[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.98,0:00:05.66,Default,,0000,0000,0000,,In this video, we're going to be\Nlooking at polar.
Dialogue: 0,0:00:06.93,0:00:13.67,Default,,0000,0000,0000,,Coordinates.\NLet's begin by actually
Dialogue: 0,0:00:13.67,0:00:20.10,Default,,0000,0000,0000,,looking at another coordinate\Nsystem. The Cartesian coordinate
Dialogue: 0,0:00:20.10,0:00:23.36,Default,,0000,0000,0000,,system. Now in that system we
Dialogue: 0,0:00:23.36,0:00:28.82,Default,,0000,0000,0000,,take 2. Axes and X axis\Nwhich is horizontal.
Dialogue: 0,0:00:29.49,0:00:36.09,Default,,0000,0000,0000,,And Y Axis which is vertical and\Na fixed .0 called the origin,
Dialogue: 0,0:00:36.09,0:00:41.68,Default,,0000,0000,0000,,which is where these two points\Ncross. These two lines cross.
Dialogue: 0,0:00:42.35,0:00:49.42,Default,,0000,0000,0000,,Now we fix a point P in\Nthe plane by saying how far it's
Dialogue: 0,0:00:49.42,0:00:54.98,Default,,0000,0000,0000,,displaced along the X axis to\Ngive us the X coordinate.
Dialogue: 0,0:00:55.55,0:01:01.72,Default,,0000,0000,0000,,And how far it's displaced along\Nthe Y access to give us the Y
Dialogue: 0,0:01:01.72,0:01:03.93,Default,,0000,0000,0000,,coordinate and so we have.
Dialogue: 0,0:01:04.49,0:01:09.20,Default,,0000,0000,0000,,A point P which is uniquely\Ndescribed by its coordinates XY
Dialogue: 0,0:01:09.20,0:01:13.91,Default,,0000,0000,0000,,and notice I said how far it's\Ndisplaced because it is
Dialogue: 0,0:01:13.91,0:01:17.76,Default,,0000,0000,0000,,displacement that we're talking\Nabout and not distance. That's
Dialogue: 0,0:01:17.76,0:01:22.89,Default,,0000,0000,0000,,what these arrowheads that we\Nput on the axes are all about
Dialogue: 0,0:01:22.89,0:01:27.60,Default,,0000,0000,0000,,their about showings, in which\Ndirection we must move so that
Dialogue: 0,0:01:27.60,0:01:31.45,Default,,0000,0000,0000,,if we're moving down this\Ndirection, it's a negative
Dialogue: 0,0:01:31.45,0:01:34.45,Default,,0000,0000,0000,,distance and negative\Ndisplacement that we're making.
Dialogue: 0,0:01:34.68,0:01:40.39,Default,,0000,0000,0000,,Now that is more than one way of\Ndescribing where a point is in
Dialogue: 0,0:01:40.39,0:01:47.65,Default,,0000,0000,0000,,the plane. And we're going to\Nbe having a look at a system
Dialogue: 0,0:01:47.65,0:01:49.24,Default,,0000,0000,0000,,called polar coordinates.
Dialogue: 0,0:01:49.25,0:01:57.09,Default,,0000,0000,0000,,So in this system of polar\Ncoordinates, we take a poll. Oh,
Dialogue: 0,0:01:57.09,0:02:01.00,Default,,0000,0000,0000,,and we take a fixed line.
Dialogue: 0,0:02:02.74,0:02:08.54,Default,,0000,0000,0000,,Now, how can we describe a point\Nin the plane using this fixed
Dialogue: 0,0:02:08.54,0:02:14.34,Default,,0000,0000,0000,,.0? The pole and this baseline.\NHere. One of the ways is to
Dialogue: 0,0:02:14.34,0:02:17.90,Default,,0000,0000,0000,,think of it as. What if we turn?
Dialogue: 0,0:02:19.38,0:02:26.75,Default,,0000,0000,0000,,Around. Centering on oh\Nfor the moment we rotate around,
Dialogue: 0,0:02:26.75,0:02:33.43,Default,,0000,0000,0000,,we can pass through a fixed\Nangle. Let's call that theater.
Dialogue: 0,0:02:33.99,0:02:40.46,Default,,0000,0000,0000,,And then along this radius we\Ncan go a set distance.
Dialogue: 0,0:02:40.97,0:02:44.13,Default,,0000,0000,0000,,And we'll end up at a point P.
Dialogue: 0,0:02:45.17,0:02:51.10,Default,,0000,0000,0000,,And so the coordinates of that\Npoint would be our theater, and
Dialogue: 0,0:02:51.10,0:02:54.06,Default,,0000,0000,0000,,this is our system of polar.
Dialogue: 0,0:02:54.60,0:02:57.41,Default,,0000,0000,0000,,Coordinates.
Dialogue: 0,0:02:59.49,0:03:04.48,Default,,0000,0000,0000,,Now, just as we've got certain\Nconventions with Cartesian
Dialogue: 0,0:03:04.48,0:03:08.35,Default,,0000,0000,0000,,coordinates, we have certain\Nconventions with polar
Dialogue: 0,0:03:08.35,0:03:13.34,Default,,0000,0000,0000,,coordinates, and these are quite\Nstrong conventions, so let's
Dialogue: 0,0:03:13.34,0:03:19.99,Default,,0000,0000,0000,,have a look at what these are.\NFirst of all, theater is
Dialogue: 0,0:03:19.99,0:03:23.26,Default,,0000,0000,0000,,measured. In
Dialogue: 0,0:03:23.26,0:03:29.15,Default,,0000,0000,0000,,radians. So\Nthat's how first convention
Dialogue: 0,0:03:29.15,0:03:32.11,Default,,0000,0000,0000,,theater is measured in radians.
Dialogue: 0,0:03:32.75,0:03:35.82,Default,,0000,0000,0000,,2nd convention
Dialogue: 0,0:03:35.82,0:03:42.63,Default,,0000,0000,0000,,well. Our second convention\Nis this that if this is our
Dialogue: 0,0:03:42.63,0:03:47.77,Default,,0000,0000,0000,,initial line and this is our\Npoll, then we measure theater
Dialogue: 0,0:03:47.77,0:03:50.57,Default,,0000,0000,0000,,positive when we go round in
Dialogue: 0,0:03:50.57,0:03:56.30,Default,,0000,0000,0000,,that direction. Anticlockwise\Nand we measure theater negative.
Dialogue: 0,0:03:56.30,0:04:02.47,Default,,0000,0000,0000,,When we go around in\Nthat direction which is
Dialogue: 0,0:04:02.47,0:04:09.63,Default,,0000,0000,0000,,clockwise. So in just the same\Nway as we had an Arrowhead on
Dialogue: 0,0:04:09.63,0:04:14.55,Default,,0000,0000,0000,,our axes X&Y. In a sense, we've\Ngot arrowheads here,
Dialogue: 0,0:04:14.55,0:04:18.49,Default,,0000,0000,0000,,distinguishing a positive\Ndirection for theater and a
Dialogue: 0,0:04:18.49,0:04:20.46,Default,,0000,0000,0000,,negative direction for measuring
Dialogue: 0,0:04:20.46,0:04:24.59,Default,,0000,0000,0000,,theater. We have 1/3\Nconvention to do with theater
Dialogue: 0,0:04:24.59,0:04:29.55,Default,,0000,0000,0000,,and that is that we never go\Nfurther round this way.
