:00:37.020
it's because one of the
angles here is 90 degrees.

00:00:37.020 --> 00:00:42.505
This right here
is a right angle.

00:00:42.505 --> 00:00:47.830
It is equal to 90 degrees.

00:00:47.830 --> 00:00:50.020
And we'll talk about
other ways to show

00:00:50.020 --> 00:00:52.992
the magnitude of angles
in future videos.

00:00:52.992 --> 00:00:54.200
So we have a 90-degree angle.

00:00:54.200 --> 00:00:55.190
It's a right triangle.

00:00:55.190 --> 00:00:57.810
And let me put some
lengths to the sides here.

00:00:57.810 --> 00:01:00.330
So this side over
here is maybe 3.

00:01:00.330 --> 00:01:02.970
This height right
over there is 3.

00:01:02.970 --> 00:01:07.120
Maybe the base of the
triangle right over here is 4.

00:01:07.120 --> 00:01:13.960
And then the hypotenuse of
the triangle over here is 5.

00:01:13.960 --> 00:01:16.610
You only have a hypotenuse
when you have a right triangle.

00:01:16.610 --> 00:01:20.130
It is the side opposite
the right angle.

00:01:20.130 --> 00:01:23.080
And it is the longest
side of a right triangle.

00:01:23.080 --> 00:01:27.280
So that right there
is the hypotenuse.

00:01:27.280 --> 00:01:29.650
You probably learned that
already from geometry.

00:01:29.650 --> 00:01:32.250
And you can verify that this
right triangle, the sides

00:01:32.250 --> 00:01:32.790
work out.

00:01:32.790 --> 00:01:34.800
We know from the
Pythagorean theorem

00:01:34.800 --> 00:01:40.550
that 3 squared
plus 4 squared has

00:01:40.550 --> 00:01:42.900
got to be equal to the
length of the longest side,

00:01:42.900 --> 00:01:46.570
the length of the hypotenuse
squared, is equal to 5 squared.

00:01:46.570 --> 00:01:48.830
So you can verify
that this works out.

00:01:48.830 --> 00:01:50.569
This satisfies the
Pythagorean theorem.

00:01:50.569 --> 00:01:52.860
Now, with that out of the
way, let's learn a little bit

00:01:52.860 --> 00:01:54.930
of trigonometry.

00:01:54.930 --> 00:01:57.627
So the core functions
of trigonometry--

00:01:57.627 --> 00:01:59.210
we're going to learn
a little bit more

00:01:59.210 --> 00:02:01.310
about what these functions mean.

00:02:01.310 --> 00:02:04.220
There is the sine function.

00:02:04.220 --> 00:02:07.680
There is the cosine function.

00:02:07.680 --> 00:02:10.789
And there is the
tangent function.

00:02:10.789 --> 00:02:15.730
And you write S-I-N,
C-O-S, and tan for short.

00:02:15.730 --> 00:02:19.930
And these really just specify--
for any angle in this triangle,

00:02:19.930 --> 00:02:22.692
it'll specify the
ratios of certain sides.

00:02:22.692 --> 00:02:24.150
So let me just
write something out.

00:02:24.150 --> 00:02:25.733
And this is a little
bit of a mnemonic

00:02:25.733 --> 00:02:27.640
here, so something just
to help you remember

00:02:27.640 --> 00:02:29.957
the definitions of
these functions.

00:02:29.957 --> 00:02:31.540
But I'm going to
write down something.

00:02:31.540 --> 00:02:39.437
It's called soh cah toa.

00:02:39.437 --> 00:02:41.770
And you'll be amazed how far
this mnemonic will take you

00:02:41.770 --> 00:02:43.120
in trigonometry.

00:02:43.120 --> 00:02:44.620
So we have soh cah toa.

00:02:44.620 --> 00:02:47.840
And what this tells
us-- soh tells us

00:02:47.840 --> 00:02:52.748
that sine is equal to
opposite over hypotenuse.

00:02:58.260 --> 00:03:00.910
It's telling us-- and this won't
make a lot of sense just yet.

00:03:00.910 --> 00:03:03.600
I'll do it a little bit
more detail in a second.

00:03:03.600 --> 00:03:08.030
And then cosine is equal to
adjacent over hypotenuse.

00:03:12.900 --> 00:03:14.355
And then you finally
have tangent.

00:03:17.530 --> 00:03:22.010
Tangent is equal to
opposite over adjacent.

00:03:22.010 --> 00:03:23.510
So you're probably
saying, hey, Sal.

00:03:23.510 --> 00:03:25.870
What is all this opposite,
hypotenuse, adjacent?

00:03:25.870 --> 00:03:27.105
What are we talking about?

00:03:27.105 --> 00:03:30.250
Well, let's take an angle here.

00:03:30.250 --> 00:03:33.920
Let's say that this
angle right over here

00:03:33.920 --> 00:03:36.080
is theta, between
the side of length 4

00:03:36.080 --> 00:03:37.580
and the side of length 5.

00:03:37.580 --> 00:03:39.430
This angle right here is theta.

00:03:39.430 --> 00:03:42.030
So let's figure out
what the sine of theta,

00:03:42.030 --> 00:03:45.920
the cosine of theta, and what
the tangent of theta are.

00:03:45.920 --> 00:03:51.230
So if we want to first
focus on the sine of theta,

00:03:51.230 --> 00:03:54.740
we just have to
remember soh cah toa.

00:03:54.740 --> 00:03:57.130
Sine is opposite
over hypotenuse.

00:03:57.130 --> 00:04:00.170
So sine of theta is
equal to the opposite.

00:04:00.170 --> 00:04:02.890
So what's the opposite
side to the angle?

00:04:02.890 --> 00:04:04.840
So this is our angle right here.

00:04:04.840 --> 00:04:10.080
The opposite side, so
not one of the sides

00:04:10.080 --> 00:04:12.770
that are kind of
adjacent to the angle.

00:04:12.770 --> 00:04:15.160
The opposite side is the 3.

00:04:15.160 --> 00:04:16.709
It's opening onto that 3.

00:04:16.709 --> 00:04:19.089
So the opposite side is 3.

00:04:19.089 --> 00:04:21.564
And then what's the hypotenuse?

00:04:21.564 --> 00:04:22.480
Well, we already know.

00:04:22.480 --> 00:04:24.150
The hypotenuse here is 5.

00:04:24.150 --> 00:04:27.185
So it's 3 over 5.

00:04:27.185 --> 00:04:28.569
The sine of theta is 3/5.

00:04:28.569 --> 00:04:30.610
So if someone says, hey,
what's the sine of that?

00:04:30.610 --> 00:04:32.059
It's 3/5.

00:04:32.059 --> 00:04:33.600
And I'm going to
show you in a second

00:04:33.600 --> 00:04:36.970
that if this angle
is a certain angle,

00:04:36.970 --> 00:04:38.840
it's always going to be 3/5.

00:04:38.840 --> 00:04:40.810
The ratio of the opposite
to the hypotenuse

00:04:40.810 --> 00:04:43.850
is always going to be the same,
even if the actual triangle

00:04:43.850 --> 00:04:45.754
were a larger triangle
or a smaller one.

00:04:45.754 --> 00:04:47.170
So I'll show you
that in a second.

00:04:47.170 --> 00:04:49.128
But let's go through all
of the trig functions.

