0:00:00.980,0:00:05.660
In this video, we're going to be[br]looking at polar.

0:00:06.930,0:00:13.672
Coordinates.[br]Let's begin by actually

0:00:13.672,0:00:20.096
looking at another coordinate[br]system. The Cartesian coordinate

0:00:20.096,0:00:23.355
system. Now in that system we

0:00:23.355,0:00:28.817
take 2. Axes and X axis[br]which is horizontal.

0:00:29.490,0:00:36.094
And Y Axis which is vertical and[br]a fixed .0 called the origin,

0:00:36.094,0:00:41.682
which is where these two points[br]cross. These two lines cross.

0:00:42.350,0:00:49.420
Now we fix a point P in[br]the plane by saying how far it's

0:00:49.420,0:00:54.975
displaced along the X axis to[br]give us the X coordinate.

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And how far it's displaced along[br]the Y access to give us the Y

0:01:01.724,0:01:03.929
coordinate and so we have.

0:01:04.490,0:01:09.198
A point P which is uniquely[br]described by its coordinates XY

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and notice I said how far it's[br]displaced because it is

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displacement that we're talking[br]about and not distance. That's

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what these arrowheads that we[br]put on the axes are all about

0:01:22.894,0:01:27.602
their about showings, in which[br]direction we must move so that

0:01:27.602,0:01:31.454
if we're moving down this[br]direction, it's a negative

0:01:31.454,0:01:34.450
distance and negative[br]displacement that we're making.

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Now that is more than one way of[br]describing where a point is in

0:01:40.392,0:01:47.651
the plane. And we're going to[br]be having a look at a system

0:01:47.651,0:01:49.244
called polar coordinates.

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So in this system of polar[br]coordinates, we take a poll. Oh,

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and we take a fixed line.

0:02:02.740,0:02:08.538
Now, how can we describe a point[br]in the plane using this fixed

0:02:08.538,0:02:14.336
.0? The pole and this baseline.[br]Here. One of the ways is to

0:02:14.336,0:02:17.904
think of it as. What if we turn?

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Around. Centering on oh[br]for the moment we rotate around,

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we can pass through a fixed[br]angle. Let's call that theater.

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And then along this radius we[br]can go a set distance.

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And we'll end up at a point P.

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And so the coordinates of that[br]point would be our theater, and

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this is our system of polar.

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Coordinates.

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Now, just as we've got certain[br]conventions with Cartesian

0:03:04.476,0:03:08.354
coordinates, we have certain[br]conventions with polar

0:03:08.354,0:03:13.340
coordinates, and these are quite[br]strong conventions, so let's

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have a look at what these are.[br]First of all, theater is

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measured. In

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radians. So[br]that's how first convention

0:03:29.150,0:03:32.110
theater is measured in radians.

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2nd convention

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well. Our second convention[br]is this that if this is our

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initial line and this is our[br]poll, then we measure theater

0:03:47.767,0:03:50.569
positive when we go round in

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that direction. Anticlockwise[br]and we measure theater negative.

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When we go around in[br]that direction which is

0:04:02.470,0:04:09.634
clockwise. So in just the same[br]way as we had an Arrowhead on

0:04:09.634,0:04:14.554
our axes X&Y. In a sense, we've[br]got arrowheads here,

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distinguishing a positive[br]direction for theater and a

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negative direction for measuring

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theater. We have 1/3[br]convention to do with theater

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and that is that we never go[br]further round this way.

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Number there's our poll. Oh, our[br]fixed point. We never go further

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around this way then there, so[br]theater is always less than or

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equal to pie and we never go[br]round further that way than

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there again. So theater is[br]always strictly greater than

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minus Π - Π ramped there plus Π[br]round to there. And notice that

0:04:59.472,0:05:06.208
we include. This bit of the line[br]if you like this extended bit of

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the line by going route to their[br]having the less than or equal to

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and having strictly greater than[br]Theta strictly greater than

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minus pie there.

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1/4 Convention 1/4[br]convention is that

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our is always

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positive. One of the things that[br]is quite important is that we be

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able to move from one system of[br]coordinates to another. So the

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question is if we have.

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A point. In our

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XY plane.[br]Who's coordinates are

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X&Y? How can we[br]change from cartesians into

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pohlers? And how can we change[br]back again, but one obvious

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thing to do is to associate the[br]pole with the origin, and then

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to associate the initial line[br]with the X axis.

