1
00:00:06,830 --> 00:00:08,339
Welcome back to recitation.
2
00:00:08,339 --> 00:00:10,880
In this video, I'd like us to
do the following problem, which
3
00:00:10,880 --> 00:00:14,720
is going to be relating polar
and Cartesian coordinates.
4
00:00:14,720 --> 00:00:17,885
So I want you to write each
of the following in Cartesian
5
00:00:17,885 --> 00:00:20,010
coordinates, and that means
our (x, y) coordinates,
6
00:00:20,010 --> 00:00:22,420
and then describe the curve.
7
00:00:22,420 --> 00:00:27,020
So the first one is r squared
equals 4r cosine theta,
8
00:00:27,020 --> 00:00:31,854
and the second one is r equals
9 tangent theta secant theta.
9
00:00:31,854 --> 00:00:34,270
So again, what I'd like you
to do is convert each of these
10
00:00:34,270 --> 00:00:37,450
to something in the Cartesian
coordinates, in the (x, y)
11
00:00:37,450 --> 00:00:39,860
coordinates, and then I
want you to describe what
12
00:00:39,860 --> 00:00:41,284
the curve actually looks like.
13
00:00:41,284 --> 00:00:43,200
So I'll give you a little
while to work on it,
14
00:00:43,200 --> 00:00:45,408
and then when I come back,
I'll show you how I do it.
15
00:00:54,390 --> 00:00:55,430
OK, welcome back.
16
00:00:55,430 --> 00:00:57,570
Well, hopefully you were
able to get pretty far
17
00:00:57,570 --> 00:01:02,300
in describing these two
curves in (x, y) coordinates.
18
00:01:02,300 --> 00:01:05,730
And I will show you how I
attacked these problems.
19
00:01:05,730 --> 00:01:08,070
So we'll start with (a).
20
00:01:08,070 --> 00:01:13,160
So for (a)-- I'm going to
rewrite the problem up here,
21
00:01:13,160 --> 00:01:17,870
so we can just be focused
on what's up here.
22
00:01:17,870 --> 00:01:22,360
So we had r squared
equals 4r cosine theta.
23
00:01:22,360 --> 00:01:24,000
Well, we know what r squared is.
24
00:01:24,000 --> 00:01:26,250
That's nice in terms
of x and y coordinates.
25
00:01:26,250 --> 00:01:28,620
That's just x squared
plus y squared.
26
00:01:28,620 --> 00:01:32,710
So we know that, so
we'll replace that.
27
00:01:32,710 --> 00:01:35,940
And then we can actually replace
all the r's and thetas over
28
00:01:35,940 --> 00:01:37,480
here pretty easily, as well.
29
00:01:37,480 --> 00:01:41,270
Because we know r cosine
theta describes x.
30
00:01:41,270 --> 00:01:44,570
So the Cartesian coordinate
x is the polar coordinate--
31
00:01:44,570 --> 00:01:47,790
or described in polar
coordinates as r cosine theta.
32
00:01:47,790 --> 00:01:50,500
So we can just write that as 4x.
33
00:01:50,500 --> 00:01:54,410
And the reason I asked you to
describe the curve is because
34
00:01:54,410 --> 00:01:56,310
from here, you could
say, oh, well I wrote it
35
00:01:56,310 --> 00:01:58,910
in the Cartesian coordinates.
36
00:01:58,910 --> 00:02:00,910
I wrote it in x, y,
and so now I'm done.
37
00:02:00,910 --> 00:02:03,450
But the point is
that you can actually
38
00:02:03,450 --> 00:02:07,172
work on this equation right
here and get into a form
39
00:02:07,172 --> 00:02:08,130
that you can recognize.
40
00:02:08,130 --> 00:02:10,320
That it'll be a
recognizable curve.
41
00:02:10,320 --> 00:02:12,710
So let's see if we can sort
of play around with this,
42
00:02:12,710 --> 00:02:15,060
and come up with something
that looks familiar.
43
00:02:15,060 --> 00:02:19,590
And what you might think to
do, would be, say, you know,
44
00:02:19,590 --> 00:02:22,170
subtract off the x squared,
or subtract off the y squared.
45
00:02:22,170 --> 00:02:24,606
Try and solve for
x or solve for y.