00:04:49.128 --> 00:04:54.030
Let's think about what
the cosine of theta is.

00:04:54.030 --> 00:04:57.260
Cosine is adjacent
over hypotenuse.

00:04:57.260 --> 00:04:58.310
So remember.

00:04:58.310 --> 00:04:59.880
Let me label them.

00:04:59.880 --> 00:05:03.310
We already figured out that
the 3 was the opposite side.

00:05:03.310 --> 00:05:04.730
This is the opposite side.

00:05:04.730 --> 00:05:06.740
And only when we're
talking about this angle.

00:05:06.740 --> 00:05:09.240
When you talk about this angle,
this side is opposite to it.

00:05:09.240 --> 00:05:13.460
When you talk about this angle,
this 4 side is adjacent to it.

00:05:13.460 --> 00:05:16.160
It's one of the sides
that kind of make up,

00:05:16.160 --> 00:05:18.940
that kind of form
the vertex here.

00:05:18.940 --> 00:05:23.050
So this right here
is an adjacent side.

00:05:23.050 --> 00:05:24.520
And I want to be very clear.

00:05:24.520 --> 00:05:26.310
This only applies to this angle.

00:05:26.310 --> 00:05:27.810
If we were talking
about that angle,

00:05:27.810 --> 00:05:29.640
then this green side
would be opposite

00:05:29.640 --> 00:05:31.520
and this yellow side
would be adjacent.

00:05:31.520 --> 00:05:34.320
But we're just focusing on
this angle right over here.

00:05:34.320 --> 00:05:37.380
So cosine of this angle--
we care about adjacent.

00:05:37.380 --> 00:05:39.970
Well, the adjacent side
to this angle is 4.

00:05:39.970 --> 00:05:42.880
So it is adjacent
over the hypotenuse.

00:05:42.880 --> 00:05:47.690
It's the adjacent, which is
4, over the hypotenuse-- 4/5.

00:05:51.224 --> 00:05:52.265
Now let's do the tangent.

00:05:55.850 --> 00:05:59.640
The tangent of theta,
opposite over adjacent.

00:05:59.640 --> 00:06:01.430
The opposite side is 3.

00:06:04.300 --> 00:06:05.754
What is the adjacent side?

00:06:05.754 --> 00:06:06.920
We already figured that out.

00:06:06.920 --> 00:06:09.540
The adjacent side is 4.

00:06:09.540 --> 00:06:11.720
So knowing the sides
of this right triangle,

00:06:11.720 --> 00:06:14.810
we were able to figure
out the major trig ratios.

00:06:14.810 --> 00:06:16.710
And we'll see there
are other trig ratios,

00:06:16.710 --> 00:06:21.510
but they can all be derived
from these three basic trig

00:06:21.510 --> 00:06:22.750
functions.

00:06:22.750 --> 00:06:25.000
Now, let's think about another
angle in this triangle.

00:06:25.000 --> 00:06:27.190
And I'll redraw it just
because my triangle

00:06:27.190 --> 00:06:28.800
is getting a little bit messy.

00:06:28.800 --> 00:06:30.540
So let's redraw the
exact same triangle.

00:06:34.090 --> 00:06:36.620
And once again, the
lengths of this triangle

00:06:36.620 --> 00:06:41.290
are we have length 4 there,
we have length 3 there,

00:06:41.290 --> 00:06:43.370
and we have length 5 there.

00:06:43.370 --> 00:06:46.370
The last example,
we used this theta.

00:06:46.370 --> 00:06:53.450
But let's do another
angle up here.

00:06:53.450 --> 00:06:55.160
And let's call this
angle-- I don't know.

00:06:55.160 --> 00:06:58.080
I'll think of something,
a random Greek letter.

00:06:58.080 --> 00:06:59.750
So let's say it's psi.

00:06:59.750 --> 00:07:01.210
I know it's a
little bit bizarre.

00:07:01.210 --> 00:07:02.501
Theta is what you normally use.

00:07:02.501 --> 00:07:04.487
But since I already used
theta, let's use psi.

00:07:04.487 --> 00:07:06.570
Actually, instead of psi,
let me just simplify it.

00:07:06.570 --> 00:07:09.990
Let me call this angle x.

00:07:09.990 --> 00:07:12.920
So let's figure out the trig
functions for that angle x.

00:07:12.920 --> 00:07:17.830
So we have sine of x is
going to be equal to what.

00:07:17.830 --> 00:07:20.820
Well, sine is opposite
over hypotenuse.

00:07:20.820 --> 00:07:23.790
So what side is opposite to x?

00:07:23.790 --> 00:07:27.170
Well, it opens onto this 4.

00:07:27.170 --> 00:07:29.330
So in this context, this
is now the opposite.

00:07:32.400 --> 00:07:37.410
Remember, 4 was adjacent to this
theta, but it's opposite to x.

00:07:37.410 --> 00:07:40.412
So it's going to be 4 over--
now, what's the hypotenuse?

00:07:40.412 --> 00:07:41.870
Well, the hypotenuse
is going to be

00:07:41.870 --> 00:07:44.340
the same regardless of
which angle you pick.

00:07:44.340 --> 00:07:46.860
So the hypotenuse is
now going to be 5.

00:07:46.860 --> 00:07:48.589
So it's 4/5.

00:07:48.589 --> 00:07:49.630
Now let's do another one.

00:07:49.630 --> 00:07:54.750
What is the cosine of x?

00:07:54.750 --> 00:07:57.450
So cosine is adjacent
over hypotenuse.

00:07:57.450 --> 00:07:59.127
What side is adjacent to x?

00:07:59.127 --> 00:08:00.210
That's not the hypotenuse.

00:08:00.210 --> 00:08:01.940
You have the hypotenuse here.

00:08:01.940 --> 00:08:04.920
Well, the 3 side-- it's
one of the sides that

00:08:04.920 --> 00:08:07.960
forms the vertex that x is at,
and it's not the hypotenuse.

00:08:07.960 --> 00:08:09.990
So this is the adjacent side.

00:08:09.990 --> 00:08:10.960
That is adjacent.

00:08:10.960 --> 00:08:14.000
So it's equal to 3
over the hypotenuse.

00:08:14.000 --> 00:08:16.550
The hypotenuse is 5.

00:08:16.550 --> 00:08:19.630
And then finally, the tangent.

00:08:19.630 --> 00:08:22.200
We want to figure
out the tangent of x.

00:08:22.200 --> 00:08:24.330
Tangent is opposite
over adjacent.

00:08:24.330 --> 00:08:29.330
Soh cah toa-- tangent is
opposite over adjacent.

00:08:29.330 --> 00:08:32.280
The opposite side is 4.

00:08:32.280 --> 00:08:34.730
I want to do it in
that blue color.

00:08:34.730 --> 00:08:42.929
The opposite side is 4,
and the adjacent side is 3.

00:08:42.929 --> 00:08:43.820
And we're done.

00:08:43.820 --> 00:08:46.230
In the next video, I'll do a
ton of more examples of this

00:08:46.230 --> 00:08:48.122
just so that we really
get a feel for it.

00:08:48.122 --> 00:08:49.580
But I'll leave you
thinking of what

00:08:49.580 --> 00:08:52.650
happens when these angles
start to approach 90 degrees,

00:08:52.650 --> 00:08:54.880
or how could they even get
larger than 90 degrees.