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And then if we draw the radius[br]out to pee.

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And that's our.

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And that is the angle theater.

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So we can see that in[br]Cartesians, we're

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describing it as XY, and[br]in Pohlers, where

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describing it as our[br]theater. So what's the

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relationship between them?

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Let's drop that perpendicular[br]down and we can see that this is

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a height. Why? Because of the Y[br]coordinate the point and this is

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at a distance X because of the X[br]coordinate of the point.

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And looking at that, we can see[br]that Y is equal to R sign

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theater and X is equal to our[br]cause theater. So given R and

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Theta, we can calculate X&Y.[br]What about moving the other way

0:07:15.440,0:07:22.200
will from Pythagoras? We can see[br]that X squared plus Y squared is

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equal to R-squared. So give now

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X. I'm now why we can[br]calculate all and we can also

0:07:30.415,0:07:34.980
see that if we take the[br]opposite over the adjacent, we

0:07:34.980,0:07:41.205
have Y over X is equal to 10[br]theater. So given AY in an X,

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we can find out what theater[br]is.

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Now.

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Always when doing these, it's[br]best to draw sketches. If we're

0:07:52.160,0:07:56.860
converting from one sort of[br]point in Cartesians to its

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equivalent in Pohlers, or if[br]we're moving back from Pohlers

0:08:01.560,0:08:06.730
to cartesians, draw a picture,[br]see where that point actually is

0:08:06.730,0:08:12.840
now. Want to have a look at some[br]examples. First of all, we're

0:08:12.840,0:08:16.600
going to have a look at how to

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plot points. Then we're going to[br]have a look at how to convert

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from one system into the other[br]and vice versa. So let's begin

0:08:27.832,0:08:30.230
with plotting. Plot.

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And what I'm going to do is I'm[br]going to plot the following

0:08:37.535,0:08:40.325
points and they're all in polar

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coordinates. I'm going to put[br]them all on the same

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picture so we can get[br]a feel for whereabouts things

0:08:55.443,0:09:02.913
are in the polar play[br]or the plane for the

0:09:02.913,0:09:08.638
polar coordinates. So we put[br]our poll, oh.

0:09:09.180,0:09:11.790
And we have our initial line.

0:09:12.710,0:09:15.314
First one that we've got to plot

0:09:15.314,0:09:22.738
is 2. Pie so we know[br]that Pi is the angle all the

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way around here, so there's pie[br]to there and we want to go

0:09:29.472,0:09:31.026
out to units.

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So it's there. This[br]is the .2.

0:09:37.940,0:09:39.490
Pie.

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Next to .1 N wealthy to is 0 so[br]we're on the initial line.

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An one will be about there,[br]so there is the .1 note.

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2 - Π by 3 - π by[br]3 means come around this

0:10:02.672,0:10:07.666
way, and so minus π by[br]three is about there, and

0:10:07.666,0:10:12.206
we're coming around there[br]minus π by three, and we

0:10:12.206,0:10:17.654
want to come out a distance[br]to, so that's roughly 2 out

0:10:17.654,0:10:22.648
there, so this would be the[br].2 - π by 3.

0:10:23.790,0:10:29.458
And finally, we've got the[br]point. One 2/3 of Π. So we take

0:10:29.458,0:10:32.074
the 2/3. That's going all the

0:10:32.074,0:10:34.640
way around. To there.

0:10:35.190,0:10:40.676
And we draw out through there,[br]and we want a distance of one

0:10:40.676,0:10:44.896
along there, which roughly[br]called the scale we're using is

0:10:44.896,0:10:47.428
about there, and so that's the

0:10:47.428,0:10:49.930
.1. 2/3 of Π.

0:10:51.680,0:10:58.100
Notice that we've taken theater[br]first to establish in which

0:10:58.100,0:11:00.668
direction were actually facing.

0:11:02.600,0:11:06.255
OK, let's now have a

0:11:06.255,0:11:11.995
look. Having got used to[br]plotting points, let's now have

0:11:11.995,0:11:16.435
a look in polar coordinates.[br]These points 2.

0:11:17.070,0:11:19.270
Minus Π by 2.

0:11:20.210,0:11:26.210
1.[br]3/4 of

0:11:26.210,0:11:33.380
Π. And 2[br]- π by three. Now these

0:11:33.380,0:11:35.840
are all in Pohlers.