46
00:02:24,606 --> 00:02:26,980
But that can be a little bit
dangerous in this situation,
47
00:02:26,980 --> 00:02:30,470
because in fact, y might
not be a function of x.
48
00:02:30,470 --> 00:02:32,310
So we might run into
some trouble there.
49
00:02:32,310 --> 00:02:34,590
But if you'll notice,
there's something kind of,
50
00:02:34,590 --> 00:02:36,940
a glaring way we should go.
51
00:02:36,940 --> 00:02:38,720
And that's because we
have this x squared
52
00:02:38,720 --> 00:02:42,880
plus y squared together-- this
maybe could look something
53
00:02:42,880 --> 00:02:45,320
like a circle or an ellipse
or something like that,
54
00:02:45,320 --> 00:02:47,540
if we could figure out
a way to put this part
55
00:02:47,540 --> 00:02:50,260
in with the x squared.
56
00:02:50,260 --> 00:02:52,910
So this is kind
of-- it's a good way
57
00:02:52,910 --> 00:02:56,320
to think about what direction
to head in this problem.
58
00:02:56,320 --> 00:02:59,607
In particular, it would be
a bad idea for this problem
59
00:02:59,607 --> 00:03:01,190
for you to subtract
x squared and take
60
00:03:01,190 --> 00:03:03,070
the square root of both sides.
61
00:03:03,070 --> 00:03:05,590
Because you would lose
some information about what
62
00:03:05,590 --> 00:03:06,860
this curve was.
63
00:03:06,860 --> 00:03:07,360
OK?
64
00:03:07,360 --> 00:03:08,810
Because when you
take the square root,
65
00:03:08,810 --> 00:03:09,990
you would have to
say, well, do I
66
00:03:09,990 --> 00:03:11,100
want the positive
square root, or do I
67
00:03:11,100 --> 00:03:12,349
want the negative square root?
68
00:03:12,349 --> 00:03:14,170
We'd lose a little
bit of information.
69
00:03:14,170 --> 00:03:16,250
So we do not want
to solve for y.
70
00:03:16,250 --> 00:03:17,850
So let's do what I said.
71
00:03:17,850 --> 00:03:21,890
Let's try and figure out a way
to get this 4x into something
72
00:03:21,890 --> 00:03:23,610
to do with this x squared term.
73
00:03:23,610 --> 00:03:31,360
So I'm going to subtract 4x
and rewrite the equation here.
74
00:03:31,360 --> 00:03:33,155
And so you might
say, well, Christine,
75
00:03:33,155 --> 00:03:35,130
this doesn't really
seem that helpful.
76
00:03:35,130 --> 00:03:37,812
It's just the same
thing moved around.
77
00:03:37,812 --> 00:03:40,020
But we're going to use one
of our favorite techniques
78
00:03:40,020 --> 00:03:43,250
from integration, which
is completing the square.
79
00:03:43,250 --> 00:03:47,230
So we can actually complete
the square on this guy right
80
00:03:47,230 --> 00:03:50,670
here, and turn it
into a perfect square.
81
00:03:50,670 --> 00:03:52,580
We'll have to add an
extra term, but once we
82
00:03:52,580 --> 00:03:55,200
do that, we'll have a perfect
square, an extra term,
83
00:03:55,200 --> 00:03:56,060
and a y squared.
84
00:03:56,060 --> 00:03:58,310
And we're getting more into
the form of something that
85
00:03:58,310 --> 00:03:59,630
actually looks like a circle.
86
00:03:59,630 --> 00:04:00,280
So let's see.
87
00:04:00,280 --> 00:04:02,380
Completing the
square on this, it's
88
00:04:02,380 --> 00:04:06,140
going to be x squared
minus 4x plus 4.
89
00:04:06,140 --> 00:04:07,110
How did I know that?
90
00:04:07,110 --> 00:04:09,430
Well, if I want to complete
the square on this,
91
00:04:09,430 --> 00:04:11,430
I need something
that, multiplied by 2,
92
00:04:11,430 --> 00:04:13,260
gives me negative 4.
93
00:04:13,260 --> 00:04:14,250
That's 2.
94
00:04:14,250 --> 00:04:16,030
And then 2 squared is 4.
95
00:04:16,030 --> 00:04:17,700
So that's where the 4 comes in.
