WEBVTT
1
00:00:00.440 --> 00:00:03.700 A:middle L:90%
Welcome to the video on converting from Cartesian to polar
2
00:00:03.700 --> 00:00:08.150 A:middle L:90%
coordinates and then back from polar to Cartesian coordinates Again
3
00:00:08.160 --> 00:00:10.160 A:middle L:90%
, Um, let's jump right in here with an
4
00:00:10.160 --> 00:00:13.869 A:middle L:90%
example. Say we have a Cartesian position. Remember
5
00:00:13.869 --> 00:00:16.539 A:middle L:90%
, Cartesian has written is X comma. Why?
6
00:00:16.539 --> 00:00:20.980 A:middle L:90%
We're gonna say that we have the position Five comma
7
00:00:20.980 --> 00:00:26.850 A:middle L:90%
three So counting that we have 12345123 We're gonna look
8
00:00:26.850 --> 00:00:28.949 A:middle L:90%
right here. It's in the first quadrant So we
9
00:00:28.949 --> 00:00:30.160 A:middle L:90%
do this. What we want to do is we
10
00:00:30.160 --> 00:00:33.619 A:middle L:90%
want to use the formulas that I gave previously where
11
00:00:33.630 --> 00:00:36.320 A:middle L:90%
X is equal to the square root of X are
12
00:00:36.320 --> 00:00:38.439 A:middle L:90%
rather is equal to the square root of X squared
13
00:00:38.439 --> 00:00:42.020 A:middle L:90%
plus y squared and Satya is going to be equal
14
00:00:42.020 --> 00:00:48.640 A:middle L:90%
to the arc Tangent of why over X. Remember
15
00:00:48.649 --> 00:00:51.189 A:middle L:90%
, you can also write this as 10 in the
16
00:00:51.189 --> 00:00:53.030 A:middle L:90%
negative one. This is how it'll pillar on your
17
00:00:53.030 --> 00:00:59.299 A:middle L:90%
calculator over X. Okay, so just plugging these
18
00:00:59.299 --> 00:01:00.030 A:middle L:90%
in very quickly we're going to see that our is
19
00:01:00.030 --> 00:01:03.969 A:middle L:90%
equal to the square root of five squared plus three
20
00:01:03.969 --> 00:01:08.609 A:middle L:90%
squared. So that's gonna be squared of 34 is
21
00:01:08.620 --> 00:01:14.680 A:middle L:90%
our radius and then r theta. It's gonna be
22
00:01:14.680 --> 00:01:21.230 A:middle L:90%
equal to in verse 10 of 3/5. So when
23
00:01:21.230 --> 00:01:23.150 A:middle L:90%
we plugged that in, we get approximately 31 degrees
24
00:01:23.840 --> 00:01:26.760 A:middle L:90%
. Now, looking at this, that that's pretty
25
00:01:26.760 --> 00:01:27.769 A:middle L:90%
simple. But what if we tried a different quarter
26
00:01:27.769 --> 00:01:34.969 A:middle L:90%
? What if we twitched it to negative 512345 123
27
00:01:36.140 --> 00:01:40.090 A:middle L:90%
So if we did negative 53 coming back. And
28
00:01:40.090 --> 00:01:41.489 A:middle L:90%
here we have negative five. Well, the negative
29
00:01:41.489 --> 00:01:42.939 A:middle L:90%
five is gonna get squared. So negative five times
30
00:01:42.939 --> 00:01:46.420 A:middle L:90%
negative. Five. That's 25. Again, we're
31
00:01:46.420 --> 00:01:48.530 A:middle L:90%
gonna end up with squared of 34 just like we
32
00:01:48.530 --> 00:01:49.849 A:middle L:90%
did before. On the other hand, when we
33
00:01:49.849 --> 00:01:53.109 A:middle L:90%
look at the data calculation, we're gonna end up
34
00:01:53.109 --> 00:01:57.269 A:middle L:90%
with three over negative five. And it turns out
35
00:01:57.269 --> 00:02:00.260 A:middle L:90%
that the inverse tangent function has this property where a
36
00:02:00.260 --> 00:02:07.159 A:middle L:90%
tangent of negative X is equal to negative inverse tangent
37
00:02:07.740 --> 00:02:13.039 A:middle L:90%
of X. So this cause a little bit of
38
00:02:13.039 --> 00:02:14.419 A:middle L:90%
problem. It means that we're going to get a
39
00:02:14.419 --> 00:02:19.800 A:middle L:90%
value of negative 31 degrees here. So, looking
40
00:02:19.800 --> 00:02:21.569 A:middle L:90%
at that, we need to interpret what that means
41
00:02:21.569 --> 00:02:23.349 A:middle L:90%
, because before I told you always, always,
42
00:02:23.349 --> 00:02:27.250 A:middle L:90%
always, we wanna measure theta and r theta format
43
00:02:27.259 --> 00:02:30.840 A:middle L:90%
as being as coming from the positive X axis going
44