0:11:36.400,0:11:41.801
What I want to do is convert[br]them into cartesian coordinates.

0:11:41.801,0:11:47.693
So first a picture whereabouts[br]are they? And I'll do them one

0:11:47.693,0:11:55.058
at a time. So let's take this[br]one 2 - π by two initial point

0:11:55.058,0:11:57.513
poll. Oh, and initial line.

0:11:58.440,0:12:02.157
Minus Π by two? Well, that's[br]coming down here.

0:12:02.730,0:12:10.060
To there. So that's minus[br]π by two, and we've

0:12:10.060,0:12:13.660
come a distance to to

0:12:13.660,0:12:18.820
there. Well, we don't need to do[br]much calculation. I don't think

0:12:18.820,0:12:23.760
to find this. If again we take[br]our origin for our cartesians as

0:12:23.760,0:12:28.700
being the pole, and we align the[br]X axis with our initial line.

0:12:33.360,0:12:40.315
And there's our X. There's RY[br]and we can see straight away the

0:12:40.315,0:12:47.805
point in Pohlers that's 2 - π[br]by two in fact, goes to the

0:12:47.805,0:12:54.225
point. In Cartesians, That's 0 -[br]2 because it's this point here

0:12:54.225,0:13:01.715
on the Y axis, and it's 2[br]units below the X axis, so it's

0:13:01.715,0:13:03.320
0 - 2.

0:13:03.370,0:13:07.759
Notice how plotting the point[br]actually saved as having to do

0:13:07.759,0:13:12.148
any of the calculations. So[br]let's take the next point now,

0:13:12.148,0:13:14.542
which was one 3/4 of Π.

0:13:15.280,0:13:23.160
1 3/4 of π. So[br]again, let's plot where it

0:13:23.160,0:13:26.560
is. Take our initial.

0:13:27.220,0:13:34.500
.0 our poll and our initial line[br]3/4 of π going round. It's

0:13:34.500,0:13:40.100
positive so it drought there be[br]somewhere out along that.

0:13:40.670,0:13:47.347
Direction there's our angle of[br]3/4 of Π, where somewhere out

0:13:47.347,0:13:54.631
here at a distance one unit.[br]So again, let's take our X&Y

0:13:54.631,0:13:57.059
axes, our X axis.

0:13:58.010,0:14:01.328
To be along the initial line.

0:14:01.850,0:14:07.823
And now why access to be[br]vertical and through the pole?

0:14:08.490,0:14:14.634
Oh So that the polo becomes our[br]origin of, and it's this point.

0:14:15.140,0:14:16.229
But where after?

0:14:17.240,0:14:24.170
Now how we going to work[br]this out that remember the

0:14:24.170,0:14:30.470
formula that we had was X[br]equals our cause theater.

0:14:32.010,0:14:39.678
Let's have a look at that.[br]Are is one an we've got

0:14:39.678,0:14:47.346
cause of 3/4 of Π and[br]the cosine of 3/4 of Π

0:14:47.346,0:14:55.014
is minus one over Route 2,[br]so that's minus one over Route

0:14:55.014,0:14:58.848
2. Why is our sign Theta?

0:14:58.880,0:15:06.155
And so this is one times the[br]sign of 3/4 of Π and the sign

0:15:06.155,0:15:13.915
of 3/4 of Π is just one over[br]Route 2, and so we have one over

0:15:13.915,0:15:19.250
Route 2 for RY coordinate. And[br]notice that these answers agree

0:15:19.250,0:15:24.585
with where the point is in this[br]particular quadrant. Negative X

0:15:24.585,0:15:28.950
and positive Y, negative X and[br]positive Y so.

0:15:28.950,0:15:33.617
Even if I've got the calculation[br]wrong in the sense that I, even

0:15:33.617,0:15:37.566
if I've done the arithmetic[br]wrong, have no, I've got the

0:15:37.566,0:15:39.361
point in the right quadrant.

0:15:39.420,0:15:46.130
Let's have a look at the[br]last one of these two.

0:15:46.900,0:15:50.386
Minus Π by

0:15:50.386,0:15:54.350
3. So again, our poll.

0:15:56.020,0:15:59.050
Our initial line.

0:15:59.050,0:16:02.179
Minus Π by three is around here.