Dialogue: 0,0:04:30.68,0:04:36.54,Default,,0000,0000,0000,,Number there's our poll. Oh, our\Nfixed point. We never go further
Dialogue: 0,0:04:36.54,0:04:42.39,Default,,0000,0000,0000,,around this way then there, so\Ntheater is always less than or
Dialogue: 0,0:04:42.39,0:04:48.25,Default,,0000,0000,0000,,equal to pie and we never go\Nround further that way than
Dialogue: 0,0:04:48.25,0:04:52.64,Default,,0000,0000,0000,,there again. So theater is\Nalways strictly greater than
Dialogue: 0,0:04:52.64,0:04:59.47,Default,,0000,0000,0000,,minus Π - Π ramped there plus Π\Nround to there. And notice that
Dialogue: 0,0:04:59.47,0:05:06.21,Default,,0000,0000,0000,,we include. This bit of the line\Nif you like this extended bit of
Dialogue: 0,0:05:06.21,0:05:12.14,Default,,0000,0000,0000,,the line by going route to their\Nhaving the less than or equal to
Dialogue: 0,0:05:12.14,0:05:15.96,Default,,0000,0000,0000,,and having strictly greater than\NTheta strictly greater than
Dialogue: 0,0:05:15.96,0:05:17.23,Default,,0000,0000,0000,,minus pie there.
Dialogue: 0,0:05:17.96,0:05:24.00,Default,,0000,0000,0000,,1/4 Convention 1/4\Nconvention is that
Dialogue: 0,0:05:24.00,0:05:27.02,Default,,0000,0000,0000,,our is always
Dialogue: 0,0:05:27.02,0:05:32.87,Default,,0000,0000,0000,,positive. One of the things that\Nis quite important is that we be
Dialogue: 0,0:05:32.87,0:05:37.70,Default,,0000,0000,0000,,able to move from one system of\Ncoordinates to another. So the
Dialogue: 0,0:05:37.70,0:05:39.72,Default,,0000,0000,0000,,question is if we have.
Dialogue: 0,0:05:40.23,0:05:44.18,Default,,0000,0000,0000,,A point. In our
Dialogue: 0,0:05:44.18,0:05:50.76,Default,,0000,0000,0000,,XY plane.\NWho's coordinates are
Dialogue: 0,0:05:50.76,0:05:56.50,Default,,0000,0000,0000,,X&Y? How can we\Nchange from cartesians into
Dialogue: 0,0:05:56.50,0:06:02.14,Default,,0000,0000,0000,,pohlers? And how can we change\Nback again, but one obvious
Dialogue: 0,0:06:02.14,0:06:08.81,Default,,0000,0000,0000,,thing to do is to associate the\Npole with the origin, and then
Dialogue: 0,0:06:08.81,0:06:13.43,Default,,0000,0000,0000,,to associate the initial line\Nwith the X axis.
Dialogue: 0,0:06:13.95,0:06:18.87,Default,,0000,0000,0000,,And then if we draw the radius\Nout to pee.
Dialogue: 0,0:06:19.92,0:06:21.70,Default,,0000,0000,0000,,And that's our.
Dialogue: 0,0:06:22.75,0:06:25.71,Default,,0000,0000,0000,,And that is the angle theater.
Dialogue: 0,0:06:26.35,0:06:30.15,Default,,0000,0000,0000,,So we can see that in\NCartesians, we're
Dialogue: 0,0:06:30.15,0:06:33.95,Default,,0000,0000,0000,,describing it as XY, and\Nin Pohlers, where
Dialogue: 0,0:06:33.95,0:06:37.75,Default,,0000,0000,0000,,describing it as our\Ntheater. So what's the
Dialogue: 0,0:06:37.75,0:06:39.18,Default,,0000,0000,0000,,relationship between them?
Dialogue: 0,0:06:40.27,0:06:45.01,Default,,0000,0000,0000,,Let's drop that perpendicular\Ndown and we can see that this is
Dialogue: 0,0:06:45.01,0:06:50.14,Default,,0000,0000,0000,,a height. Why? Because of the Y\Ncoordinate the point and this is
Dialogue: 0,0:06:50.14,0:06:54.88,Default,,0000,0000,0000,,at a distance X because of the X\Ncoordinate of the point.
Dialogue: 0,0:06:55.68,0:07:02.96,Default,,0000,0000,0000,,And looking at that, we can see\Nthat Y is equal to R sign
Dialogue: 0,0:07:02.96,0:07:09.72,Default,,0000,0000,0000,,theater and X is equal to our\Ncause theater. So given R and
Dialogue: 0,0:07:09.72,0:07:15.44,Default,,0000,0000,0000,,Theta, we can calculate X&Y.\NWhat about moving the other way
Dialogue: 0,0:07:15.44,0:07:22.20,Default,,0000,0000,0000,,will from Pythagoras? We can see\Nthat X squared plus Y squared is
Dialogue: 0,0:07:22.20,0:07:25.32,Default,,0000,0000,0000,,equal to R-squared. So give now
Dialogue: 0,0:07:25.32,0:07:30.42,Default,,0000,0000,0000,,X. I'm now why we can\Ncalculate all and we can also
Dialogue: 0,0:07:30.42,0:07:34.98,Default,,0000,0000,0000,,see that if we take the\Nopposite over the adjacent, we
Dialogue: 0,0:07:34.98,0:07:41.20,Default,,0000,0000,0000,,have Y over X is equal to 10\Ntheater. So given AY in an X,
Dialogue: 0,0:07:41.20,0:07:44.11,Default,,0000,0000,0000,,we can find out what theater\Nis.
Dialogue: 0,0:07:45.37,0:07:45.97,Default,,0000,0000,0000,,Now.
Dialogue: 0,0:07:46.99,0:07:52.16,Default,,0000,0000,0000,,Always when doing these, it's\Nbest to draw sketches. If we're
Dialogue: 0,0:07:52.16,0:07:56.86,Default,,0000,0000,0000,,converting from one sort of\Npoint in Cartesians to its
Dialogue: 0,0:07:56.86,0:08:01.56,Default,,0000,0000,0000,,equivalent in Pohlers, or if\Nwe're moving back from Pohlers
Dialogue: 0,0:08:01.56,0:08:06.73,Default,,0000,0000,0000,,to cartesians, draw a picture,\Nsee where that point actually is
Dialogue: 0,0:08:06.73,0:08:12.84,Default,,0000,0000,0000,,now. Want to have a look at some\Nexamples. First of all, we're
Dialogue: 0,0:08:12.84,0:08:16.60,Default,,0000,0000,0000,,going to have a look at how to
Dialogue: 0,0:08:16.60,0:08:22.50,Default,,0000,0000,0000,,plot points. Then we're going to\Nhave a look at how to convert
Dialogue: 0,0:08:22.50,0:08:27.83,Default,,0000,0000,0000,,from one system into the other\Nand vice versa. So let's begin
Dialogue: 0,0:08:27.83,0:08:30.23,Default,,0000,0000,0000,,with plotting. Plot.
Dialogue: 0,0:08:31.49,0:08:37.54,Default,,0000,0000,0000,,And what I'm going to do is I'm\Ngoing to plot the following
Dialogue: 0,0:08:37.54,0:08:40.32,Default,,0000,0000,0000,,points and they're all in polar
Dialogue: 0,0:08:40.32,0:08:47.97,Default,,0000,0000,0000,,coordinates. I'm going to put\Nthem all on the same
Dialogue: 0,0:08:47.97,0:08:55.44,Default,,0000,0000,0000,,picture so we can get\Na feel for whereabouts things
Dialogue: 0,0:08:55.44,0:09:02.91,Default,,0000,0000,0000,,are in the polar play\Nor the plane for the
Dialogue: 0,0:09:02.91,0:09:08.64,Default,,0000,0000,0000,,polar coordinates. So we put\Nour poll, oh.
Dialogue: 0,0:09:09.18,0:09:11.79,Default,,0000,0000,0000,,And we have our initial line.