96
00:04:17,700 --> 00:04:21,960
To keep this equal, I'll add
4 to the other side, as well.
97
00:04:21,960 --> 00:04:25,010
So if I add 4 to both sides, I
haven't changed the equality,
98
00:04:25,010 --> 00:04:28,269
and I keep my y squared
along for the ride.
99
00:04:28,269 --> 00:04:29,560
So now I have a perfect square.
100
00:04:29,560 --> 00:04:30,640
What does this give me?
101
00:04:30,640 --> 00:04:35,620
This gives me x minus 2
quantity squared plus 4-- plus 4
102
00:04:35,620 --> 00:04:38,380
squared-- plus y squared.
103
00:04:38,380 --> 00:04:40,450
So x minus 2 quantity
squared-- that came
104
00:04:40,450 --> 00:04:44,850
from these three terms--
plus y squared equals four.
105
00:04:44,850 --> 00:04:48,070
And now it's a curve we
can describe, clearly.
106
00:04:48,070 --> 00:04:49,110
What curve is this?
107
00:04:49,110 --> 00:04:52,370
Well, it's obviously a circle.
108
00:04:52,370 --> 00:04:57,280
It's centered at the point 2
comma 0, and it has radius 2.
109
00:04:57,280 --> 00:05:00,920
We've talked, or you've seen
this in the lecture videos,
110
00:05:00,920 --> 00:05:04,600
I believe, what the
form for a circle is.
111
00:05:04,600 --> 00:05:07,970
x minus a quality squared plus
y minus b quantity squared
112
00:05:07,970 --> 00:05:09,290
equals r squared.
113
00:05:09,290 --> 00:05:14,800
So this is, a is 2,
b is 0, and r is 2.
114
00:05:14,800 --> 00:05:18,230
so it's a circle of radius
2, centered at 2 comma 0.
115
00:05:18,230 --> 00:05:22,070
So we have a good way to
describe what started off
116
00:05:22,070 --> 00:05:23,630
in polar coordinates.
117
00:05:23,630 --> 00:05:27,030
We can now describe it
in (x, y) coordinates.
118
00:05:27,030 --> 00:05:28,320
OK.
119
00:05:28,320 --> 00:05:31,590
So now let's move on to (b).
120
00:05:31,590 --> 00:05:34,340
And I'm going to rewrite
(b) over here as well,
121
00:05:34,340 --> 00:05:38,220
so we don't have to worry
about it, looking back.
122
00:05:38,220 --> 00:05:45,311
r equals 9 tan
theta secant theta.
123
00:05:45,311 --> 00:05:45,810
OK.
124
00:05:45,810 --> 00:05:46,726
So let's look at this.
125
00:05:46,726 --> 00:05:49,280
Now, there's some
information buried in here,
126
00:05:49,280 --> 00:05:51,120
in terms of (x, y) coordinates.
127
00:05:51,120 --> 00:05:53,020
And one thing that
should stand out to you
128
00:05:53,020 --> 00:05:55,580
is, what is secant theta?
129
00:05:55,580 --> 00:05:58,291
Secant theta is 1
over cosine theta.
130
00:05:58,291 --> 00:05:58,790
Right?
131
00:05:58,790 --> 00:06:01,500
And if we have 1 over
cosine theta over here,
132
00:06:01,500 --> 00:06:03,800
we can multiply both
sides by cosine theta,
133
00:06:03,800 --> 00:06:06,257
and we get an r cosine
theta over here.
134
00:06:06,257 --> 00:06:07,590
So I'm going to write that down.
135
00:06:07,590 --> 00:06:10,320
That this actually
is in the same--
136
00:06:10,320 --> 00:06:16,750
this is the same as r cosine
theta equals 9 tan theta.
137
00:06:16,750 --> 00:06:19,370
Right?
138
00:06:19,370 --> 00:06:22,050
I mean, you could get
mad at me about where
139
00:06:22,050 --> 00:06:23,700
this is defined
in terms of theta,
140
00:06:23,700 --> 00:06:26,500
but I'm not worrying about
that in this situation,
141
00:06:26,500 --> 00:06:27,540
just right now.
142
00:06:27,540 --> 00:06:29,160
We're just trying
to figure out how
143
00:06:29,160 --> 00:06:31,320
we could write this in x and y.