00:08:54.880 --> 00:08:57.379
And what we're going to see is
that this definition, the soh

00:08:57.379 --> 00:08:59.900
cah toa definition, takes us
a long way for angles that

00:08:59.900 --> 00:09:04.000
are between 0 and 90 degrees, or
that are less than 90 degrees.

00:09:04.000 --> 00:09:06.730
But they kind of start to mess
up really at the boundaries.

00:09:06.730 --> 00:09:08.938
And we're going to introduce
a new definition, that's

00:09:08.938 --> 00:09:11.090
kind of derived from the
soh cah toa definition,

00:09:11.090 --> 00:09:14.040
for finding the sine,
cosine, and tangent of really

00:09:14.040 --> 00:09:15.872
any angle.


Pangaea

WEBVTT
Kind: captions
Language: en

00:00:00.700 --> 00:00:03.310
We know that new plate
material is being formed,

00:00:03.310 --> 00:00:07.090
and these lithosphere plates
on the surface of the Earth

00:00:07.090 --> 00:00:08.420
are moving around.

00:00:08.420 --> 00:00:10.820
And that might raise the
question in your brain--

00:00:10.820 --> 00:00:13.130
what happens if we
kind of reverse things?

00:00:13.130 --> 00:00:15.900
We know the direction
they're moving in.

00:00:15.900 --> 00:00:18.480
What does that tell us
about where they came from?

00:00:18.480 --> 00:00:20.340
So let's just do the
thought experiment.

00:00:20.340 --> 00:00:22.280
Right now, South
America and Africa

00:00:22.280 --> 00:00:26.850
are moving away from each other,
because of new plate material

00:00:26.850 --> 00:00:28.750
being created at the
mid-Atlantic rift.

00:00:28.750 --> 00:00:29.660
Let's rewind it.

00:00:29.660 --> 00:00:30.955
Let's bring them back together.

00:00:33.890 --> 00:00:37.480
We know that India is jamming
into the Eurasian Plate right

00:00:37.480 --> 00:00:40.810
now, causing the Himalayas
to get higher and higher.

00:00:40.810 --> 00:00:42.310
What if we rewind that?

00:00:42.310 --> 00:00:45.280
Let's bring India back
down towards Antarctica.

00:00:45.280 --> 00:00:46.710
Same thing with Australia.

00:00:46.710 --> 00:00:52.100
We have new plate
material being formed

00:00:52.100 --> 00:00:54.550
between Australia
and Antarctica that's

00:00:54.550 --> 00:00:56.480
making the continents
move apart.

00:00:56.480 --> 00:00:58.050
Let's bring them back together.

00:00:58.050 --> 00:01:01.550
Let's rewind the clock.

00:01:01.550 --> 00:01:04.090
Even North America-- it's not
as obvious from this diagram,

00:01:04.090 --> 00:01:06.170
but if you actually
look at the GPS data,

00:01:06.170 --> 00:01:09.290
it becomes pretty obvious
that North America, right now

00:01:09.290 --> 00:01:13.990
is moving in a
counterclockwise rotation.

00:01:13.990 --> 00:01:18.610
So let's rewind it
into a-- let's go back,

00:01:18.610 --> 00:01:21.000
moving it in a
clockwise direction.

00:01:21.000 --> 00:01:23.640
Let's, instead of Eurasia
going further away

00:01:23.640 --> 00:01:26.110
from North America, let's
bring it back together.

00:01:26.110 --> 00:01:27.890
And so what you could
imagine is a reality

00:01:27.890 --> 00:01:29.890
where India, Australia
are jammed down

00:01:29.890 --> 00:01:33.000
into South America--
sorry, into Antarctica.

00:01:33.000 --> 00:01:35.420
South America and Africa
are jammed together.

00:01:35.420 --> 00:01:37.320
North America is
jammed in there.

00:01:37.320 --> 00:01:39.960
And essentially, Eurasia
is also jammed in there.

00:01:39.960 --> 00:01:42.580
So it looks like they
all would clump together

00:01:42.580 --> 00:01:45.370
if you go back a few
hundred million years.

00:01:45.370 --> 00:01:48.890
And based on, literally--
based on just that thought

00:01:48.890 --> 00:01:51.930
experiment, you could
imagine at one point,

00:01:51.930 --> 00:01:53.580
all of the continents
on the world

00:01:53.580 --> 00:01:57.070
were merged into
one supercontinent.

00:01:57.070 --> 00:02:02.460
And that supercontinent
is called Pangaea-- pan

00:02:02.460 --> 00:02:05.710
for entire, or whole, and
gaea, coming from Gaia,

00:02:05.710 --> 00:02:07.370
for the world.

00:02:07.370 --> 00:02:10.240
And it turns out that all of
the evidence we've seen actually

00:02:10.240 --> 00:02:14.360
does make us believe that there
was a supercontinent called--

00:02:14.360 --> 00:02:15.796
well, we call it Pangaea, now.

00:02:15.796 --> 00:02:17.170
Obviously, there
probably weren't

00:02:17.170 --> 00:02:20.012
things on the planet calling
it anything back then.

00:02:20.012 --> 00:02:22.525
Or, there were things
back then, but not

00:02:22.525 --> 00:02:24.900
things that would actually go
and try to label continents

00:02:24.900 --> 00:02:26.310
that we know of.

00:02:26.310 --> 00:02:30.370
But all of the evidence tells us
that Pangaea existed about 200

00:02:30.370 --> 00:02:34.540
to 300 million years ago,
roughly maybe 250 million,

00:02:34.540 --> 00:02:36.342
give or take, years ago.

00:02:36.342 --> 00:02:37.300
And I want to be clear.

00:02:37.300 --> 00:02:42.990
This was not the
first supercontinent.

00:02:42.990 --> 00:02:46.490
To a large degree, it's kind of
the most recent supercontinent.

00:02:46.490 --> 00:02:48.800
And it's easiest
for us to construct

00:02:48.800 --> 00:02:51.100
because it was the
most recent one.

00:02:51.100 --> 00:02:53.480
But we believe that there
were other supercontinents

00:02:53.480 --> 00:02:54.830
before this.

00:02:54.830 --> 00:02:56.510
That if you rewind
even more that you

00:02:56.510 --> 00:02:58.750
would have to break up
Pangaea and it would reform.

00:02:58.750 --> 00:03:00.170
But we're now
going back in time.

00:03:00.170 --> 00:03:01.920
Or that there were
several supercontinents

00:03:01.920 --> 00:03:04.260
in the past that broke up,
reformed, broke up, reformed.

00:03:04.260 --> 00:03:06.380
And the last time we
had a supercontinent

00:03:06.380 --> 00:03:10.210
was Pangaea, about
250 million years ago.

00:03:10.210 --> 00:03:16.050
And now it's broken up into
our current day geography.

00:03:16.050 --> 00:03:17.840
Now, I won't go into
all of the detail

00:03:17.840 --> 00:03:22.540
why we believe that there was a
Pangaea about 250 million years

00:03:22.540 --> 00:03:25.970
ago-- or, this diagram tells
us, about 225 million years ago,

00:03:25.970 --> 00:03:26.980
give or take.

00:03:26.980 --> 00:03:29.610
But I'll go into some of
the interesting evidence.