Dialogue: 0,0:09:12.71,0:09:15.31,Default,,0000,0000,0000,,First one that we've got to plot
Dialogue: 0,0:09:15.31,0:09:22.74,Default,,0000,0000,0000,,is 2. Pie so we know\Nthat Pi is the angle all the
Dialogue: 0,0:09:22.74,0:09:29.47,Default,,0000,0000,0000,,way around here, so there's pie\Nto there and we want to go
Dialogue: 0,0:09:29.47,0:09:31.03,Default,,0000,0000,0000,,out to units.
Dialogue: 0,0:09:31.63,0:09:37.38,Default,,0000,0000,0000,,So it's there. This\Nis the .2.
Dialogue: 0,0:09:37.94,0:09:39.49,Default,,0000,0000,0000,,Pie.
Dialogue: 0,0:09:41.49,0:09:46.95,Default,,0000,0000,0000,,Next to .1 N wealthy to is 0 so\Nwe're on the initial line.
Dialogue: 0,0:09:48.11,0:09:55.25,Default,,0000,0000,0000,,An one will be about there,\Nso there is the .1 note.
Dialogue: 0,0:09:56.77,0:10:02.67,Default,,0000,0000,0000,,2 - Π by 3 - π by\N3 means come around this
Dialogue: 0,0:10:02.67,0:10:07.67,Default,,0000,0000,0000,,way, and so minus π by\Nthree is about there, and
Dialogue: 0,0:10:07.67,0:10:12.21,Default,,0000,0000,0000,,we're coming around there\Nminus π by three, and we
Dialogue: 0,0:10:12.21,0:10:17.65,Default,,0000,0000,0000,,want to come out a distance\Nto, so that's roughly 2 out
Dialogue: 0,0:10:17.65,0:10:22.65,Default,,0000,0000,0000,,there, so this would be the\N.2 - π by 3.
Dialogue: 0,0:10:23.79,0:10:29.46,Default,,0000,0000,0000,,And finally, we've got the\Npoint. One 2/3 of Π. So we take
Dialogue: 0,0:10:29.46,0:10:32.07,Default,,0000,0000,0000,,the 2/3. That's going all the
Dialogue: 0,0:10:32.07,0:10:34.64,Default,,0000,0000,0000,,way around. To there.
Dialogue: 0,0:10:35.19,0:10:40.68,Default,,0000,0000,0000,,And we draw out through there,\Nand we want a distance of one
Dialogue: 0,0:10:40.68,0:10:44.90,Default,,0000,0000,0000,,along there, which roughly\Ncalled the scale we're using is
Dialogue: 0,0:10:44.90,0:10:47.43,Default,,0000,0000,0000,,about there, and so that's the
Dialogue: 0,0:10:47.43,0:10:49.93,Default,,0000,0000,0000,,.1. 2/3 of Π.
Dialogue: 0,0:10:51.68,0:10:58.10,Default,,0000,0000,0000,,Notice that we've taken theater\Nfirst to establish in which
Dialogue: 0,0:10:58.10,0:11:00.67,Default,,0000,0000,0000,,direction were actually facing.
Dialogue: 0,0:11:02.60,0:11:06.26,Default,,0000,0000,0000,,OK, let's now have a
Dialogue: 0,0:11:06.26,0:11:11.100,Default,,0000,0000,0000,,look. Having got used to\Nplotting points, let's now have
Dialogue: 0,0:11:11.100,0:11:16.44,Default,,0000,0000,0000,,a look in polar coordinates.\NThese points 2.
Dialogue: 0,0:11:17.07,0:11:19.27,Default,,0000,0000,0000,,Minus Π by 2.
Dialogue: 0,0:11:20.21,0:11:26.21,Default,,0000,0000,0000,,1.\N3/4 of
Dialogue: 0,0:11:26.21,0:11:33.38,Default,,0000,0000,0000,,Π. And 2\N- π by three. Now these
Dialogue: 0,0:11:33.38,0:11:35.84,Default,,0000,0000,0000,,are all in Pohlers.
Dialogue: 0,0:11:36.40,0:11:41.80,Default,,0000,0000,0000,,What I want to do is convert\Nthem into cartesian coordinates.
Dialogue: 0,0:11:41.80,0:11:47.69,Default,,0000,0000,0000,,So first a picture whereabouts\Nare they? And I'll do them one
Dialogue: 0,0:11:47.69,0:11:55.06,Default,,0000,0000,0000,,at a time. So let's take this\None 2 - π by two initial point
Dialogue: 0,0:11:55.06,0:11:57.51,Default,,0000,0000,0000,,poll. Oh, and initial line.
Dialogue: 0,0:11:58.44,0:12:02.16,Default,,0000,0000,0000,,Minus Π by two? Well, that's\Ncoming down here.
Dialogue: 0,0:12:02.73,0:12:10.06,Default,,0000,0000,0000,,To there. So that's minus\Nπ by two, and we've
Dialogue: 0,0:12:10.06,0:12:13.66,Default,,0000,0000,0000,,come a distance to to
Dialogue: 0,0:12:13.66,0:12:18.82,Default,,0000,0000,0000,,there. Well, we don't need to do\Nmuch calculation. I don't think
Dialogue: 0,0:12:18.82,0:12:23.76,Default,,0000,0000,0000,,to find this. If again we take\Nour origin for our cartesians as
Dialogue: 0,0:12:23.76,0:12:28.70,Default,,0000,0000,0000,,being the pole, and we align the\NX axis with our initial line.
Dialogue: 0,0:12:33.36,0:12:40.32,Default,,0000,0000,0000,,And there's our X. There's RY\Nand we can see straight away the
Dialogue: 0,0:12:40.32,0:12:47.80,Default,,0000,0000,0000,,point in Pohlers that's 2 - π\Nby two in fact, goes to the
Dialogue: 0,0:12:47.80,0:12:54.22,Default,,0000,0000,0000,,point. In Cartesians, That's 0 -\N2 because it's this point here
Dialogue: 0,0:12:54.22,0:13:01.72,Default,,0000,0000,0000,,on the Y axis, and it's 2\Nunits below the X axis, so it's
Dialogue: 0,0:13:01.72,0:13:03.32,Default,,0000,0000,0000,,0 - 2.
Dialogue: 0,0:13:03.37,0:13:07.76,Default,,0000,0000,0000,,Notice how plotting the point\Nactually saved as having to do
Dialogue: 0,0:13:07.76,0:13:12.15,Default,,0000,0000,0000,,any of the calculations. So\Nlet's take the next point now,
Dialogue: 0,0:13:12.15,0:13:14.54,Default,,0000,0000,0000,,which was one 3/4 of Π.
Dialogue: 0,0:13:15.28,0:13:23.16,Default,,0000,0000,0000,,1 3/4 of π. So\Nagain, let's plot where it
Dialogue: 0,0:13:23.16,0:13:26.56,Default,,0000,0000,0000,,is. Take our initial.
Dialogue: 0,0:13:27.22,0:13:34.50,Default,,0000,0000,0000,,.0 our poll and our initial line\N3/4 of π going round. It's
Dialogue: 0,0:13:34.50,0:13:40.10,Default,,0000,0000,0000,,positive so it drought there be\Nsomewhere out along that.
Dialogue: 0,0:13:40.67,0:13:47.35,Default,,0000,0000,0000,,Direction there's our angle of\N3/4 of Π, where somewhere out
Dialogue: 0,0:13:47.35,0:13:54.63,Default,,0000,0000,0000,,here at a distance one unit.\NSo again, let's take our X&Y
Dialogue: 0,0:13:54.63,0:13:57.06,Default,,0000,0000,0000,,axes, our X axis.
Dialogue: 0,0:13:58.01,0:14:01.33,Default,,0000,0000,0000,,To be along the initial line.
Dialogue: 0,0:14:01.85,0:14:07.82,Default,,0000,0000,0000,,And now why access to be\Nvertical and through the pole?