0:16:04.760,0:16:11.533
So we've come around there minus[br]π by three, and we're out a

0:16:11.533,0:16:13.617
distance, two along there.

0:16:15.440,0:16:20.550
Take our X axis to coincide with[br]the initial line.

0:16:21.090,0:16:22.500
And now origin.

0:16:23.410,0:16:25.910
Coincide with the pole.

0:16:28.360,0:16:35.170
Let's write down our equations[br]that tell us X is

0:16:35.170,0:16:41.980
our cause theater, which is[br]2 times the cosine of

0:16:41.980,0:16:48.109
minus π by three, which[br]is equal to 2.

0:16:49.720,0:16:56.980
Times Now we want the cause[br]of minus π by three and

0:16:56.980,0:17:04.240
the cosine of minus π by[br]three is 1/2, and so that

0:17:04.240,0:17:05.450
gives US1.

0:17:06.830,0:17:10.520
Why is equal to our

0:17:10.520,0:17:17.585
sign theater? Which is 2 times[br]the sign of minus π by three,

0:17:17.585,0:17:24.865
which is 2 Times Now we want the[br]sign of my minus Pi π three, and

0:17:24.865,0:17:31.235
that is minus Route 3 over 2.[br]The two is cancelled to give us

0:17:31.235,0:17:32.600
minus Route 3.

0:17:33.130,0:17:37.628
And so again, notice we know[br]that we've got it in the right

0:17:37.628,0:17:41.434
quadrant. 'cause when we drew[br]the diagram, we have positive X

0:17:41.434,0:17:44.894
and negative Y, and that's how[br]we've ended up here.

0:17:46.070,0:17:51.110
What do we do about going back[br]the other way?

0:17:51.650,0:17:59.054
Well, let's have a look at[br]some examples that will do that

0:17:59.054,0:18:05.238
for us. What I'm going to look[br]at as these points, which are

0:18:05.238,0:18:11.230
cartesians. The .22 point[br]minus 3 four.

0:18:12.010,0:18:17.890
The point minus 2 -[br]2 Route 3.

0:18:18.730,0:18:25.150
And the .1 - 1 now these[br]are all points in Cartesian's.

0:18:25.150,0:18:28.360
So let's begin with this one.

0:18:29.060,0:18:32.228
Show where it is.

0:18:32.740,0:18:38.920
To begin with, on the cartesian[br]axes so it's at 2 for X and two

0:18:38.920,0:18:40.980
for Y. So it's there.

0:18:42.220,0:18:48.110
So again.[br]We'll associate the origin in

0:18:48.110,0:18:53.236
Cartesians with the pole in[br]polar's, and the X axis, with

0:18:53.236,0:18:58.828
the initial line and what we[br]want to calculate is what's that

0:18:58.828,0:19:01.158
angle there an what's that

0:19:01.158,0:19:03.550
radius there? Well.

0:19:04.060,0:19:09.400
All squared is equal to X[br]squared plus Y squared.

0:19:10.890,0:19:17.845
So that's 2 squared +2 squared,[br]keeps us 8 and so are is

0:19:17.845,0:19:20.520
equal to 2 Route 2.

0:19:21.260,0:19:23.416
When we take the square root of

0:19:23.416,0:19:31.065
8. What about theater? Well,[br]tan Theta is equal to

0:19:31.065,0:19:33.270
Y over X.

0:19:33.870,0:19:41.010
In this case it's two over[br]2, which is one, and so

0:19:41.010,0:19:46.960
theater is π by 4, and[br]so therefore the polar

0:19:46.960,0:19:52.910
coordinates of this point are[br]two route 2π over 4.

0:19:55.470,0:19:59.782
Let's have a look at this one[br]now, minus 3 four.

0:20:00.750,0:20:05.310
Let's

0:20:05.310,0:20:13.954
begin.[br]By establishing whereabouts it

0:20:13.954,0:20:17.238
is on our cartesian

0:20:17.238,0:20:22.318
axes. Minus 3 means it's back[br]here somewhere, so there's

0:20:22.318,0:20:28.454
minus three and the four on[br]the Y. It's up there, so I'll

0:20:28.454,0:20:29.870
point is there.

