WEBVTT

00:00:06.830 --> 00:00:08.339
Welcome back to recitation.

00:00:08.339 --> 00:00:10.880
In this video, I'd like us to
do the following problem, which

00:00:10.880 --> 00:00:14.720
is going to be relating polar
and Cartesian coordinates.

00:00:14.720 --> 00:00:17.885
So I want you to write each
of the following in Cartesian

00:00:17.885 --> 00:00:20.010
coordinates, and that means
our (x, y) coordinates,

00:00:20.010 --> 00:00:22.420
and then describe the curve.

00:00:22.420 --> 00:00:27.020
So the first one is r squared
equals 4r cosine theta,

00:00:27.020 --> 00:00:31.854
and the second one is r equals
9 tangent theta secant theta.

00:00:31.854 --> 00:00:34.270
So again, what I'd like you
to do is convert each of these

00:00:34.270 --> 00:00:37.450
to something in the Cartesian
coordinates, in the (x, y)

00:00:37.450 --> 00:00:39.860
coordinates, and then I
want you to describe what

00:00:39.860 --> 00:00:41.284
the curve actually looks like.

00:00:41.284 --> 00:00:43.200
So I'll give you a little
while to work on it,

00:00:43.200 --> 00:00:45.408
and then when I come back,
I'll show you how I do it.

00:00:54.390 --> 00:00:55.430
OK, welcome back.

00:00:55.430 --> 00:00:57.570
Well, hopefully you were
able to get pretty far

00:00:57.570 --> 00:01:02.300
in describing these two
curves in (x, y) coordinates.

00:01:02.300 --> 00:01:05.730
And I will show you how I
attacked these problems.

00:01:05.730 --> 00:01:08.070
So we'll start with (a).

00:01:08.070 --> 00:01:13.160
So for (a)-- I'm going to
rewrite the problem up here,

00:01:13.160 --> 00:01:17.870
so we can just be focused
on what's up here.

00:01:17.870 --> 00:01:22.360
So we had r squared
equals 4r cosine theta.

00:01:22.360 --> 00:01:24.000
Well, we know what r squared is.

00:01:24.000 --> 00:01:26.250
That's nice in terms
of x and y coordinates.

00:01:26.250 --> 00:01:28.620
That's just x squared
plus y squared.

00:01:28.620 --> 00:01:32.710
So we know that, so
we'll replace that.

00:01:32.710 --> 00:01:35.940
And then we can actually replace
all the r's and thetas over

00:01:35.940 --> 00:01:37.480
here pretty easily, as well.

00:01:37.480 --> 00:01:41.270
Because we know r cosine
theta describes x.

00:01:41.270 --> 00:01:44.570
So the Cartesian coordinate
x is the polar coordinate--

00:01:44.570 --> 00:01:47.790
or described in polar
coordinates as r cosine theta.

00:01:47.790 --> 00:01:50.500
So we can just write that as 4x.

00:01:50.500 --> 00:01:54.410
And the reason I asked you to
describe the curve is because

00:01:54.410 --> 00:01:56.310
from here, you could
say, oh, well I wrote it

00:01:56.310 --> 00:01:58.910
in the Cartesian coordinates.

00:01:58.910 --> 00:02:00.910
I wrote it in x, y,
and so now I'm done.

00:02:00.910 --> 00:02:03.450
But the point is
that you can actually

00:02:03.450 --> 00:02:07.172
work on this equation right
here and get into a form

00:02:07.172 --> 00:02:08.130
that you can recognize.

00:02:08.130 --> 00:02:10.320
That it'll be a
recognizable curve.

00:02:10.320 --> 00:02:12.710
So let's see if we can sort
of play around with this,

00:02:12.710 --> 00:02:15.060
and come up with something
that looks familiar.

00:02:15.060 --> 00:02:19.590
And what you might think to
do, would be, say, you know,

00:02:19.590 --> 00:02:22.170
subtract off the x squared,
or subtract off the y squared.

00:02:22.170 --> 00:02:24.606
Try and solve for
x or solve for y.