Dialogue: 0,0:14:08.49,0:14:14.63,Default,,0000,0000,0000,,Oh So that the polo becomes our\Norigin of, and it's this point.
Dialogue: 0,0:14:15.14,0:14:16.23,Default,,0000,0000,0000,,But where after?
Dialogue: 0,0:14:17.24,0:14:24.17,Default,,0000,0000,0000,,Now how we going to work\Nthis out that remember the
Dialogue: 0,0:14:24.17,0:14:30.47,Default,,0000,0000,0000,,formula that we had was X\Nequals our cause theater.
Dialogue: 0,0:14:32.01,0:14:39.68,Default,,0000,0000,0000,,Let's have a look at that.\NAre is one an we've got
Dialogue: 0,0:14:39.68,0:14:47.35,Default,,0000,0000,0000,,cause of 3/4 of Π and\Nthe cosine of 3/4 of Π
Dialogue: 0,0:14:47.35,0:14:55.01,Default,,0000,0000,0000,,is minus one over Route 2,\Nso that's minus one over Route
Dialogue: 0,0:14:55.01,0:14:58.85,Default,,0000,0000,0000,,2. Why is our sign Theta?
Dialogue: 0,0:14:58.88,0:15:06.16,Default,,0000,0000,0000,,And so this is one times the\Nsign of 3/4 of Π and the sign
Dialogue: 0,0:15:06.16,0:15:13.92,Default,,0000,0000,0000,,of 3/4 of Π is just one over\NRoute 2, and so we have one over
Dialogue: 0,0:15:13.92,0:15:19.25,Default,,0000,0000,0000,,Route 2 for RY coordinate. And\Nnotice that these answers agree
Dialogue: 0,0:15:19.25,0:15:24.58,Default,,0000,0000,0000,,with where the point is in this\Nparticular quadrant. Negative X
Dialogue: 0,0:15:24.58,0:15:28.95,Default,,0000,0000,0000,,and positive Y, negative X and\Npositive Y so.
Dialogue: 0,0:15:28.95,0:15:33.62,Default,,0000,0000,0000,,Even if I've got the calculation\Nwrong in the sense that I, even
Dialogue: 0,0:15:33.62,0:15:37.57,Default,,0000,0000,0000,,if I've done the arithmetic\Nwrong, have no, I've got the
Dialogue: 0,0:15:37.57,0:15:39.36,Default,,0000,0000,0000,,point in the right quadrant.
Dialogue: 0,0:15:39.42,0:15:46.13,Default,,0000,0000,0000,,Let's have a look at the\Nlast one of these two.
Dialogue: 0,0:15:46.90,0:15:50.39,Default,,0000,0000,0000,,Minus Π by
Dialogue: 0,0:15:50.39,0:15:54.35,Default,,0000,0000,0000,,3. So again, our poll.
Dialogue: 0,0:15:56.02,0:15:59.05,Default,,0000,0000,0000,,Our initial line.
Dialogue: 0,0:15:59.05,0:16:02.18,Default,,0000,0000,0000,,Minus Π by three is around here.
Dialogue: 0,0:16:04.76,0:16:11.53,Default,,0000,0000,0000,,So we've come around there minus\Nπ by three, and we're out a
Dialogue: 0,0:16:11.53,0:16:13.62,Default,,0000,0000,0000,,distance, two along there.
Dialogue: 0,0:16:15.44,0:16:20.55,Default,,0000,0000,0000,,Take our X axis to coincide with\Nthe initial line.
Dialogue: 0,0:16:21.09,0:16:22.50,Default,,0000,0000,0000,,And now origin.
Dialogue: 0,0:16:23.41,0:16:25.91,Default,,0000,0000,0000,,Coincide with the pole.
Dialogue: 0,0:16:28.36,0:16:35.17,Default,,0000,0000,0000,,Let's write down our equations\Nthat tell us X is
Dialogue: 0,0:16:35.17,0:16:41.98,Default,,0000,0000,0000,,our cause theater, which is\N2 times the cosine of
Dialogue: 0,0:16:41.98,0:16:48.11,Default,,0000,0000,0000,,minus π by three, which\Nis equal to 2.
Dialogue: 0,0:16:49.72,0:16:56.98,Default,,0000,0000,0000,,Times Now we want the cause\Nof minus π by three and
Dialogue: 0,0:16:56.98,0:17:04.24,Default,,0000,0000,0000,,the cosine of minus π by\Nthree is 1/2, and so that
Dialogue: 0,0:17:04.24,0:17:05.45,Default,,0000,0000,0000,,gives US1.
Dialogue: 0,0:17:06.83,0:17:10.52,Default,,0000,0000,0000,,Why is equal to our
Dialogue: 0,0:17:10.52,0:17:17.58,Default,,0000,0000,0000,,sign theater? Which is 2 times\Nthe sign of minus π by three,
Dialogue: 0,0:17:17.58,0:17:24.86,Default,,0000,0000,0000,,which is 2 Times Now we want the\Nsign of my minus Pi π three, and
Dialogue: 0,0:17:24.86,0:17:31.24,Default,,0000,0000,0000,,that is minus Route 3 over 2.\NThe two is cancelled to give us
Dialogue: 0,0:17:31.24,0:17:32.60,Default,,0000,0000,0000,,minus Route 3.
Dialogue: 0,0:17:33.13,0:17:37.63,Default,,0000,0000,0000,,And so again, notice we know\Nthat we've got it in the right
Dialogue: 0,0:17:37.63,0:17:41.43,Default,,0000,0000,0000,,quadrant. 'cause when we drew\Nthe diagram, we have positive X
Dialogue: 0,0:17:41.43,0:17:44.89,Default,,0000,0000,0000,,and negative Y, and that's how\Nwe've ended up here.
Dialogue: 0,0:17:46.07,0:17:51.11,Default,,0000,0000,0000,,What do we do about going back\Nthe other way?
Dialogue: 0,0:17:51.65,0:17:59.05,Default,,0000,0000,0000,,Well, let's have a look at\Nsome examples that will do that
Dialogue: 0,0:17:59.05,0:18:05.24,Default,,0000,0000,0000,,for us. What I'm going to look\Nat as these points, which are
Dialogue: 0,0:18:05.24,0:18:11.23,Default,,0000,0000,0000,,cartesians. The .22 point\Nminus 3 four.
Dialogue: 0,0:18:12.01,0:18:17.89,Default,,0000,0000,0000,,The point minus 2 -\N2 Route 3.
Dialogue: 0,0:18:18.73,0:18:25.15,Default,,0000,0000,0000,,And the .1 - 1 now these\Nare all points in Cartesian's.
Dialogue: 0,0:18:25.15,0:18:28.36,Default,,0000,0000,0000,,So let's begin with this one.
Dialogue: 0,0:18:29.06,0:18:32.23,Default,,0000,0000,0000,,Show where it is.
Dialogue: 0,0:18:32.74,0:18:38.92,Default,,0000,0000,0000,,To begin with, on the cartesian\Naxes so it's at 2 for X and two
Dialogue: 0,0:18:38.92,0:18:40.98,Default,,0000,0000,0000,,for Y. So it's there.
Dialogue: 0,0:18:42.22,0:18:48.11,Default,,0000,0000,0000,,So again.\NWe'll associate the origin in
Dialogue: 0,0:18:48.11,0:18:53.24,Default,,0000,0000,0000,,Cartesians with the pole in\Npolar's, and the X axis, with
Dialogue: 0,0:18:53.24,0:18:58.83,Default,,0000,0000,0000,,the initial line and what we\Nwant to calculate is what's that
Dialogue: 0,0:18:58.83,0:19:01.16,Default,,0000,0000,0000,,angle there an what's that
Dialogue: 0,0:19:01.16,0:19:03.55,Default,,0000,0000,0000,,radius there? Well.