0:20:32.450,0:20:39.665
Join it up to the origin as our[br]point P and we are after. Now

0:20:39.665,0:20:44.956
the polar coordinates for this[br]point. So again we associate the

0:20:44.956,0:20:50.728
pole with the origin and the[br]initial line with the X axis,

0:20:50.728,0:20:56.019
and so there's the value of[br]theater that we're after. And

0:20:56.019,0:21:01.791
this opie is the length are that[br]were after, so R-squared is

0:21:01.791,0:21:03.715
equal to X squared.

0:21:03.740,0:21:10.592
Plus Y squared, which in this[br]case is minus 3 squared, +4

0:21:10.592,0:21:18.586
squared. That's 9 + 16, gives us[br]25, and so R is the square

0:21:18.586,0:21:22.583
root of 25, which is just five.

0:21:23.210,0:21:27.560
What about finding[br]theater now well?

0:21:28.640,0:21:30.968
Tan Theta is.

0:21:32.100,0:21:34.839
Y over X.

0:21:35.880,0:21:37.488
Which gives us.

0:21:38.480,0:21:41.948
4 over minus three.

0:21:42.510,0:21:47.119
Now when you put that into your[br]Calculator, you will get.

0:21:47.720,0:21:53.110
A slightly odd answers. It will[br]actually give you a negative

0:21:53.110,0:21:56.130
answer. That might be[br]difficult for you to

0:21:56.130,0:21:58.470
interpret. It sits actually[br]telling you this angle out

0:21:58.470,0:21:58.730
here.

0:21:59.890,0:22:06.415
And we want to be all the way[br]around there now the way that I

0:22:06.415,0:22:11.635
think these are best done is[br]actually to look at a right

0:22:11.635,0:22:16.420
angle triangle like this and[br]call that angle Alpha. Now let's

0:22:16.420,0:22:23.380
have a look at what an Alpha is.[br]Tan Alpha is 4 over 3 and when

0:22:23.380,0:22:28.165
you put that into your[br]Calculator it will tell you that

0:22:28.165,0:22:29.905
Alpha is nought .9.

0:22:29.920,0:22:36.300
Three radians. Remember, theater[br]has to be in radians and

0:22:36.300,0:22:43.074
therefore. Theater here is[br]equal to π minus

0:22:43.074,0:22:49.490
Alpha, and so that's[br]π - 4.9. Three,

0:22:49.490,0:22:52.698
which gives us 2.2

0:22:52.698,0:22:58.202
one radians. And that's[br]theater so you can see that

0:22:58.202,0:23:02.292
the calculation of our is[br]always going to be relatively

0:23:02.292,0:23:05.155
straightforward, but the[br]calculation this angle theater

0:23:05.155,0:23:10.063
is going to be quite tricky,[br]and that's one of the reasons

0:23:10.063,0:23:14.562
why it's best to plot these[br]points before you try and

0:23:14.562,0:23:16.198
workout what theater is.

0:23:17.850,0:23:21.765
Now the next example was the[br]point minus 2.

0:23:22.580,0:23:25.980
Minus 2 Route 3.

0:23:26.730,0:23:28.100
So again.

0:23:30.830,0:23:36.147
Let's have a look where it is in[br]the cartesian plane. These are

0:23:36.147,0:23:40.646
its cartesian coordinates, so[br]we've minus two for X. So we

0:23:40.646,0:23:45.554
somewhere back here and minus 2[br]route 3 four Y. So where

0:23:45.554,0:23:48.826
somewhere down here? So I'll[br]point is here.

0:23:49.690,0:23:52.690
Join it up to our origin.

0:23:53.600,0:23:55.828
Marking our point P.

0:23:56.380,0:24:02.395
Again, we'll take the origin to[br]be the pole and the X axis to be

0:24:02.395,0:24:07.207
the initial line, and we can see[br]that the theater were looking

0:24:07.207,0:24:08.811
for is around there.

0:24:09.510,0:24:12.825
That's our theater, and here's

0:24:12.825,0:24:18.876
our. So again, let's[br]calculate R-squared that's X

0:24:18.876,0:24:21.508
squared plus Y squared.

0:24:22.200,0:24:29.328
Is equal to. Well, in this[br]case we've got minus two all

0:24:29.328,0:24:36.456
squared plus minus 2 route 3[br]all squared, which gives us 4

0:24:36.456,0:24:43.536
+ 12. 16 and so R is equal[br]to the square root of 16, which

0:24:43.536,0:24:44.892
is just 4.