00:02:24.606 --> 00:02:26.980
But that can be a little bit
dangerous in this situation,

00:02:26.980 --> 00:02:30.470
because in fact, y might
not be a function of x.

00:02:30.470 --> 00:02:32.310
So we might run into
some trouble there.

00:02:32.310 --> 00:02:34.590
But if you'll notice,
there's something kind of,

00:02:34.590 --> 00:02:36.940
a glaring way we should go.

00:02:36.940 --> 00:02:38.720
And that's because we
have this x squared

00:02:38.720 --> 00:02:42.880
plus y squared together-- this
maybe could look something

00:02:42.880 --> 00:02:45.320
like a circle or an ellipse
or something like that,

00:02:45.320 --> 00:02:47.540
if we could figure out
a way to put this part

00:02:47.540 --> 00:02:50.260
in with the x squared.

00:02:50.260 --> 00:02:52.910
So this is kind
of-- it's a good way

00:02:52.910 --> 00:02:56.320
to think about what direction
to head in this problem.

00:02:56.320 --> 00:02:59.607
In particular, it would be
a bad idea for this problem

00:02:59.607 --> 00:03:01.190
for you to subtract
x squared and take

00:03:01.190 --> 00:03:03.070
the square root of both sides.

00:03:03.070 --> 00:03:05.590
Because you would lose
some information about what

00:03:05.590 --> 00:03:06.860
this curve was.

00:03:06.860 --> 00:03:07.360
OK?

00:03:07.360 --> 00:03:08.810
Because when you
take the square root,

00:03:08.810 --> 00:03:09.990
you would have to
say, well, do I

00:03:09.990 --> 00:03:11.100
want the positive
square root, or do I

00:03:11.100 --> 00:03:12.349
want the negative square root?

00:03:12.349 --> 00:03:14.170
We'd lose a little
bit of information.

00:03:14.170 --> 00:03:16.250
So we do not want
to solve for y.

00:03:16.250 --> 00:03:17.850
So let's do what I said.

00:03:17.850 --> 00:03:21.890
Let's try and figure out a way
to get this 4x into something

00:03:21.890 --> 00:03:23.610
to do with this x squared term.

00:03:23.610 --> 00:03:31.360
So I'm going to subtract 4x
and rewrite the equation here.

00:03:31.360 --> 00:03:33.155
And so you might
say, well, Christine,

00:03:33.155 --> 00:03:35.130
this doesn't really
seem that helpful.

00:03:35.130 --> 00:03:37.812
It's just the same
thing moved around.

00:03:37.812 --> 00:03:40.020
But we're going to use one
of our favorite techniques

00:03:40.020 --> 00:03:43.250
from integration, which
is completing the square.

00:03:43.250 --> 00:03:47.230
So we can actually complete
the square on this guy right

00:03:47.230 --> 00:03:50.670
here, and turn it
into a perfect square.

00:03:50.670 --> 00:03:52.580
We'll have to add an
extra term, but once we

00:03:52.580 --> 00:03:55.200
do that, we'll have a perfect
square, an extra term,

00:03:55.200 --> 00:03:56.060
and a y squared.

00:03:56.060 --> 00:03:58.310
And we're getting more into
the form of something that

00:03:58.310 --> 00:03:59.630
actually looks like a circle.

00:03:59.630 --> 00:04:00.280
So let's see.

00:04:00.280 --> 00:04:02.380
Completing the
square on this, it's

00:04:02.380 --> 00:04:06.140
going to be x squared
minus 4x plus 4.

00:04:06.140 --> 00:04:07.110
How did I know that?

00:04:07.110 --> 00:04:09.430
Well, if I want to complete
the square on this,

00:04:09.430 --> 00:04:11.430
I need something
that, multiplied by 2,

00:04:11.430 --> 00:04:13.260
gives me negative 4.

00:04:13.260 --> 00:04:14.250
That's 2.

00:04:14.250 --> 00:04:16.030
And then 2 squared is 4.

00:04:16.030 --> 00:04:17.700
So that's where the 4 comes in.