00:03:29.610 --> 00:03:31.350
On a very high
level, you have a lot

00:03:31.350 --> 00:03:33.710
of rock commonalities
between things

00:03:33.710 --> 00:03:36.156
that would have had to
combine during Pangaea.

00:03:36.156 --> 00:03:37.780
And probably the most
interesting thing

00:03:37.780 --> 00:03:38.770
is the fossil evidence.

00:03:41.340 --> 00:03:42.840
There are a whole
bunch of fossils.

00:03:42.840 --> 00:03:45.940
And here are examples
of it, from species

00:03:45.940 --> 00:03:50.110
that were around between 200
and 300 million years ago.

00:03:50.110 --> 00:03:52.740
And their fossils are found
in a very specific place.

00:03:52.740 --> 00:03:56.230
This animal right
here, cynognathus--

00:03:56.230 --> 00:03:59.770
I hope I'm pronouncing
that right-- cynognathus.

00:03:59.770 --> 00:04:02.070
This animal's fossils
are only found

00:04:02.070 --> 00:04:04.870
in this area of South
America on a nice clean band

00:04:04.870 --> 00:04:07.620
here, and in this
part of Africa.

00:04:07.620 --> 00:04:09.710
So not only does
South America look

00:04:09.710 --> 00:04:13.490
like it fits very
nicely into Africa.

00:04:13.490 --> 00:04:15.220
But the fossil
evidence also makes

00:04:15.220 --> 00:04:17.810
it look like there was
a nice clean band where

00:04:17.810 --> 00:04:20.600
this animal lived and
where we find the fossils.

00:04:20.600 --> 00:04:23.810
So it really makes it seem
like these were connected,

00:04:23.810 --> 00:04:25.870
at least when this
animal lived, maybe

00:04:25.870 --> 00:04:28.840
on the order of 250
million years ago.

00:04:28.840 --> 00:04:31.070
This species right
over here, its fossils

00:04:31.070 --> 00:04:33.310
are found in this
area-- let me do it

00:04:33.310 --> 00:04:36.720
in a color that has more
contrast-- in this area

00:04:36.720 --> 00:04:38.060
right over here.

00:04:38.060 --> 00:04:40.060
This plant, its
fossils-- now, this

00:04:40.060 --> 00:04:42.670
starts to connect to a lot of
dots between a lot of cont--

00:04:42.670 --> 00:04:46.390
its fossils are found
in this entire area,

00:04:46.390 --> 00:04:50.100
across South America,
Africa, Antarctica,

00:04:50.100 --> 00:04:52.230
India, and Australia.

00:04:52.230 --> 00:04:55.151
And so not only does it
look like the continents fit

00:04:55.151 --> 00:04:56.650
together in a puzzle
piece, not only

00:04:56.650 --> 00:04:58.436
do we get it to a
configuration like this

00:04:58.436 --> 00:05:00.310
if we essentially just
rewind to the movement

00:05:00.310 --> 00:05:02.990
that we're seeing now-- but
the fossil evidence also

00:05:02.990 --> 00:05:07.030
kind of confirms that they
fit together in this way,

00:05:07.030 --> 00:05:10.340
This animal right
here, we find fossils

00:05:10.340 --> 00:05:13.180
on this nice stripe that goes
from Africa through India,

00:05:13.180 --> 00:05:14.920
all the way to Antarctica.

00:05:14.920 --> 00:05:19.880
Now, this only gives us evidence
of the Southern Hemisphere

00:05:19.880 --> 00:05:21.780
of Pangaea.

00:05:21.780 --> 00:05:23.030
But there is other evidence.

00:05:23.030 --> 00:05:26.590
We find kind of
continuing mountain chains

00:05:26.590 --> 00:05:28.960
between North
America and Europe.

00:05:28.960 --> 00:05:31.650
We find rock evidence,
where just the way

00:05:31.650 --> 00:05:33.760
we see the fossils
line up nicely.

00:05:33.760 --> 00:05:36.980
We see common rock that lines
up nicely between South America

00:05:36.980 --> 00:05:40.300
and Africa and other continents
that were at once connected.

00:05:40.300 --> 00:05:42.450
So all the evidence, as
far as we can tell now,

00:05:42.450 --> 00:05:46.970
does make us think that there
at one time was a Pangaea.

00:05:46.970 --> 00:05:49.889
And, for all we know,
all the continents

00:05:49.889 --> 00:05:50.930
are going to keep moving.

00:05:50.930 --> 00:05:53.640
And maybe in a few
hundred million years,

00:05:53.640 --> 00:05:55.740
we'll have another
supercontinent.

00:05:55.740 --> 00:05:57.202
Who knows?


Finding dimensions of triangle from area

WEBVTT
Kind: captions
Language: en

00:00:00.670 --> 00:00:03.490
The height of a triangle
is four inches less

00:00:03.490 --> 00:00:06.750
than the length of the base.

00:00:06.750 --> 00:00:09.300
The area of the triangle
is 30 inches squared.

00:00:09.300 --> 00:00:11.200
Find the height and base.

00:00:11.200 --> 00:00:14.170
Use the formula area
equals one half base times

00:00:14.170 --> 00:00:17.521
height for the
area of a triangle.

00:00:17.521 --> 00:00:18.020
OK.

00:00:18.020 --> 00:00:19.640
So let's think about
it a little bit.

00:00:19.640 --> 00:00:22.630
We have the-- let me
draw a triangle here.

00:00:22.630 --> 00:00:25.240
So this is our triangle.

00:00:25.240 --> 00:00:28.060
And let's say that the
length of this bottom side,

00:00:28.060 --> 00:00:30.083
that's the base,
let's call that b.

00:00:30.083 --> 00:00:32.400
And then this is the height.

00:00:32.400 --> 00:00:35.390
This is the height
right over here.

00:00:35.390 --> 00:00:38.090
And then the area is equal to
one half base times height.

00:00:41.270 --> 00:00:42.770
Now in this first
sentence they tell

00:00:42.770 --> 00:00:45.900
us at the height
of a triangle is

00:00:45.900 --> 00:00:50.460
four inch is less than
the length of the base.

00:00:50.460 --> 00:00:54.740
So the height is equal
to the base minus 4.

00:00:54.740 --> 00:00:56.540
That's what that first
sentence tells us.

00:00:56.540 --> 00:00:59.220
The area of the triangle
is 30 inches squared.

00:00:59.220 --> 00:01:01.240
So if we take one half
the base times the height

00:01:01.240 --> 00:01:02.810
we'll get 30 inches squared.

00:01:02.810 --> 00:01:08.460
Or we could say that
30 inches squared

00:01:08.460 --> 00:01:17.850
is equal to one half times
the base, times the height.

00:01:17.850 --> 00:01:19.852
Now instead of putting
an h in for height,

00:01:19.852 --> 00:01:21.560
we know that the height
is the same thing

00:01:21.560 --> 00:01:23.850
as 4 less than the base.

00:01:23.850 --> 00:01:25.380
So let's put that in there.

00:01:25.380 --> 00:01:27.741
4 less than the base.

00:01:27.741 --> 00:01:29.240
And then let's see
what we get here.

00:01:29.240 --> 00:01:31.480
We get-- let me
do this in yellow.