0:24:45.730,0:24:50.867
Now, what about this? We can[br]see that theater should be

0:24:50.867,0:24:55.070
negative, so let's just[br]calculate this angle as an

0:24:55.070,0:24:59.273
angle in a right angle[br]triangle. So tan Alpha

0:24:59.273,0:25:03.943
equals, well, it's going to[br]be the opposite, which is

0:25:03.943,0:25:05.344
this side here.

0:25:07.300,0:25:13.324
2 Route 3 in length over the[br]adjacent, which is just two

0:25:13.324,0:25:20.352
which gives us Route 3. So Alpha[br]just calculated as an angle is π

0:25:20.352,0:25:27.380
by three. So if that's pie by[br]three this angle in size is 2π

0:25:27.380,0:25:32.902
by three, but of course we must[br]measure theater negatively when

0:25:32.902,0:25:37.922
we come clockwise from the[br]initial line, and so Theta.

0:25:37.950,0:25:44.255
Is minus 2π by three the 2π[br]three giving us the size the

0:25:44.255,0:25:50.075
minus sign giving us the[br]direction so we can see that the

0:25:50.075,0:25:55.895
point we've got described as[br]minus 2 - 2 route 3 in

0:25:55.895,0:26:03.170
Cartesians is the .4 - 2π over 3[br]or minus 2/3 of Π in Pollas.

0:26:03.970,0:26:08.470
Now we've taken a point in this[br]quadrant. A point in this

0:26:08.470,0:26:12.220
quadrant appointing this[br]quadrant. Let's have a look at a

0:26:12.220,0:26:16.720
point in the fourth quadrant[br]just to finish off this set of

0:26:16.720,0:26:20.470
examples and the point we chose[br]was 1 - 1.

0:26:21.490,0:26:22.650
So again.

0:26:24.320,0:26:29.060
Let's have a look at where it is[br]in our cartesian system.

0:26:30.020,0:26:35.090
So we've a value of one 4X[br]and the value of minus one

0:26:35.090,0:26:39.770
for Y. So there's our point[br]P. Join it to the origin.

0:26:40.790,0:26:45.510
And again will associate the[br]origin in the Cartesian's with

0:26:45.510,0:26:50.230
the pole of the polar[br]coordinates and the initial line

0:26:50.230,0:26:55.894
will be the X axis, so we're[br]looking for this angle theater.

0:26:57.010,0:27:00.290
And this length of OP.

0:27:02.020,0:27:07.487
So all squared is equal to X[br]squared plus Y squared.

0:27:08.130,0:27:14.478
So that's one squared plus minus[br]one squared, and that's one plus

0:27:14.478,0:27:19.768
one is 2 so far is equal[br]to Route 2.

0:27:20.510,0:27:24.713
Let's not worry about the[br]direction here. Let's just

0:27:24.713,0:27:28.916
calculate the magnitude of[br]theater well. The magnitude of

0:27:28.916,0:27:34.053
theater, in fact, to do that,[br]I'd rather actually call it

0:27:34.053,0:27:39.190
Alpha, just want to calculate[br]the magnitude. So tan Alpha is.

0:27:39.860,0:27:45.896
Opposite, which is one over the[br]adjacent, which is one which is

0:27:45.896,0:27:52.938
just one. So Alpha is in fact[br]Π by 4. That means that my

0:27:52.938,0:27:58.471
angle theater for the coordinate[br]coming around this way is minus

0:27:58.471,0:28:04.507
π by 4, and so my polar[br]coordinates for this point, our

0:28:04.507,0:28:08.028
Route 2 and minus π by 4.

0:28:08.760,0:28:15.126
So. We've seen here why it's[br]so important to plot your points

0:28:15.126,0:28:17.406
before you do any calculation.

0:28:18.110,0:28:21.854
Having looked at what happens[br]with points, let's see if we can

0:28:21.854,0:28:25.910
now have a look at what happens[br]to a collection of points. In

0:28:25.910,0:28:27.158
other words, a curve.

0:28:28.000,0:28:31.210
Let's take a very simple curve

0:28:31.210,0:28:37.398
in Cartesians. X squared plus Y[br]squared equals a squared.

0:28:37.950,0:28:42.430
Now this is a circle, a[br]circle centered on the

0:28:42.430,0:28:44.222
origin of Radius A.