00:04:17.700 --> 00:04:21.960
To keep this equal, I'll add
4 to the other side, as well.

00:04:21.960 --> 00:04:25.010
So if I add 4 to both sides, I
haven't changed the equality,

00:04:25.010 --> 00:04:28.269
and I keep my y squared
along for the ride.

00:04:28.269 --> 00:04:29.560
So now I have a perfect square.

00:04:29.560 --> 00:04:30.640
What does this give me?

00:04:30.640 --> 00:04:35.620
This gives me x minus 2
quantity squared plus 4-- plus 4

00:04:35.620 --> 00:04:38.380
squared-- plus y squared.

00:04:38.380 --> 00:04:40.450
So x minus 2 quantity
squared-- that came

00:04:40.450 --> 00:04:44.850
from these three terms--
plus y squared equals four.

00:04:44.850 --> 00:04:48.070
And now it's a curve we
can describe, clearly.

00:04:48.070 --> 00:04:49.110
What curve is this?

00:04:49.110 --> 00:04:52.370
Well, it's obviously a circle.

00:04:52.370 --> 00:04:57.280
It's centered at the point 2
comma 0, and it has radius 2.

00:04:57.280 --> 00:05:00.920
We've talked, or you've seen
this in the lecture videos,

00:05:00.920 --> 00:05:04.600
I believe, what the
form for a circle is.

00:05:04.600 --> 00:05:07.970
x minus a quality squared plus
y minus b quantity squared

00:05:07.970 --> 00:05:09.290
equals r squared.

00:05:09.290 --> 00:05:14.800
So this is, a is 2,
b is 0, and r is 2.

00:05:14.800 --> 00:05:18.230
so it's a circle of radius
2, centered at 2 comma 0.

00:05:18.230 --> 00:05:22.070
So we have a good way to
describe what started off

00:05:22.070 --> 00:05:23.630
in polar coordinates.

00:05:23.630 --> 00:05:27.030
We can now describe it
in (x, y) coordinates.

00:05:27.030 --> 00:05:28.320
OK.

00:05:28.320 --> 00:05:31.590
So now let's move on to (b).

00:05:31.590 --> 00:05:34.340
And I'm going to rewrite
(b) over here as well,

00:05:34.340 --> 00:05:38.220
so we don't have to worry
about it, looking back.

00:05:38.220 --> 00:05:45.311
r equals 9 tan
theta secant theta.

00:05:45.311 --> 00:05:45.810
OK.

00:05:45.810 --> 00:05:46.726
So let's look at this.

00:05:46.726 --> 00:05:49.280
Now, there's some
information buried in here,

00:05:49.280 --> 00:05:51.120
in terms of (x, y) coordinates.

00:05:51.120 --> 00:05:53.020
And one thing that
should stand out to you

00:05:53.020 --> 00:05:55.580
is, what is secant theta?

00:05:55.580 --> 00:05:58.291
Secant theta is 1
over cosine theta.

00:05:58.291 --> 00:05:58.790
Right?

00:05:58.790 --> 00:06:01.500
And if we have 1 over
cosine theta over here,

00:06:01.500 --> 00:06:03.800
we can multiply both
sides by cosine theta,

00:06:03.800 --> 00:06:06.257
and we get an r cosine
theta over here.

00:06:06.257 --> 00:06:07.590
So I'm going to write that down.

00:06:07.590 --> 00:06:10.320
That this actually
is in the same--

00:06:10.320 --> 00:06:16.750
this is the same as r cosine
theta equals 9 tan theta.

00:06:16.750 --> 00:06:19.370
Right?

00:06:19.370 --> 00:06:22.050
I mean, you could get
mad at me about where

00:06:22.050 --> 00:06:23.700
this is defined
in terms of theta,

00:06:23.700 --> 00:06:26.500
but I'm not worrying about
that in this situation,

00:06:26.500 --> 00:06:27.540
just right now.

00:06:27.540 --> 00:06:29.160
We're just trying
to figure out how

00:06:29.160 --> 00:06:31.320
we could write this in x and y.