00:01:31.480 --> 00:01:35.490
We get 30 is equal to
one half times-- let's

00:01:35.490 --> 00:01:39.560
distribute the b-- times
b, let me make it clear.

00:01:39.560 --> 00:01:41.690
So let's do it this way.

00:01:41.690 --> 00:01:45.830
Times b over 2,
times b, minus 4.

00:01:45.830 --> 00:01:47.760
I just multiplied the
one half times the b.

00:01:47.760 --> 00:01:49.690
Now let's distribute
the b over 2.

00:01:49.690 --> 00:01:56.320
So 30 is equal to b
squared over 2, be careful.

00:01:56.320 --> 00:01:58.980
b over 2 times b is
just b squared over 2.

00:01:58.980 --> 00:02:05.300
And then b over 2, times
negative 4 is negative 2b.

00:02:05.300 --> 00:02:07.650
Now just to get rid
of this fraction

00:02:07.650 --> 00:02:10.880
here let's multiply both
sides of this equation by 2.

00:02:10.880 --> 00:02:13.040
So let's multiply
that side by 2.

00:02:13.040 --> 00:02:15.400
And let's multiply
that side by 2.

00:02:15.400 --> 00:02:17.390
On the left hand
side you get 60.

00:02:17.390 --> 00:02:19.740
On the right hand side
2 times b squared over 2

00:02:19.740 --> 00:02:21.350
is just b squared.

00:02:21.350 --> 00:02:24.870
Negative 2b times
2 is negative 4b.

00:02:24.870 --> 00:02:27.450
And now we have
a quadratic here.

00:02:27.450 --> 00:02:29.537
And the best way to
solve a quadratic-- we

00:02:29.537 --> 00:02:31.120
have a second degree
term right here--

00:02:31.120 --> 00:02:34.360
is to get all of the terms
on one side of the equation,

00:02:34.360 --> 00:02:36.280
having them equal 0.

00:02:36.280 --> 00:02:39.240
So let's subtract 60 from
both sides of this equation.

00:02:43.620 --> 00:02:51.327
And we get 0 equal to b
squared, minus 4b, minus 60.

00:02:51.327 --> 00:02:52.910
And so what we need
to do here is just

00:02:52.910 --> 00:02:56.180
factor this thing right
now, or factor it.

00:02:56.180 --> 00:02:59.250
And then, no-- if I have
the product of some things,

00:02:59.250 --> 00:03:02.080
and that equals 0, that
means that either one or both

00:03:02.080 --> 00:03:03.950
of those things need
to be equal to 0.

00:03:03.950 --> 00:03:09.150
So we need to factor b
squared, minus 4b, minus 60.

00:03:09.150 --> 00:03:12.470
So what we want to do, we
want to find two numbers

00:03:12.470 --> 00:03:19.505
whose sum is negative 4 and
whose product is negative 60.

00:03:28.070 --> 00:03:30.430
Now, given that the
product is negative,

00:03:30.430 --> 00:03:32.480
we know there are
different signs.

00:03:32.480 --> 00:03:34.920
And this tells us that
their absolute values

00:03:34.920 --> 00:03:36.530
are going to be four apart.

00:03:36.530 --> 00:03:39.510
That one is going to be
four less than the others.

00:03:39.510 --> 00:03:42.746
So you could look at the
products of the factors of 60.

00:03:42.746 --> 00:03:44.900
1 and 60 are too far apart.

00:03:44.900 --> 00:03:46.920
Even if you made
one of the negative,

00:03:46.920 --> 00:03:50.350
you would either get positive
59 as the sum or negative 59

00:03:50.350 --> 00:03:51.480
as the sum.

00:03:51.480 --> 00:03:54.450
2 and 30, still too far apart.

00:03:54.450 --> 00:03:59.800
3 and 20, still too far apart.

00:03:59.800 --> 00:04:01.260
If you had made
one negative you'd

00:04:01.260 --> 00:04:05.280
either get negative
17 or positive 17.

00:04:05.280 --> 00:04:08.986
Then you could have 4 and
15, still too far apart.

00:04:08.986 --> 00:04:10.360
If you made one
of them negative,

00:04:10.360 --> 00:04:13.940
their sum would be either
negative 11 or positive 11.

00:04:13.940 --> 00:04:18.040
Then you have 5 and a 12,
still seems too far apart

00:04:18.040 --> 00:04:18.844
from each other.

00:04:18.844 --> 00:04:20.510
One of them is negative,
then you either

00:04:20.510 --> 00:04:24.410
have their sum being
positive 7 or negative 7.

00:04:24.410 --> 00:04:26.586
Then you have 6 and 10.

00:04:26.586 --> 00:04:27.710
Now this looks interesting.

00:04:27.710 --> 00:04:29.180
They are four apart.

00:04:29.180 --> 00:04:33.094
So if we make-- and we want
the larger absolute magnitude

00:04:33.094 --> 00:04:35.260
number to be negative so
that their sum is negative.

00:04:35.260 --> 00:04:37.180
So if we make it
6 and negative 10

00:04:37.180 --> 00:04:40.130
their sum will be negative
4, and their product

00:04:40.130 --> 00:04:41.340
is negative 60.

00:04:41.340 --> 00:04:42.270
So that works.

00:04:42.270 --> 00:04:43.770
So you could literally
say that this

00:04:43.770 --> 00:04:55.260
is equal to b plus 6, times
b, minus 10. b plus the a,

00:04:55.260 --> 00:04:57.210
plus b minus the b.

00:04:57.210 --> 00:04:59.210
And let me be very careful here.

00:04:59.210 --> 00:05:01.230
This b over here, I want
to make it very clear,

00:05:01.230 --> 00:05:05.740
is different than the b that
we're using in the equation.

00:05:05.740 --> 00:05:07.500
I just used this b
here to say, look,

00:05:07.500 --> 00:05:11.314
we're looking for two numbers
that add up to this second term

00:05:11.314 --> 00:05:11.980
right over here.

00:05:11.980 --> 00:05:12.771
It's a different b.

00:05:12.771 --> 00:05:15.720
I could have said x plus
y is equal to negative 4,

00:05:15.720 --> 00:05:18.470
and x times y is
equal to negative 60.

00:05:18.470 --> 00:05:21.070
In fact, let me do it that way
just so we don't get confused.

00:05:21.070 --> 00:05:25.090
So we could write x plus
y is equal to negative 4.

00:05:25.090 --> 00:05:28.600
And then we have x times
y is equal to negative 60.

00:05:28.600 --> 00:05:33.390
So we have b plus 6,
times b plus y. x is 6,

00:05:33.390 --> 00:05:35.300
y is negative 10.

00:05:35.300 --> 00:05:39.446
And that is equal to 0.

00:05:39.446 --> 00:05:40.820
Let's just solve
this right here.

00:05:40.820 --> 00:05:42.319
And then we'll go
back and show you.

00:05:42.319 --> 00:05:44.980
You could also factor
this by grouping.

00:05:44.980 --> 00:05:47.240
But just from this, we know
that either one of these

00:05:47.240 --> 00:05:48.120
is equal to zero.

00:05:48.120 --> 00:05:57.320
Either b plus 6 is equal to 0,
or b minus 10 is equal to 0.

00:05:57.320 --> 00:05:59.530
If we subtract 6 from both
sides of this equation,

00:05:59.530 --> 00:06:01.480
we get b is equal to negative 6.