0:28:45.260,0:28:47.096
So if we think about that.

0:28:49.650,0:28:56.982
Circle centered on the origin of[br]radius a, so it will go through.

0:28:57.930,0:29:01.470
These points on the axis.

0:29:09.150,0:29:10.080
Like so.

0:29:11.580,0:29:16.910
If we think about what that[br]tells us, it tells us that no

0:29:16.910,0:29:18.960
matter what the angle is.

0:29:22.220,0:29:24.770
For any one of our points.

0:29:26.640,0:29:32.412
If we were thinking in Pohlers,[br]the radius is always a constant.

0:29:32.412,0:29:39.146
So if we were to guess at the[br]polar equation, it would be our

0:29:39.146,0:29:44.918
equals A and it wouldn't involve[br]theater at all. Well, it just

0:29:44.918,0:29:51.171
check that we know that X is[br]equal to our cause theater, and

0:29:51.171,0:29:55.019
we know that Y is equal to our

0:29:55.019,0:30:01.549
sign Theta. So we can[br]substitute these in R-squared

0:30:01.549,0:30:07.642
cost, Square theater plus[br]R-squared. Sine squared Theta is

0:30:07.642,0:30:14.412
equal to a squared. We[br]can take out the R-squared.

0:30:16.770,0:30:24.100
And that leaves us with[br]this factor of Cos squared

0:30:24.100,0:30:26.299
plus sign squared.

0:30:26.410,0:30:31.162
Now cost squared plus sign[br]squared is a well known identity

0:30:31.162,0:30:36.346
cost squared plus sign squared[br]at the same angle is always one,

0:30:36.346,0:30:41.098
so this just reduces two[br]R-squared equals a squared or R

0:30:41.098,0:30:46.282
equals AR is a constant, which[br]is what we predicted for looking

0:30:46.282,0:30:49.738
at the situation there now.[br]Another very straightforward

0:30:49.738,0:30:54.922
curve is the straight line Y[br]equals MX. Let's just have a

0:30:54.922,0:30:56.218
look at that.

0:30:56.760,0:31:01.140
Y equals MX is a straight line[br]that goes through the origin.

0:31:02.550,0:31:07.518
Think about it, is it has a[br]constant gradient and of course

0:31:07.518,0:31:12.900
M. The gradient is defined to be[br]the tangent of the angle that

0:31:12.900,0:31:17.454
the line makes with the positive[br]direction of the X axis.

0:31:18.060,0:31:22.361
So if the gradient is a[br]constant, the tangent of the

0:31:22.361,0:31:27.444
angle is a constant, and so this[br]angle theater is a constant. So

0:31:27.444,0:31:31.745
let's just have a look at that.[br]Why is we know?

0:31:32.350,0:31:39.630
All. Sign[br]Theta equals M times

0:31:39.630,0:31:43.534
by our cause theater.

0:31:44.490,0:31:51.224
The ask cancel out and so I[br]have sign theater over Cos Theta

0:31:51.224,0:31:58.423
equals M. And so I[br]have tan Theta equals M and

0:31:58.423,0:32:01.558
so theater does equal a

0:32:01.558,0:32:08.256
constant. But and here there is[br]a big bot for Y equals MX.

0:32:08.256,0:32:13.129
That's the picture that we get[br]if we're working in Cartesians.

0:32:13.840,0:32:20.033
But if we're working in Pohlers,[br]there's our poll. There's our

0:32:20.033,0:32:25.663
initial line Theta equals a[br]constant. There is the angle

0:32:25.663,0:32:33.056
Theta. And remember, we do not[br]have negative values of R and so

0:32:33.056,0:32:39.888
we get a half line. In other[br]words, we only get this bit of

0:32:39.888,0:32:45.084
the line. The half line there.[br]That simple example should

0:32:45.084,0:32:49.957
warnors that whenever we are[br]moving between one sort of curve

0:32:49.957,0:32:53.944
in cartesians into its[br]equivalent in polar's, we need

0:32:53.944,0:32:58.374
to be very careful about the[br]results that we get.

0:32:59.140,0:33:05.185
So let's just have a look at a[br]couple more examples. Let's take

0:33:05.185,0:33:11.695
X squared plus. Y squared is[br]equal to 9. We know that X is

0:33:11.695,0:33:15.880
our cause theater. And why is[br]our sign theater?