00:06:31.320 --> 00:06:33.870
We know what our
cosine theta is.

00:06:33.870 --> 00:06:36.830
Again, it's x, as it was before.

00:06:36.830 --> 00:06:39.090
What about tan theta?

00:06:39.090 --> 00:06:41.950
Tangent theta,
remember, if you recall,

00:06:41.950 --> 00:06:46.610
this tangent theta is
opposite over adjacent, right?

00:06:46.610 --> 00:06:50.100
And in this case, opposite is
the y, and adjacent is the x.

00:06:50.100 --> 00:06:51.860
This is something
you saw a picture of,

00:06:51.860 --> 00:06:54.190
you can see a picture
of pretty easily.

00:06:54.190 --> 00:06:58.841
So this is x is equal
to 9 times y over x.

00:06:58.841 --> 00:06:59.340
Right?

00:06:59.340 --> 00:07:05.770
Which is x squared is equal 9y.

00:07:05.770 --> 00:07:07.920
So this is in fact
how you could write

00:07:07.920 --> 00:07:12.740
this expression that's in r
and theta in terms of x and y.

00:07:12.740 --> 00:07:14.630
And so this, if you
look at it, is actually

00:07:14.630 --> 00:07:17.730
a parabola that goes
through the point (0, 0),

00:07:17.730 --> 00:07:22.814
and is stretched by a
factor of 9, or 1/9.

00:07:22.814 --> 00:07:24.230
Well, I guess you
can say, there's

00:07:24.230 --> 00:07:27.302
a vertical stretch or
horizontal stretch,

00:07:27.302 --> 00:07:28.510
you can pick which one it is.

00:07:28.510 --> 00:07:31.940
And in one case, it's
going to be by 3 or 1/3.

00:07:31.940 --> 00:07:33.100
I always mix those up.

00:07:33.100 --> 00:07:33.990
I'd have to check.

00:07:33.990 --> 00:07:35.520
Or by 9 or 1/9.

00:07:35.520 --> 00:07:39.230
So essentially, it's going to be
a parabola with some stretching

00:07:39.230 --> 00:07:40.240
on it.

00:07:40.240 --> 00:07:44.274
Now, the problem is that
you might say, well,

00:07:44.274 --> 00:07:45.440
it's not really all of that.

00:07:45.440 --> 00:07:46.840
Because secant
theta is not going

00:07:46.840 --> 00:07:50.170
to be defined for all
theta the way cosine is.

00:07:50.170 --> 00:07:52.350
So you do potentially
run into some problems.

00:07:52.350 --> 00:07:55.710
You might have to worry
about what part of the domain

00:07:55.710 --> 00:07:58.306
makes sense for theta, so
that this is well-defined.

00:07:58.306 --> 00:07:59.680
And so that this
is well-defined,

00:07:59.680 --> 00:08:01.510
what part of the
curve is carved out.

00:08:01.510 --> 00:08:03.110
That's a little more
technical than I

00:08:03.110 --> 00:08:04.680
want to go in this video.

00:08:04.680 --> 00:08:07.560
But some of you might look
at it and say, oh, she's

00:08:07.560 --> 00:08:08.720
missing something.

00:08:08.720 --> 00:08:12.730
Yeah, you caught something that
I'm intentionally ignoring.

00:08:12.730 --> 00:08:14.700
So the main point
of this was just so

00:08:14.700 --> 00:08:20.310
that you could see how you
can take these functions of r

00:08:20.310 --> 00:08:23.820
and theta and turn them
into functions of x and y,

00:08:23.820 --> 00:08:27.330
and then figure out kind of
what the curves might look like.

00:08:27.330 --> 00:08:29.600
So I'm going to stop there.

00:08:29.600 --> 00:08:31.080
Hopefully this was
a good exercise

00:08:31.080 --> 00:08:35.680
to get you understanding how
these different coordinates

00:08:35.680 --> 00:08:37.240
relate to one another.

00:08:37.240 --> 00:08:40.810
And yeah, that's
where we'll leave it.