00:02:30.840 --> 00:02:32.629 A:middle L:90%
in the counter clockwise direction. So we got that
45
00:02:32.629 --> 00:02:35.400 A:middle L:90%
correct when we were in first Quadrant. But now
46
00:02:35.400 --> 00:02:37.979 A:middle L:90%
we're getting negative. 31 degrees. Well, what
47
00:02:37.979 --> 00:02:38.569 A:middle L:90%
does that mean? Well, what is telling us
48
00:02:38.580 --> 00:02:44.250 A:middle L:90%
is that this angle is 31 degrees, which means
49
00:02:44.250 --> 00:02:47.389 A:middle L:90%
that we're actually gonna need 180 minus 31 in order
50
00:02:47.389 --> 00:02:50.490 A:middle L:90%
to get the correct answer. So that's going to
51
00:02:50.490 --> 00:02:54.789 A:middle L:90%
be 149 degrees is the correct answer from here to
52
00:02:54.789 --> 00:02:58.599 A:middle L:90%
here. But you're never gonna get your calculator to
53
00:02:58.599 --> 00:03:00.810 A:middle L:90%
say that, because the way that Arc Tangent marks
54
00:03:00.810 --> 00:03:02.550 A:middle L:90%
So what we need to do then, is keep
55
00:03:02.550 --> 00:03:07.580 A:middle L:90%
track of which quadrant you are in. And what
56
00:03:07.580 --> 00:03:08.800 A:middle L:90%
is your Why and what is your ex, right
57
00:03:08.800 --> 00:03:13.439 A:middle L:90%
? So if we are taking the arc tangent of
58
00:03:13.449 --> 00:03:17.650 A:middle L:90%
an angle and we say why over X, then
59
00:03:17.650 --> 00:03:20.830 A:middle L:90%
it means we're always going to be measuring the angle
60
00:03:20.830 --> 00:03:24.310 A:middle L:90%
from the x axis. Okay, so make sure
61
00:03:24.310 --> 00:03:27.800 A:middle L:90%
make sure make sure that you always realize whatever quadrant
62
00:03:27.800 --> 00:03:30.159 A:middle L:90%
in your amazing from the x axis. So if
63
00:03:30.159 --> 00:03:30.810 A:middle L:90%
you were in the third quadrant, it gives you
64
00:03:30.810 --> 00:03:34.340 A:middle L:90%
this angle If you're in the fourth quadrant, it
65
00:03:34.340 --> 00:03:37.629 A:middle L:90%
gives you this angle and then convert so that your
66
00:03:37.629 --> 00:03:39.810 A:middle L:90%
theta that you report to your teacher is always being
67
00:03:39.810 --> 00:03:44.229 A:middle L:90%
measured from the positive X axis in the counter clockwise
68
00:03:44.229 --> 00:03:46.500 A:middle L:90%
direction. As long as you do that, you'll
69
00:03:46.500 --> 00:03:49.110 A:middle L:90%
be fine. I recommend that you practice that a
70
00:03:49.110 --> 00:03:50.990 A:middle L:90%
few times to make sure that you can do it
71
00:03:51.000 --> 00:03:53.819 A:middle L:90%
correctly. So let's go do a quick example in
72
00:03:53.819 --> 00:03:58.689 A:middle L:90%
the opposite direction. Say that we have our that
73
00:03:58.699 --> 00:04:03.120 A:middle L:90%
is three and a tha tha that is 140 degrees
74
00:04:03.120 --> 00:04:04.469 A:middle L:90%
. So if I were to look at that here
75
00:04:04.469 --> 00:04:08.969 A:middle L:90%
, it would be in this quadrant somewhere around here
76
00:04:09.639 --> 00:04:12.699 A:middle L:90%
. So what we wanna do is convert to X
77
00:04:12.699 --> 00:04:15.529 A:middle L:90%
and y. So remember, our conversions are R
78
00:04:15.529 --> 00:04:20.439 A:middle L:90%
cosine theta and R sine theta. Luckily, because
79
00:04:20.439 --> 00:04:23.180 A:middle L:90%
we're using cosine and sine, we don't have the
80
00:04:23.180 --> 00:04:26.680 A:middle L:90%
same issue that we did with the inverse tangent function
81
00:04:26.689 --> 00:04:28.819 A:middle L:90%
. So X then is going to be equal to
82
00:04:28.819 --> 00:04:32.819 A:middle L:90%
three times to co sign of 140 degrees. Why
83
00:04:32.819 --> 00:04:36.689 A:middle L:90%
will be equal to three times to sign of 140
84
00:04:36.689 --> 00:04:43.120 A:middle L:90%
degrees, which gives us answers of approximately negative 2.3
85
00:04:43.129 --> 00:04:46.810 A:middle L:90%
and 1.9. Uh, try this a few times
86
00:04:46.810 --> 00:04:49.019 A:middle L:90%
. Practice going back and forth. It's a pretty
87
00:04:49.019 --> 00:04:50.060 A:middle L:90%
simple exercise.