Dialogue: 0,0:19:04.06,0:19:09.40,Default,,0000,0000,0000,,All squared is equal to X\Nsquared plus Y squared.
Dialogue: 0,0:19:10.89,0:19:17.84,Default,,0000,0000,0000,,So that's 2 squared +2 squared,\Nkeeps us 8 and so are is
Dialogue: 0,0:19:17.84,0:19:20.52,Default,,0000,0000,0000,,equal to 2 Route 2.
Dialogue: 0,0:19:21.26,0:19:23.42,Default,,0000,0000,0000,,When we take the square root of
Dialogue: 0,0:19:23.42,0:19:31.06,Default,,0000,0000,0000,,8. What about theater? Well,\Ntan Theta is equal to
Dialogue: 0,0:19:31.06,0:19:33.27,Default,,0000,0000,0000,,Y over X.
Dialogue: 0,0:19:33.87,0:19:41.01,Default,,0000,0000,0000,,In this case it's two over\N2, which is one, and so
Dialogue: 0,0:19:41.01,0:19:46.96,Default,,0000,0000,0000,,theater is π by 4, and\Nso therefore the polar
Dialogue: 0,0:19:46.96,0:19:52.91,Default,,0000,0000,0000,,coordinates of this point are\Ntwo route 2π over 4.
Dialogue: 0,0:19:55.47,0:19:59.78,Default,,0000,0000,0000,,Let's have a look at this one\Nnow, minus 3 four.
Dialogue: 0,0:20:00.75,0:20:05.31,Default,,0000,0000,0000,,Let's
Dialogue: 0,0:20:05.31,0:20:13.95,Default,,0000,0000,0000,,begin.\NBy establishing whereabouts it
Dialogue: 0,0:20:13.95,0:20:17.24,Default,,0000,0000,0000,,is on our cartesian
Dialogue: 0,0:20:17.24,0:20:22.32,Default,,0000,0000,0000,,axes. Minus 3 means it's back\Nhere somewhere, so there's
Dialogue: 0,0:20:22.32,0:20:28.45,Default,,0000,0000,0000,,minus three and the four on\Nthe Y. It's up there, so I'll
Dialogue: 0,0:20:28.45,0:20:29.87,Default,,0000,0000,0000,,point is there.
Dialogue: 0,0:20:32.45,0:20:39.66,Default,,0000,0000,0000,,Join it up to the origin as our\Npoint P and we are after. Now
Dialogue: 0,0:20:39.66,0:20:44.96,Default,,0000,0000,0000,,the polar coordinates for this\Npoint. So again we associate the
Dialogue: 0,0:20:44.96,0:20:50.73,Default,,0000,0000,0000,,pole with the origin and the\Ninitial line with the X axis,
Dialogue: 0,0:20:50.73,0:20:56.02,Default,,0000,0000,0000,,and so there's the value of\Ntheater that we're after. And
Dialogue: 0,0:20:56.02,0:21:01.79,Default,,0000,0000,0000,,this opie is the length are that\Nwere after, so R-squared is
Dialogue: 0,0:21:01.79,0:21:03.72,Default,,0000,0000,0000,,equal to X squared.
Dialogue: 0,0:21:03.74,0:21:10.59,Default,,0000,0000,0000,,Plus Y squared, which in this\Ncase is minus 3 squared, +4
Dialogue: 0,0:21:10.59,0:21:18.59,Default,,0000,0000,0000,,squared. That's 9 + 16, gives us\N25, and so R is the square
Dialogue: 0,0:21:18.59,0:21:22.58,Default,,0000,0000,0000,,root of 25, which is just five.
Dialogue: 0,0:21:23.21,0:21:27.56,Default,,0000,0000,0000,,What about finding\Ntheater now well?
Dialogue: 0,0:21:28.64,0:21:30.97,Default,,0000,0000,0000,,Tan Theta is.
Dialogue: 0,0:21:32.10,0:21:34.84,Default,,0000,0000,0000,,Y over X.
Dialogue: 0,0:21:35.88,0:21:37.49,Default,,0000,0000,0000,,Which gives us.
Dialogue: 0,0:21:38.48,0:21:41.95,Default,,0000,0000,0000,,4 over minus three.
Dialogue: 0,0:21:42.51,0:21:47.12,Default,,0000,0000,0000,,Now when you put that into your\NCalculator, you will get.
Dialogue: 0,0:21:47.72,0:21:53.11,Default,,0000,0000,0000,,A slightly odd answers. It will\Nactually give you a negative
Dialogue: 0,0:21:53.11,0:21:56.13,Default,,0000,0000,0000,,answer. That might be\Ndifficult for you to
Dialogue: 0,0:21:56.13,0:21:58.47,Default,,0000,0000,0000,,interpret. It sits actually\Ntelling you this angle out
Dialogue: 0,0:21:58.47,0:21:58.73,Default,,0000,0000,0000,,here.
Dialogue: 0,0:21:59.89,0:22:06.42,Default,,0000,0000,0000,,And we want to be all the way\Naround there now the way that I
Dialogue: 0,0:22:06.42,0:22:11.64,Default,,0000,0000,0000,,think these are best done is\Nactually to look at a right
Dialogue: 0,0:22:11.64,0:22:16.42,Default,,0000,0000,0000,,angle triangle like this and\Ncall that angle Alpha. Now let's
Dialogue: 0,0:22:16.42,0:22:23.38,Default,,0000,0000,0000,,have a look at what an Alpha is.\NTan Alpha is 4 over 3 and when
Dialogue: 0,0:22:23.38,0:22:28.16,Default,,0000,0000,0000,,you put that into your\NCalculator it will tell you that
Dialogue: 0,0:22:28.16,0:22:29.90,Default,,0000,0000,0000,,Alpha is nought .9.
Dialogue: 0,0:22:29.92,0:22:36.30,Default,,0000,0000,0000,,Three radians. Remember, theater\Nhas to be in radians and
Dialogue: 0,0:22:36.30,0:22:43.07,Default,,0000,0000,0000,,therefore. Theater here is\Nequal to π minus
Dialogue: 0,0:22:43.07,0:22:49.49,Default,,0000,0000,0000,,Alpha, and so that's\Nπ - 4.9. Three,
Dialogue: 0,0:22:49.49,0:22:52.70,Default,,0000,0000,0000,,which gives us 2.2
Dialogue: 0,0:22:52.70,0:22:58.20,Default,,0000,0000,0000,,one radians. And that's\Ntheater so you can see that
Dialogue: 0,0:22:58.20,0:23:02.29,Default,,0000,0000,0000,,the calculation of our is\Nalways going to be relatively
Dialogue: 0,0:23:02.29,0:23:05.16,Default,,0000,0000,0000,,straightforward, but the\Ncalculation this angle theater
Dialogue: 0,0:23:05.16,0:23:10.06,Default,,0000,0000,0000,,is going to be quite tricky,\Nand that's one of the reasons
Dialogue: 0,0:23:10.06,0:23:14.56,Default,,0000,0000,0000,,why it's best to plot these\Npoints before you try and
Dialogue: 0,0:23:14.56,0:23:16.20,Default,,0000,0000,0000,,workout what theater is.
Dialogue: 0,0:23:17.85,0:23:21.76,Default,,0000,0000,0000,,Now the next example was the\Npoint minus 2.
Dialogue: 0,0:23:22.58,0:23:25.98,Default,,0000,0000,0000,,Minus 2 Route 3.
Dialogue: 0,0:23:26.73,0:23:28.10,Default,,0000,0000,0000,,So again.