0:33:16.720,0:33:23.170
We can plug those in R-squared,[br]Cos squared Theta plus

0:33:23.170,0:33:25.750
R-squared, sine squared Theta

0:33:25.750,0:33:29.300
equals 9. All squared is a

0:33:29.300,0:33:35.246
common factor. So we can take it[br]out and we've got cost squared

0:33:35.246,0:33:40.238
Theta plus sign squared. Theta[br]is equal to 9 cost squared plus

0:33:40.238,0:33:45.230
sign squared is an identity cost[br]squared plus sign squared of the

0:33:45.230,0:33:50.638
same angle is always one, and so[br]R-squared equals 9. R is equal

0:33:50.638,0:33:53.550
to three IE a circle of radius

0:33:53.550,0:33:56.455
3. Let's

0:33:56.455,0:34:03.265
take. The[br]rectangular hyperbola XY is

0:34:03.265,0:34:05.830
equal to 4.

0:34:06.280,0:34:13.168
And again, we're going to use[br]X equals our cause theater and

0:34:13.168,0:34:18.908
Y equals R sign theater. So[br]we're multiplying X&Y together.

0:34:18.908,0:34:25.222
So when we do that, we're[br]going to have our squared.

0:34:25.290,0:34:32.870
Sign theater[br]Cos Theta equals 4.

0:34:34.350,0:34:41.936
Now. Sign Theta Cos[br]Theta will twice sign tita cost

0:34:41.936,0:34:45.156
theater would be signed to

0:34:45.156,0:34:51.006
theater. But I've taken 2 lots[br]there, so if I've taken 2 lots

0:34:51.006,0:34:54.889
there, it's the equivalent of[br]multiplying that side by two. So

0:34:54.889,0:34:57.007
I've got to multiply that side

0:34:57.007,0:34:59.386
by two. So I end up with that.

0:35:00.580,0:35:01.819
For my equation.

0:35:02.670,0:35:06.856
I still some the other way round[br]now, but one point to notice

0:35:06.856,0:35:10.730
before we do. Knowledge of[br]trig identity's is very

0:35:10.730,0:35:14.105
important. We've used cost[br]squared plus sign. Squared is

0:35:14.105,0:35:18.605
one and we've now used sign to[br]Theta is equal to two

0:35:18.605,0:35:21.980
scientist accosts theater. So[br]knowledge of those is very

0:35:21.980,0:35:26.855
important. So as I said, let's[br]see if we can turn this around

0:35:26.855,0:35:31.355
now and have a look at some[br]examples going the other way.

0:35:32.660,0:35:39.954
First one will take is 2 over,[br]R is equal to 1 plus cause

0:35:39.954,0:35:45.930
theater. I don't like really the[br]way it's written, so let's

0:35:45.930,0:35:51.762
multiply up by R so I get R[br]Plus R cause theater.

0:35:52.540,0:35:57.530
Now, because I've done that,[br]let's just remember that are

0:35:57.530,0:36:02.021
squared is equal to X squared[br]plus Y squared.

0:36:03.080,0:36:09.968
So that means I can replace[br]this are here by the square

0:36:09.968,0:36:13.412
root of X squared plus Y

0:36:13.412,0:36:19.595
squared. Our costs theater.[br]Will, our Cos Theta is equal

0:36:19.595,0:36:25.145
to X so I can replace this[br]bit by X.

0:36:26.420,0:36:31.136
Now it looks untidy's got a[br]square root in it, so naturally

0:36:31.136,0:36:36.638
we would want to get rid of that[br]square root. So let's take X

0:36:36.638,0:36:38.210
away from each side.

0:36:38.370,0:36:44.090
And then let's Square both[br]sides. So that gives us X

0:36:44.090,0:36:50.850
squared plus Y squared there and[br]on this side it's 2 minus X

0:36:50.850,0:36:57.610
all squared, which will give us[br]4 - 4 X plus X squared.

0:36:58.170,0:37:04.570
So I've got an X squared on each[br]side that will go out and so I'm

0:37:04.570,0:37:08.970
left with Y squared is equal to[br]4 - 4 X.

0:37:10.410,0:37:15.503
And what you should notice there[br]is that actually a parabola.

0:37:16.180,0:37:22.251
So this would seem to be the[br]way in which we define a

0:37:22.251,0:37:24.119
parabola in polar coordinates.