00:06:01.480 --> 00:06:04.290
Or if you add 10 to both
sides of this equation,

00:06:04.290 --> 00:06:06.970
you get b is equal to 10.

00:06:06.970 --> 00:06:09.660
And those are our two solutions.

00:06:09.660 --> 00:06:11.740
You could put them
back in and verify

00:06:11.740 --> 00:06:14.200
that they satisfy
our constraints.

00:06:14.200 --> 00:06:16.492
Now the other way that
you could solve this,

00:06:16.492 --> 00:06:18.200
and we're going to
get exact same answer.

00:06:18.200 --> 00:06:21.110
Is you could just break
up this negative 4b

00:06:21.110 --> 00:06:22.550
into its constituents.

00:06:22.550 --> 00:06:28.080
So you could have broken this
up into 0 is equal to b squared.

00:06:28.080 --> 00:06:34.860
And then you could have broken
it up into plus 6b, minus 10b,

00:06:34.860 --> 00:06:36.210
minus 60.

00:06:36.210 --> 00:06:38.050
And then factor it by grouping.

00:06:38.050 --> 00:06:39.530
Group these first two terms.

00:06:39.530 --> 00:06:41.680
Group these second two terms.

00:06:41.680 --> 00:06:43.390
Just going to add them together.

00:06:43.390 --> 00:06:45.280
The first one you
could factor out a b.

00:06:45.280 --> 00:06:48.240
So you have b times b, plus 6.

00:06:48.240 --> 00:06:50.440
The second one you can
factor out a negative 10.

00:06:50.440 --> 00:06:54.210
So minus 10 times b, plus 6.

00:06:54.210 --> 00:06:55.680
All that's equal to 0.

00:06:55.680 --> 00:06:58.180
And now you can
factor out a b plus 6.

00:06:58.180 --> 00:07:01.280
So if you factor
out a b plus 6 here,

00:07:01.280 --> 00:07:09.020
you get 0 is equal to b
minus 10, times b, plus 6.

00:07:09.020 --> 00:07:16.090
We're literally just factoring
out this out of the expression.

00:07:16.090 --> 00:07:18.770
You're just left
with a b minus 10.

00:07:18.770 --> 00:07:21.930
You get the same thing that
we did in one step over here.

00:07:21.930 --> 00:07:23.020
Whatever works for you.

00:07:23.020 --> 00:07:24.920
But either way,
the solutions are

00:07:24.920 --> 00:07:27.529
either b is equal to negative
6, or b is equal to 10.

00:07:27.529 --> 00:07:28.820
And we have to be careful here.

00:07:28.820 --> 00:07:30.440
Remember, this is
a word problem.

00:07:30.440 --> 00:07:33.580
We can't just state, oh b could
be negative 6 or b could be 10.

00:07:33.580 --> 00:07:36.510
We have to think about
whether this makes sense

00:07:36.510 --> 00:07:39.480
in the context of
the actual problem.

00:07:39.480 --> 00:07:42.500
We're talking about
lengths of triangles,

00:07:42.500 --> 00:07:44.500
or lengths of the
sides of triangles.

00:07:44.500 --> 00:07:48.720
We can't have a negative length.

00:07:48.720 --> 00:07:50.910
So because of that,
the base of a triangle

00:07:50.910 --> 00:07:52.840
can't have length of negative 6.

00:07:52.840 --> 00:07:54.210
So we can cross that out.

00:07:54.210 --> 00:07:56.340
So we actually only
have one solution here.

00:07:56.340 --> 00:07:57.697
Almost made a careless mistake.

00:07:57.697 --> 00:07:59.780
Forgot that we were dealing
with the word problem.

00:07:59.780 --> 00:08:02.720
The only possible base is 10.

00:08:02.720 --> 00:08:06.130
And let's see, they say find
the height and the base.

00:08:06.130 --> 00:08:07.320
Once again, done.

00:08:07.320 --> 00:08:09.590
So the base we're saying is 10.

00:08:09.590 --> 00:08:11.350
The height is four inches less.

00:08:11.350 --> 00:08:12.620
It's b minus 4.

00:08:12.620 --> 00:08:14.810
So the height is 6.

00:08:14.810 --> 00:08:15.870
And then you can verify.

00:08:15.870 --> 00:08:20.469
The area is 6 times 10
times one half, which is 30.


Factoring trinomials with a non-1 leading coefficient by grouping

WEBVTT
Kind: captions
Language: en

00:00:00.680 --> 00:00:04.890
Factor 4y squared
plus 4y minus 15.

00:00:04.890 --> 00:00:08.039
So we don't have a leading
coefficient of 1 here.

00:00:08.039 --> 00:00:10.640
And it also doesn't look
like we can factor out a 4,

00:00:10.640 --> 00:00:13.099
since 15 is not divisible by 4.

00:00:13.099 --> 00:00:15.640
So what we're going to try to
do, is factor this by grouping.

00:00:15.640 --> 00:00:17.120
In the factor by
grouping, we want

00:00:17.120 --> 00:00:19.770
to look for two numbers
whose product is

00:00:19.770 --> 00:00:23.490
equal to the product
of 4 and negative 15.

00:00:23.490 --> 00:00:27.130
We explained it in other
videos why this will work.

00:00:27.130 --> 00:00:29.360
But let's just look
for those two numbers.

00:00:29.360 --> 00:00:35.510
So the product is going to be
equal to 4 times negative 15,

00:00:35.510 --> 00:00:37.240
which is negative 60.

00:00:37.240 --> 00:00:42.720
And their sum needs to be equal
to this 4 right over here.

00:00:42.720 --> 00:00:46.150
So a plus b has got
to be equal to 4.

00:00:46.150 --> 00:00:48.670
And if we can figure out an
a and b that satisfy this,

00:00:48.670 --> 00:00:52.550
so we can break up this
4y into an ay plus a by.

00:00:52.550 --> 00:00:55.370
And then we can
factor by grouping.

00:00:55.370 --> 00:00:58.800
Now the first clue here is
that the product of the numbers

00:00:58.800 --> 00:00:59.650
is negative.

00:00:59.650 --> 00:01:02.900
So that means that a and b
must have different signs.

00:01:05.870 --> 00:01:09.376
And since a plus
b is equal to 4,

00:01:09.376 --> 00:01:10.750
we know that the--
and I want you

00:01:10.750 --> 00:01:11.820
to think about
this a little bit.

00:01:11.820 --> 00:01:14.210
We know that the difference
of their absolute values

00:01:14.210 --> 00:01:15.660
must be equal to 4.

00:01:15.660 --> 00:01:18.160
Let me just make that concrete,
that little statement I just

00:01:18.160 --> 00:01:18.960
told you.

00:01:18.960 --> 00:01:21.810
If I were to say, well,
let's just say that I have,

00:01:21.810 --> 00:01:23.160
let's ignore this for a second.

00:01:23.160 --> 00:01:24.659
Let me just take
two simple numbers.

00:01:24.659 --> 00:01:28.190
If I have negative 2 times 3.

00:01:28.190 --> 00:01:32.480
Those two, the product is going
to be equal to negative 6,

00:01:32.480 --> 00:01:37.014
and then their sum, negative
2 plus 3 is equal to one.