Dialogue: 0,0:23:30.83,0:23:36.15,Default,,0000,0000,0000,,Let's have a look where it is in\Nthe cartesian plane. These are
Dialogue: 0,0:23:36.15,0:23:40.65,Default,,0000,0000,0000,,its cartesian coordinates, so\Nwe've minus two for X. So we
Dialogue: 0,0:23:40.65,0:23:45.55,Default,,0000,0000,0000,,somewhere back here and minus 2\Nroute 3 four Y. So where
Dialogue: 0,0:23:45.55,0:23:48.83,Default,,0000,0000,0000,,somewhere down here? So I'll\Npoint is here.
Dialogue: 0,0:23:49.69,0:23:52.69,Default,,0000,0000,0000,,Join it up to our origin.
Dialogue: 0,0:23:53.60,0:23:55.83,Default,,0000,0000,0000,,Marking our point P.
Dialogue: 0,0:23:56.38,0:24:02.40,Default,,0000,0000,0000,,Again, we'll take the origin to\Nbe the pole and the X axis to be
Dialogue: 0,0:24:02.40,0:24:07.21,Default,,0000,0000,0000,,the initial line, and we can see\Nthat the theater were looking
Dialogue: 0,0:24:07.21,0:24:08.81,Default,,0000,0000,0000,,for is around there.
Dialogue: 0,0:24:09.51,0:24:12.82,Default,,0000,0000,0000,,That's our theater, and here's
Dialogue: 0,0:24:12.82,0:24:18.88,Default,,0000,0000,0000,,our. So again, let's\Ncalculate R-squared that's X
Dialogue: 0,0:24:18.88,0:24:21.51,Default,,0000,0000,0000,,squared plus Y squared.
Dialogue: 0,0:24:22.20,0:24:29.33,Default,,0000,0000,0000,,Is equal to. Well, in this\Ncase we've got minus two all
Dialogue: 0,0:24:29.33,0:24:36.46,Default,,0000,0000,0000,,squared plus minus 2 route 3\Nall squared, which gives us 4
Dialogue: 0,0:24:36.46,0:24:43.54,Default,,0000,0000,0000,,+ 12. 16 and so R is equal\Nto the square root of 16, which
Dialogue: 0,0:24:43.54,0:24:44.89,Default,,0000,0000,0000,,is just 4.
Dialogue: 0,0:24:45.73,0:24:50.87,Default,,0000,0000,0000,,Now, what about this? We can\Nsee that theater should be
Dialogue: 0,0:24:50.87,0:24:55.07,Default,,0000,0000,0000,,negative, so let's just\Ncalculate this angle as an
Dialogue: 0,0:24:55.07,0:24:59.27,Default,,0000,0000,0000,,angle in a right angle\Ntriangle. So tan Alpha
Dialogue: 0,0:24:59.27,0:25:03.94,Default,,0000,0000,0000,,equals, well, it's going to\Nbe the opposite, which is
Dialogue: 0,0:25:03.94,0:25:05.34,Default,,0000,0000,0000,,this side here.
Dialogue: 0,0:25:07.30,0:25:13.32,Default,,0000,0000,0000,,2 Route 3 in length over the\Nadjacent, which is just two
Dialogue: 0,0:25:13.32,0:25:20.35,Default,,0000,0000,0000,,which gives us Route 3. So Alpha\Njust calculated as an angle is π
Dialogue: 0,0:25:20.35,0:25:27.38,Default,,0000,0000,0000,,by three. So if that's pie by\Nthree this angle in size is 2π
Dialogue: 0,0:25:27.38,0:25:32.90,Default,,0000,0000,0000,,by three, but of course we must\Nmeasure theater negatively when
Dialogue: 0,0:25:32.90,0:25:37.92,Default,,0000,0000,0000,,we come clockwise from the\Ninitial line, and so Theta.
Dialogue: 0,0:25:37.95,0:25:44.26,Default,,0000,0000,0000,,Is minus 2π by three the 2π\Nthree giving us the size the
Dialogue: 0,0:25:44.26,0:25:50.08,Default,,0000,0000,0000,,minus sign giving us the\Ndirection so we can see that the
Dialogue: 0,0:25:50.08,0:25:55.90,Default,,0000,0000,0000,,point we've got described as\Nminus 2 - 2 route 3 in
Dialogue: 0,0:25:55.90,0:26:03.17,Default,,0000,0000,0000,,Cartesians is the .4 - 2π over 3\Nor minus 2/3 of Π in Pollas.
Dialogue: 0,0:26:03.97,0:26:08.47,Default,,0000,0000,0000,,Now we've taken a point in this\Nquadrant. A point in this
Dialogue: 0,0:26:08.47,0:26:12.22,Default,,0000,0000,0000,,quadrant appointing this\Nquadrant. Let's have a look at a
Dialogue: 0,0:26:12.22,0:26:16.72,Default,,0000,0000,0000,,point in the fourth quadrant\Njust to finish off this set of
Dialogue: 0,0:26:16.72,0:26:20.47,Default,,0000,0000,0000,,examples and the point we chose\Nwas 1 - 1.
Dialogue: 0,0:26:21.49,0:26:22.65,Default,,0000,0000,0000,,So again.
Dialogue: 0,0:26:24.32,0:26:29.06,Default,,0000,0000,0000,,Let's have a look at where it is\Nin our cartesian system.
Dialogue: 0,0:26:30.02,0:26:35.09,Default,,0000,0000,0000,,So we've a value of one 4X\Nand the value of minus one
Dialogue: 0,0:26:35.09,0:26:39.77,Default,,0000,0000,0000,,for Y. So there's our point\NP. Join it to the origin.
Dialogue: 0,0:26:40.79,0:26:45.51,Default,,0000,0000,0000,,And again will associate the\Norigin in the Cartesian's with
Dialogue: 0,0:26:45.51,0:26:50.23,Default,,0000,0000,0000,,the pole of the polar\Ncoordinates and the initial line
Dialogue: 0,0:26:50.23,0:26:55.89,Default,,0000,0000,0000,,will be the X axis, so we're\Nlooking for this angle theater.
Dialogue: 0,0:26:57.01,0:27:00.29,Default,,0000,0000,0000,,And this length of OP.
Dialogue: 0,0:27:02.02,0:27:07.49,Default,,0000,0000,0000,,So all squared is equal to X\Nsquared plus Y squared.
Dialogue: 0,0:27:08.13,0:27:14.48,Default,,0000,0000,0000,,So that's one squared plus minus\None squared, and that's one plus
Dialogue: 0,0:27:14.48,0:27:19.77,Default,,0000,0000,0000,,one is 2 so far is equal\Nto Route 2.
Dialogue: 0,0:27:20.51,0:27:24.71,Default,,0000,0000,0000,,Let's not worry about the\Ndirection here. Let's just
Dialogue: 0,0:27:24.71,0:27:28.92,Default,,0000,0000,0000,,calculate the magnitude of\Ntheater well. The magnitude of
Dialogue: 0,0:27:28.92,0:27:34.05,Default,,0000,0000,0000,,theater, in fact, to do that,\NI'd rather actually call it
Dialogue: 0,0:27:34.05,0:27:39.19,Default,,0000,0000,0000,,Alpha, just want to calculate\Nthe magnitude. So tan Alpha is.
Dialogue: 0,0:27:39.86,0:27:45.90,Default,,0000,0000,0000,,Opposite, which is one over the\Nadjacent, which is one which is
Dialogue: 0,0:27:45.90,0:27:52.94,Default,,0000,0000,0000,,just one. So Alpha is in fact\NΠ by 4. That means that my
Dialogue: 0,0:27:52.94,0:27:58.47,Default,,0000,0000,0000,,angle theater for the coordinate\Ncoming around this way is minus
Dialogue: 0,0:27:58.47,0:28:04.51,Default,,0000,0000,0000,,π by 4, and so my polar\Ncoordinates for this point, our
Dialogue: 0,0:28:04.51,0:28:08.03,Default,,0000,0000,0000,,Route 2 and minus π by 4.