144
00:06:31,320 --> 00:06:33,870
We know what our
cosine theta is.
145
00:06:33,870 --> 00:06:36,830
Again, it's x, as it was before.
146
00:06:36,830 --> 00:06:39,090
What about tan theta?
147
00:06:39,090 --> 00:06:41,950
Tangent theta,
remember, if you recall,
148
00:06:41,950 --> 00:06:46,610
this tangent theta is
opposite over adjacent, right?
149
00:06:46,610 --> 00:06:50,100
And in this case, opposite is
the y, and adjacent is the x.
150
00:06:50,100 --> 00:06:51,860
This is something
you saw a picture of,
151
00:06:51,860 --> 00:06:54,190
you can see a picture
of pretty easily.
152
00:06:54,190 --> 00:06:58,841
So this is x is equal
to 9 times y over x.
153
00:06:58,841 --> 00:06:59,340
Right?
154
00:06:59,340 --> 00:07:05,770
Which is x squared is equal 9y.
155
00:07:05,770 --> 00:07:07,920
So this is in fact
how you could write
156
00:07:07,920 --> 00:07:12,740
this expression that's in r
and theta in terms of x and y.
157
00:07:12,740 --> 00:07:14,630
And so this, if you
look at it, is actually
158
00:07:14,630 --> 00:07:17,730
a parabola that goes
through the point (0, 0),
159
00:07:17,730 --> 00:07:22,814
and is stretched by a
factor of 9, or 1/9.
160
00:07:22,814 --> 00:07:24,230
Well, I guess you
can say, there's
161
00:07:24,230 --> 00:07:27,302
a vertical stretch or
horizontal stretch,
162
00:07:27,302 --> 00:07:28,510
you can pick which one it is.
163
00:07:28,510 --> 00:07:31,940
And in one case, it's
going to be by 3 or 1/3.
164
00:07:31,940 --> 00:07:33,100
I always mix those up.
165
00:07:33,100 --> 00:07:33,990
I'd have to check.
166
00:07:33,990 --> 00:07:35,520
Or by 9 or 1/9.
167
00:07:35,520 --> 00:07:39,230
So essentially, it's going to be
a parabola with some stretching
168
00:07:39,230 --> 00:07:40,240
on it.
169
00:07:40,240 --> 00:07:44,274
Now, the problem is that
you might say, well,
170
00:07:44,274 --> 00:07:45,440
it's not really all of that.
171
00:07:45,440 --> 00:07:46,840
Because secant
theta is not going
172
00:07:46,840 --> 00:07:50,170
to be defined for all
theta the way cosine is.
173
00:07:50,170 --> 00:07:52,350
So you do potentially
run into some problems.
174
00:07:52,350 --> 00:07:55,710
You might have to worry
about what part of the domain
175
00:07:55,710 --> 00:07:58,306
makes sense for theta, so
that this is well-defined.
176
00:07:58,306 --> 00:07:59,680
And so that this
is well-defined,
177
00:07:59,680 --> 00:08:01,510
what part of the
curve is carved out.
178
00:08:01,510 --> 00:08:03,110
That's a little more
technical than I
179
00:08:03,110 --> 00:08:04,680
want to go in this video.
180
00:08:04,680 --> 00:08:07,560
But some of you might look
at it and say, oh, she's
181
00:08:07,560 --> 00:08:08,720
missing something.
182
00:08:08,720 --> 00:08:12,730
Yeah, you caught something that
I'm intentionally ignoring.
183
00:08:12,730 --> 00:08:14,700
So the main point
of this was just so
184
00:08:14,700 --> 00:08:20,310
that you could see how you
can take these functions of r
185
00:08:20,310 --> 00:08:23,820
and theta and turn them
into functions of x and y,
186
00:08:23,820 --> 00:08:27,330
and then figure out kind of
what the curves might look like.
187
00:08:27,330 --> 00:08:29,600
So I'm going to stop there.
188
00:08:29,600 --> 00:08:31,080
Hopefully this was
a good exercise
189
00:08:31,080 --> 00:08:35,680
to get you understanding how
these different coordinates
190
00:08:35,680 --> 00:08:37,240
relate to one another.
191
00:08:37,240 --> 00:08:40,810
And yeah, that's
where we'll leave it.