00:01:37.014 --> 00:01:39.120
Notice, their
product is negative,

00:01:39.120 --> 00:01:41.210
so they had to have
different signs.

00:01:41.210 --> 00:01:43.740
And their sum is 1 and
they have different signs.

00:01:43.740 --> 00:01:47.060
And one is also the difference
of their absolute values.

00:01:47.060 --> 00:01:49.090
The difference between
the absolute value of 3

00:01:49.090 --> 00:01:52.620
and the absolute
value of 2 is one.

00:01:52.620 --> 00:01:53.870
That's what I'm talking about.

00:01:53.870 --> 00:01:55.453
So this gives us a
clue of essentially

00:01:55.453 --> 00:01:58.830
how far apart are the
absolute values of the number.

00:01:58.830 --> 00:02:01.470
So let's think about all
of the factors of 60.

00:02:01.470 --> 00:02:03.090
And then remind
ourselves that we're

00:02:03.090 --> 00:02:06.720
looking for numbers
that are 4 apart,

00:02:06.720 --> 00:02:09.400
numbers whose absolute
values are 4 apart.

00:02:09.400 --> 00:02:11.540
One has to be positive,
and one has to be negative.

00:02:11.540 --> 00:02:15.420
So the factors of 60,
you could do 1 and 60,

00:02:15.420 --> 00:02:18.070
you could do 2 and 30.

00:02:18.070 --> 00:02:20.070
I'm going to run out of
space doing it this way,

00:02:20.070 --> 00:02:21.236
let me write it as a column.

00:02:21.236 --> 00:02:29.450
So you have 1 and 60, you have
2 and 30, you have 3 and 20,

00:02:29.450 --> 00:02:36.360
you have 4 and 15, you have
5 and 12, you have 6 and 10.

00:02:36.360 --> 00:02:39.850
This looks pretty tantalizing
since these are 4 apart.

00:02:39.850 --> 00:02:42.890
And then that looks
like we are about done.

00:02:42.890 --> 00:02:46.740
7 doesn't go into it, 8 doesn't
go into it, 9 doesn't go do it.

00:02:46.740 --> 00:02:47.620
We're done.

00:02:47.620 --> 00:02:49.830
Now already, we already
talked about the difference

00:02:49.830 --> 00:02:52.020
of the numbers, the absolute
value of the difference

00:02:52.020 --> 00:02:54.530
has to be 4, so 6 and 10
looks pretty tempting.

00:02:54.530 --> 00:02:57.340
But let's just go through all
of these numbers over here.

00:02:57.340 --> 00:03:01.510
So the possible sums here,
if we make 1 positive and 60

00:03:01.510 --> 00:03:04.769
negative, then it's going
to be a negative 59.

00:03:04.769 --> 00:03:07.060
Remember, I'm making sure
that we have different signs.

00:03:07.060 --> 00:03:09.400
If this is positive, this
could be positive or negative.

00:03:09.400 --> 00:03:10.911
And this is negative
or positive.

00:03:10.911 --> 00:03:12.410
I wrote it upside
down here, to show

00:03:12.410 --> 00:03:13.820
that their signs are different.

00:03:13.820 --> 00:03:16.970
So it's either going to be
negative 59 if this is positive

00:03:16.970 --> 00:03:22.060
and this is negative, or it's
going to be positive 69 if this

00:03:22.060 --> 00:03:23.590
is positive and
this is negative.

00:03:23.590 --> 00:03:27.080
So it's negative or
positive 59, I should say.

00:03:27.080 --> 00:03:32.870
This right over here is going
to be negative or positive 28.

00:03:32.870 --> 00:03:35.530
You see that the absolute
value of, the difference

00:03:35.530 --> 00:03:42.115
of the absolute values of
these numbers keeps showing up.

00:03:42.115 --> 00:03:43.990
Now we know that we want
to go to 4, so let's

00:03:43.990 --> 00:03:46.340
just cut to the chase
and go to 6 and 10.

00:03:46.340 --> 00:03:49.400
If we take positive--
and we want

00:03:49.400 --> 00:03:51.010
them to add up to a positive 4.

00:03:51.010 --> 00:03:53.490
So the larger number
should be positive.

00:03:53.490 --> 00:03:56.590
So if we take positive
10 and negative 6,

00:03:56.590 --> 00:03:59.820
their sum is definitely
going to be equal to 4,

00:03:59.820 --> 00:04:03.680
and positive 10 times
negative 6 is negative 60.

00:04:03.680 --> 00:04:05.900
So let's break it up like that.

00:04:05.900 --> 00:04:08.200
So what we could do is
break up the 4 as a sum,

00:04:08.200 --> 00:04:13.490
break up the 4y as a sum
of a negative 6y and a 10y.

00:04:13.490 --> 00:04:14.490
So let's do that.

00:04:14.490 --> 00:04:22.870
So 4y, I'm going to write
as negative 6y plus 10y.

00:04:22.870 --> 00:04:25.280
If you add these two
together, clearly you get 4y,

00:04:25.280 --> 00:04:27.490
and then we have the
rest of the problem here.

00:04:27.490 --> 00:04:31.390
We have the 4y squared minus 15.

00:04:31.390 --> 00:04:34.054
Now we're ready to
factor by grouping.

00:04:34.054 --> 00:04:35.470
Let's group these
first two terms,

00:04:35.470 --> 00:04:37.710
and we can because they
have common factors here.

00:04:37.710 --> 00:04:39.224
They're both
divisible by 2 and y.

00:04:39.224 --> 00:04:41.640
And then we're going to group
these two guys, both of them

00:04:41.640 --> 00:04:43.680
are divisible by 5.

00:04:43.680 --> 00:04:45.970
So if we factor this
first one over here,

00:04:45.970 --> 00:04:51.730
let's factor out a 2y over here.

00:04:51.730 --> 00:04:56.580
4y squared divided by
2y just gives us a 2y.

00:04:56.580 --> 00:05:01.870
And then negative 6y divided
by 2y is negative 3, right?

00:05:01.870 --> 00:05:06.540
y divided by y is 1, negative
6 divided by 2 is negative 3.

00:05:06.540 --> 00:05:10.030
Then let's factor
this guy over here.

00:05:10.030 --> 00:05:17.560
We can factor out a 5, so this
is plus 5 times 2y minus 3.

00:05:17.560 --> 00:05:21.660
And now lucky for us, we
can factor out a 2y minus 3.

00:05:25.250 --> 00:05:31.805
So we have 2y minus
3, times 2y, plus 5.

00:05:35.210 --> 00:05:37.605
And we are done.


Factoring sum of cubes

WEBVTT
Kind: captions
Language: en

00:00:00.499 --> 00:00:05.230
Factor 27x to the
sixth plus 125.

00:00:05.230 --> 00:00:07.400
So this is a pretty
interesting problem.

00:00:07.400 --> 00:00:08.900
And frankly, the
only way to do this

00:00:08.900 --> 00:00:11.590
is if you recognize
it as a special form.

00:00:11.590 --> 00:00:13.260
And what I want to
do is kind of show

00:00:13.260 --> 00:00:15.320
you the special form first.

00:00:15.320 --> 00:00:17.150
And then we can kind
of pattern match.

00:00:17.150 --> 00:00:19.960
So the special form
is if I were to take--

00:00:19.960 --> 00:00:22.250
and this is really just
something you need to know.