Dialogue: 0,0:28:08.76,0:28:15.13,Default,,0000,0000,0000,,So. We've seen here why it's\Nso important to plot your points
Dialogue: 0,0:28:15.13,0:28:17.41,Default,,0000,0000,0000,,before you do any calculation.
Dialogue: 0,0:28:18.11,0:28:21.85,Default,,0000,0000,0000,,Having looked at what happens\Nwith points, let's see if we can
Dialogue: 0,0:28:21.85,0:28:25.91,Default,,0000,0000,0000,,now have a look at what happens\Nto a collection of points. In
Dialogue: 0,0:28:25.91,0:28:27.16,Default,,0000,0000,0000,,other words, a curve.
Dialogue: 0,0:28:28.00,0:28:31.21,Default,,0000,0000,0000,,Let's take a very simple curve
Dialogue: 0,0:28:31.21,0:28:37.40,Default,,0000,0000,0000,,in Cartesians. X squared plus Y\Nsquared equals a squared.
Dialogue: 0,0:28:37.95,0:28:42.43,Default,,0000,0000,0000,,Now this is a circle, a\Ncircle centered on the
Dialogue: 0,0:28:42.43,0:28:44.22,Default,,0000,0000,0000,,origin of Radius A.
Dialogue: 0,0:28:45.26,0:28:47.10,Default,,0000,0000,0000,,So if we think about that.
Dialogue: 0,0:28:49.65,0:28:56.98,Default,,0000,0000,0000,,Circle centered on the origin of\Nradius a, so it will go through.
Dialogue: 0,0:28:57.93,0:29:01.47,Default,,0000,0000,0000,,These points on the axis.
Dialogue: 0,0:29:09.15,0:29:10.08,Default,,0000,0000,0000,,Like so.
Dialogue: 0,0:29:11.58,0:29:16.91,Default,,0000,0000,0000,,If we think about what that\Ntells us, it tells us that no
Dialogue: 0,0:29:16.91,0:29:18.96,Default,,0000,0000,0000,,matter what the angle is.
Dialogue: 0,0:29:22.22,0:29:24.77,Default,,0000,0000,0000,,For any one of our points.
Dialogue: 0,0:29:26.64,0:29:32.41,Default,,0000,0000,0000,,If we were thinking in Pohlers,\Nthe radius is always a constant.
Dialogue: 0,0:29:32.41,0:29:39.15,Default,,0000,0000,0000,,So if we were to guess at the\Npolar equation, it would be our
Dialogue: 0,0:29:39.15,0:29:44.92,Default,,0000,0000,0000,,equals A and it wouldn't involve\Ntheater at all. Well, it just
Dialogue: 0,0:29:44.92,0:29:51.17,Default,,0000,0000,0000,,check that we know that X is\Nequal to our cause theater, and
Dialogue: 0,0:29:51.17,0:29:55.02,Default,,0000,0000,0000,,we know that Y is equal to our
Dialogue: 0,0:29:55.02,0:30:01.55,Default,,0000,0000,0000,,sign Theta. So we can\Nsubstitute these in R-squared
Dialogue: 0,0:30:01.55,0:30:07.64,Default,,0000,0000,0000,,cost, Square theater plus\NR-squared. Sine squared Theta is
Dialogue: 0,0:30:07.64,0:30:14.41,Default,,0000,0000,0000,,equal to a squared. We\Ncan take out the R-squared.
Dialogue: 0,0:30:16.77,0:30:24.10,Default,,0000,0000,0000,,And that leaves us with\Nthis factor of Cos squared
Dialogue: 0,0:30:24.10,0:30:26.30,Default,,0000,0000,0000,,plus sign squared.
Dialogue: 0,0:30:26.41,0:30:31.16,Default,,0000,0000,0000,,Now cost squared plus sign\Nsquared is a well known identity
Dialogue: 0,0:30:31.16,0:30:36.35,Default,,0000,0000,0000,,cost squared plus sign squared\Nat the same angle is always one,
Dialogue: 0,0:30:36.35,0:30:41.10,Default,,0000,0000,0000,,so this just reduces two\NR-squared equals a squared or R
Dialogue: 0,0:30:41.10,0:30:46.28,Default,,0000,0000,0000,,equals AR is a constant, which\Nis what we predicted for looking
Dialogue: 0,0:30:46.28,0:30:49.74,Default,,0000,0000,0000,,at the situation there now.\NAnother very straightforward
Dialogue: 0,0:30:49.74,0:30:54.92,Default,,0000,0000,0000,,curve is the straight line Y\Nequals MX. Let's just have a
Dialogue: 0,0:30:54.92,0:30:56.22,Default,,0000,0000,0000,,look at that.
Dialogue: 0,0:30:56.76,0:31:01.14,Default,,0000,0000,0000,,Y equals MX is a straight line\Nthat goes through the origin.
Dialogue: 0,0:31:02.55,0:31:07.52,Default,,0000,0000,0000,,Think about it, is it has a\Nconstant gradient and of course
Dialogue: 0,0:31:07.52,0:31:12.90,Default,,0000,0000,0000,,M. The gradient is defined to be\Nthe tangent of the angle that
Dialogue: 0,0:31:12.90,0:31:17.45,Default,,0000,0000,0000,,the line makes with the positive\Ndirection of the X axis.
Dialogue: 0,0:31:18.06,0:31:22.36,Default,,0000,0000,0000,,So if the gradient is a\Nconstant, the tangent of the
Dialogue: 0,0:31:22.36,0:31:27.44,Default,,0000,0000,0000,,angle is a constant, and so this\Nangle theater is a constant. So
Dialogue: 0,0:31:27.44,0:31:31.74,Default,,0000,0000,0000,,let's just have a look at that.\NWhy is we know?
Dialogue: 0,0:31:32.35,0:31:39.63,Default,,0000,0000,0000,,All. Sign\NTheta equals M times
Dialogue: 0,0:31:39.63,0:31:43.53,Default,,0000,0000,0000,,by our cause theater.
Dialogue: 0,0:31:44.49,0:31:51.22,Default,,0000,0000,0000,,The ask cancel out and so I\Nhave sign theater over Cos Theta
Dialogue: 0,0:31:51.22,0:31:58.42,Default,,0000,0000,0000,,equals M. And so I\Nhave tan Theta equals M and
Dialogue: 0,0:31:58.42,0:32:01.56,Default,,0000,0000,0000,,so theater does equal a
Dialogue: 0,0:32:01.56,0:32:08.26,Default,,0000,0000,0000,,constant. But and here there is\Na big bot for Y equals MX.
Dialogue: 0,0:32:08.26,0:32:13.13,Default,,0000,0000,0000,,That's the picture that we get\Nif we're working in Cartesians.
Dialogue: 0,0:32:13.84,0:32:20.03,Default,,0000,0000,0000,,But if we're working in Pohlers,\Nthere's our poll. There's our
Dialogue: 0,0:32:20.03,0:32:25.66,Default,,0000,0000,0000,,initial line Theta equals a\Nconstant. There is the angle
Dialogue: 0,0:32:25.66,0:32:33.06,Default,,0000,0000,0000,,Theta. And remember, we do not\Nhave negative values of R and so
Dialogue: 0,0:32:33.06,0:32:39.89,Default,,0000,0000,0000,,we get a half line. In other\Nwords, we only get this bit of
Dialogue: 0,0:32:39.89,0:32:45.08,Default,,0000,0000,0000,,the line. The half line there.\NThat simple example should
Dialogue: 0,0:32:45.08,0:32:49.96,Default,,0000,0000,0000,,warnors that whenever we are\Nmoving between one sort of curve
Dialogue: 0,0:32:49.96,0:32:53.94,Default,,0000,0000,0000,,in cartesians into its\Nequivalent in polar's, we need
Dialogue: 0,0:32:53.94,0:32:58.37,Default,,0000,0000,0000,,to be very careful about the\Nresults that we get.
Dialogue: 0,0:32:59.14,0:33:05.18,Default,,0000,0000,0000,,So let's just have a look at a\Ncouple more examples. Let's take
Dialogue: 0,0:33:05.18,0:33:11.70,Default,,0000,0000,0000,,X squared plus. Y squared is\Nequal to 9. We know that X is
Dialogue: 0,0:33:11.70,0:33:15.88,Default,,0000,0000,0000,,our cause theater. And why is\Nour sign theater?
Dialogue: 0,0:33:16.72,0:33:23.17,Default,,0000,0000,0000,,We can plug those in R-squared,\NCos squared Theta plus
Dialogue: 0,0:33:23.17,0:33:25.75,Default,,0000,0000,0000,,R-squared, sine squared Theta
Dialogue: 0,0:33:25.75,0:33:29.30,Default,,0000,0000,0000,,equals 9. All squared is a
Dialogue: 0,0:33:29.30,0:33:35.25,Default,,0000,0000,0000,,common factor. So we can take it\Nout and we've got cost squared
Dialogue: 0,0:33:35.25,0:33:40.24,Default,,0000,0000,0000,,Theta plus sign squared. Theta\Nis equal to 9 cost squared plus
Dialogue: 0,0:33:40.24,0:33:45.23,Default,,0000,0000,0000,,sign squared is an identity cost\Nsquared plus sign squared of the
Dialogue: 0,0:33:45.23,0:33:50.64,Default,,0000,0000,0000,,same angle is always one, and so\NR-squared equals 9. R is equal
Dialogue: 0,0:33:50.64,0:33:53.55,Default,,0000,0000,0000,,to three IE a circle of radius
Dialogue: 0,0:33:53.55,0:33:56.46,Default,,0000,0000,0000,,3. Let's
Dialogue: 0,0:33:56.46,0:34:03.26,Default,,0000,0000,0000,,take. The\Nrectangular hyperbola XY is
Dialogue: 0,0:34:03.26,0:34:05.83,Default,,0000,0000,0000,,equal to 4.
Dialogue: 0,0:34:06.28,0:34:13.17,Default,,0000,0000,0000,,And again, we're going to use\NX equals our cause theater and
Dialogue: 0,0:34:13.17,0:34:18.91,Default,,0000,0000,0000,,Y equals R sign theater. So\Nwe're multiplying X&Y together.
Dialogue: 0,0:34:18.91,0:34:25.22,Default,,0000,0000,0000,,So when we do that, we're\Ngoing to have our squared.
Dialogue: 0,0:34:25.29,0:34:32.87,Default,,0000,0000,0000,,Sign theater\NCos Theta equals 4.
Dialogue: 0,0:34:34.35,0:34:41.94,Default,,0000,0000,0000,,Now. Sign Theta Cos\NTheta will twice sign tita cost
Dialogue: 0,0:34:41.94,0:34:45.16,Default,,0000,0000,0000,,theater would be signed to
Dialogue: 0,0:34:45.16,0:34:51.01,Default,,0000,0000,0000,,theater. But I've taken 2 lots\Nthere, so if I've taken 2 lots
Dialogue: 0,0:34:51.01,0:34:54.89,Default,,0000,0000,0000,,there, it's the equivalent of\Nmultiplying that side by two. So
Dialogue: 0,0:34:54.89,0:34:57.01,Default,,0000,0000,0000,,I've got to multiply that side
Dialogue: 0,0:34:57.01,0:34:59.39,Default,,0000,0000,0000,,by two. So I end up with that.
Dialogue: 0,0:35:00.58,0:35:01.82,Default,,0000,0000,0000,,For my equation.
Dialogue: 0,0:35:02.67,0:35:06.86,Default,,0000,0000,0000,,I still some the other way round\Nnow, but one point to notice
Dialogue: 0,0:35:06.86,0:35:10.73,Default,,0000,0000,0000,,before we do. Knowledge of\Ntrig identity's is very
Dialogue: 0,0:35:10.73,0:35:14.10,Default,,0000,0000,0000,,important. We've used cost\Nsquared plus sign. Squared is
Dialogue: 0,0:35:14.10,0:35:18.60,Default,,0000,0000,0000,,one and we've now used sign to\NTheta is equal to two
Dialogue: 0,0:35:18.60,0:35:21.98,Default,,0000,0000,0000,,scientist accosts theater. So\Nknowledge of those is very
Dialogue: 0,0:35:21.98,0:35:26.86,Default,,0000,0000,0000,,important. So as I said, let's\Nsee if we can turn this around
Dialogue: 0,0:35:26.86,0:35:31.36,Default,,0000,0000,0000,,now and have a look at some\Nexamples going the other way.
Dialogue: 0,0:35:32.66,0:35:39.95,Default,,0000,0000,0000,,First one will take is 2 over,\NR is equal to 1 plus cause
Dialogue: 0,0:35:39.95,0:35:45.93,Default,,0000,0000,0000,,theater. I don't like really the\Nway it's written, so let's
Dialogue: 0,0:35:45.93,0:35:51.76,Default,,0000,0000,0000,,multiply up by R so I get R\NPlus R cause theater.
Dialogue: 0,0:35:52.54,0:35:57.53,Default,,0000,0000,0000,,Now, because I've done that,\Nlet's just remember that are
Dialogue: 0,0:35:57.53,0:36:02.02,Default,,0000,0000,0000,,squared is equal to X squared\Nplus Y squared.
Dialogue: 0,0:36:03.08,0:36:09.97,Default,,0000,0000,0000,,So that means I can replace\Nthis are here by the square
Dialogue: 0,0:36:09.97,0:36:13.41,Default,,0000,0000,0000,,root of X squared plus Y
Dialogue: 0,0:36:13.41,0:36:19.60,Default,,0000,0000,0000,,squared. Our costs theater.\NWill, our Cos Theta is equal
Dialogue: 0,0:36:19.60,0:36:25.14,Default,,0000,0000,0000,,to X so I can replace this\Nbit by X.
Dialogue: 0,0:36:26.42,0:36:31.14,Default,,0000,0000,0000,,Now it looks untidy's got a\Nsquare root in it, so naturally
Dialogue: 0,0:36:31.14,0:36:36.64,Default,,0000,0000,0000,,we would want to get rid of that\Nsquare root. So let's take X
Dialogue: 0,0:36:36.64,0:36:38.21,Default,,0000,0000,0000,,away from each side.
Dialogue: 0,0:36:38.37,0:36:44.09,Default,,0000,0000,0000,,And then let's Square both\Nsides. So that gives us X
Dialogue: 0,0:36:44.09,0:36:50.85,Default,,0000,0000,0000,,squared plus Y squared there and\Non this side it's 2 minus X
Dialogue: 0,0:36:50.85,0:36:57.61,Default,,0000,0000,0000,,all squared, which will give us\N4 - 4 X plus X squared.
Dialogue: 0,0:36:58.17,0:37:04.57,Default,,0000,0000,0000,,So I've got an X squared on each\Nside that will go out and so I'm
Dialogue: 0,0:37:04.57,0:37:08.97,Default,,0000,0000,0000,,left with Y squared is equal to\N4 - 4 X.
Dialogue: 0,0:37:10.41,0:37:15.50,Default,,0000,0000,0000,,And what you should notice there\Nis that actually a parabola.
Dialogue: 0,0:37:16.18,0:37:22.25,Default,,0000,0000,0000,,So this would seem to be the\Nway in which we define a
Dialogue: 0,0:37:22.25,0:37:24.12,Default,,0000,0000,0000,,parabola in polar coordinates.