00:00:22.250 --> 00:00:25.329
And I'd argue whether you
really need to know this.

00:00:25.329 --> 00:00:26.870
But actually to do
this problem, it's

00:00:26.870 --> 00:00:28.320
something you just need to know.

00:00:28.320 --> 00:00:33.560
And that's if you have a
squared minus ab plus b squared

00:00:33.560 --> 00:00:36.331
and you multiply
that times a plus b,

00:00:36.331 --> 00:00:38.080
let's think about what
we're going to get.

00:00:38.080 --> 00:00:39.590
So we're going to take
a product right here.

00:00:39.590 --> 00:00:40.580
We're multiplying.

00:00:40.580 --> 00:00:42.980
So let's do some
algebraic multiplication.

00:00:42.980 --> 00:00:46.230
So let's multiply. b times
b squared is b to the third.

00:00:46.230 --> 00:00:51.000
b times negative ab is
negative ab squared.

00:00:51.000 --> 00:00:55.300
b times a squared
is a squared b.

00:00:55.300 --> 00:00:58.340
Now let's multiply
this top term times a.

00:00:58.340 --> 00:01:02.090
a times b squared is ab squared.

00:01:02.090 --> 00:01:08.480
a times negative ab is
negative a squared b.

00:01:08.480 --> 00:01:11.660
And then a times a
squared is a to the third.

00:01:11.660 --> 00:01:16.164
And then we just have to
add up all of the terms.

00:01:16.164 --> 00:01:18.580
We have a positive a squared
b and a negative a squared b.

00:01:18.580 --> 00:01:19.760
So these guys cancel out.

00:01:19.760 --> 00:01:23.350
We have a negative ab squared
and a positive ab squared.

00:01:23.350 --> 00:01:24.710
These guys cancel out.

00:01:24.710 --> 00:01:29.630
So all we're left with is an
a to the third here and then

00:01:29.630 --> 00:01:34.560
plus this b to the third.

00:01:34.560 --> 00:01:36.110
Or another way to
think about it--

00:01:36.110 --> 00:01:39.530
if someone gives you a to the
third plus b to the third,

00:01:39.530 --> 00:01:42.480
this can be factored into
these two expressions.

00:01:42.480 --> 00:01:48.980
That can be factored into a
plus b times a squared minus

00:01:48.980 --> 00:01:51.180
ab plus b squared.

00:01:51.180 --> 00:01:53.190
So this is essentially
the special form.

00:01:53.190 --> 00:01:56.630
If you have a sum of cubes,
it can be factored out

00:01:56.630 --> 00:02:02.670
as the sum of the
cube roots times

00:02:02.670 --> 00:02:04.290
this expression right here.

00:02:04.290 --> 00:02:05.950
And we just showed
that it works.

00:02:05.950 --> 00:02:08.630
So let's see if we have
that special form here.

00:02:08.630 --> 00:02:12.820
Well, 27 is definitely
the cube of 3.

00:02:12.820 --> 00:02:14.760
3 to the third power is 27.

00:02:14.760 --> 00:02:18.500
x to the sixth is also
the cube of x squared.

00:02:18.500 --> 00:02:21.100
If you raise x to the
sixth to the 1/3 power,

00:02:21.100 --> 00:02:22.590
you get x squared.

00:02:22.590 --> 00:02:24.440
So this first term
right over here

00:02:24.440 --> 00:02:28.930
can be rewritten as 3x
squared to the third power.

00:02:28.930 --> 00:02:32.580
And the second term right here--
that's 5 to the third power,

00:02:32.580 --> 00:02:35.050
so plus 5 to the third power.

00:02:35.050 --> 00:02:38.680
And this might be a little
bit confusing for you.

00:02:38.680 --> 00:02:40.080
It never hurts to review.

00:02:40.080 --> 00:02:46.030
Let's multiply 3x squared times
3x squared times 3x squared.

00:02:46.030 --> 00:02:49.650
That is literally equal
to 3 times 3 times 3 times

00:02:49.650 --> 00:02:53.850
x squared times x
squared times x squared.

00:02:53.850 --> 00:02:56.020
This part right here is 27.

00:02:56.020 --> 00:02:59.100
x squared times x squared is x
to the fourth, times x squared

00:02:59.100 --> 00:03:01.006
is x to the sixth.

00:03:01.006 --> 00:03:03.380
Or you could just raise both
of these to the third power.

00:03:03.380 --> 00:03:04.960
3 to the third is 27.

00:03:04.960 --> 00:03:08.000
x squared to the third
power-- you take an exponent

00:03:08.000 --> 00:03:09.742
to an exponent, and
you're going to take

00:03:09.742 --> 00:03:10.950
the product of the exponents.

00:03:10.950 --> 00:03:15.000
So it'll be x to the 2 times
3, or x to the sixth power.

00:03:15.000 --> 00:03:17.830
So now we know that
we have this pattern.

00:03:17.830 --> 00:03:19.280
So we can just use this.

00:03:19.280 --> 00:03:21.040
We have the sum of cubes.

00:03:21.040 --> 00:03:23.270
So just by using this
pattern right over here,

00:03:23.270 --> 00:03:25.620
that means that we
can factor it as.

00:03:25.620 --> 00:03:31.285
This is going to be equal to
3x squared-- that's our a.

00:03:31.285 --> 00:03:32.160
Let me make it clear.

00:03:32.160 --> 00:03:34.200
This right here is our a.

00:03:34.200 --> 00:03:36.450
This right here is our b.

00:03:36.450 --> 00:03:38.000
So it's going to be a plus b.

00:03:38.000 --> 00:03:44.360
So it's going to be 3x
squared plus b, plus 5,

00:03:44.360 --> 00:03:48.000
times a squared.

00:03:48.000 --> 00:03:49.250
Let me do this in a new color.

00:03:49.250 --> 00:03:53.900
So 3x squared squared-- let's
think about that for a second.

00:03:53.900 --> 00:03:56.630
3x squared squared--
well, that's going

00:03:56.630 --> 00:03:59.000
to be 9x to the fourth.

00:03:59.000 --> 00:04:01.710
So it's going to be
times 9x to the fourth

00:04:01.710 --> 00:04:04.720
minus the product
of these two things.

00:04:04.720 --> 00:04:11.500
So minus the product of 5 and 3x
squared, so minus 15x squared.

00:04:11.500 --> 00:04:14.590
And then finally plus
b squared. b is 5.

00:04:14.590 --> 00:04:18.790
So it's going to be 5
squared, so plus 25.

00:04:18.790 --> 00:04:21.139
And when I said b
is-- this is b, not

00:04:21.139 --> 00:04:22.180
the whole 5 to the third.

00:04:22.180 --> 00:04:24.570
And when I say a,
just this part is a.

00:04:24.570 --> 00:04:25.650
And we're done.

00:04:25.650 --> 00:04:28.600
And I won't explain it
in detail in this video.

00:04:28.600 --> 00:04:31.200
But this right here, if we're
thinking about real numbers,

00:04:31.200 --> 00:04:33.540
we can't actually
factor this any more.

00:04:33.540 --> 00:04:35.200
So we are done factoring this.

00:04:35.200 --> 00:04:39.090
And remember, this is really
just a very, very, very special

00:04:39.090 --> 00:04:44.942
case of being able to
recognize the sum of cubes.