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## www.mathcentre.ac.uk/.../Polar%20Co-ordinates.mp4

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In this video, we're going to be
looking at polar.
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Coordinates.
Let's begin by actually
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looking at another coordinate
system. The Cartesian coordinate
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system. Now in that system we
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take 2. Axes and X axis
which is horizontal.
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And Y Axis which is vertical and
a fixed .0 called the origin,
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which is where these two points
cross. These two lines cross.
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Now we fix a point P in
the plane by saying how far it's
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displaced along the X axis to
give us the X coordinate.
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And how far it's displaced along
the Y access to give us the Y
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coordinate and so we have.
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A point P which is uniquely
described by its coordinates XY
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and notice I said how far it's
displaced because it is
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displacement that we're talking
about and not distance. That's
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what these arrowheads that we
put on the axes are all about
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their about showings, in which
direction we must move so that
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if we're moving down this
direction, it's a negative
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distance and negative
displacement that we're making.
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Now that is more than one way of
describing where a point is in
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the plane. And we're going to
be having a look at a system
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called polar coordinates.
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So in this system of polar
coordinates, we take a poll. Oh,
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and we take a fixed line.
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Now, how can we describe a point
in the plane using this fixed
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.0? The pole and this baseline.
Here. One of the ways is to
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think of it as. What if we turn?
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Around. Centering on oh
for the moment we rotate around,
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we can pass through a fixed
angle. Let's call that theater.
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And then along this radius we
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
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
conventions with Cartesian
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coordinates, we have certain
conventions with polar
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coordinates, and these are quite
strong conventions, so let's
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have a look at what these are.
First of all, theater is
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measured. In
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radians. So
that's how first convention
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theater is measured in radians.
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2nd convention
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well. Our second convention
is this that if this is our
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initial line and this is our
poll, then we measure theater
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positive when we go round in
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that direction. Anticlockwise
and we measure theater negative.
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When we go around in
that direction which is
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clockwise. So in just the same
way as we had an Arrowhead on
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our axes X&Y. In a sense, we've
got arrowheads here,
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distinguishing a positive
direction for theater and a
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negative direction for measuring
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theater. We have 1/3
convention to do with theater
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and that is that we never go
further round this way.
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Number there's our poll. Oh, our
fixed point. We never go further
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around this way then there, so
theater is always less than or
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equal to pie and we never go
round further that way than
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there again. So theater is
always strictly greater than
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minus Π - Π ramped there plus Π
round to there. And notice that
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we include. This bit of the line
if you like this extended bit of
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the line by going route to their
having the less than or equal to
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and having strictly greater than
Theta strictly greater than
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minus pie there.
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1/4 Convention 1/4
convention is that
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our is always
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positive. One of the things that
is quite important is that we be
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able to move from one system of
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.
Who's coordinates are
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X&Y? How can we
change from cartesians into
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pohlers? And how can we change
back again, but one obvious
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thing to do is to associate the
pole with the origin, and then
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to associate the initial line
with the X axis.
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And then if we draw the radius
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
Cartesians, we're
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describing it as XY, and
in Pohlers, where
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describing it as our
theater. So what's the
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relationship between them?
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Let's drop that perpendicular
down and we can see that this is
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a height. Why? Because of the Y
coordinate the point and this is
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at a distance X because of the X
coordinate of the point.
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And looking at that, we can see
that Y is equal to R sign
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theater and X is equal to our
cause theater. So given R and
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Theta, we can calculate X&Y.
What about moving the other way
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will from Pythagoras? We can see
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
calculate all and we can also
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see that if we take the
opposite over the adjacent, we
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have Y over X is equal to 10
theater. So given AY in an X,
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we can find out what theater
is.
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Now.
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Always when doing these, it's
best to draw sketches. If we're
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converting from one sort of
point in Cartesians to its
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equivalent in Pohlers, or if
we're moving back from Pohlers
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to cartesians, draw a picture,
see where that point actually is
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now. Want to have a look at some
examples. First of all, we're
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going to have a look at how to
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plot points. Then we're going to
have a look at how to convert
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from one system into the other
and vice versa. So let's begin
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with plotting. Plot.
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And what I'm going to do is I'm
going to plot the following
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points and they're all in polar
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coordinates. I'm going to put
them all on the same
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picture so we can get
a feel for whereabouts things
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are in the polar play
or the plane for the
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polar coordinates. So we put
our poll, oh.
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And we have our initial line.
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First one that we've got to plot
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is 2. Pie so we know
that Pi is the angle all the
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way around here, so there's pie
to there and we want to go
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out to units.
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So it's there. This
is the .2.
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Pie.
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Next to .1 N wealthy to is 0 so
we're on the initial line.
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An one will be about there,
so there is the .1 note.
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2 - Π by 3 - π by
3 means come around this
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way, and so minus π by
three is about there, and
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we're coming around there
minus π by three, and we
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want to come out a distance
to, so that's roughly 2 out
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there, so this would be the
.2 - π by 3.
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And finally, we've got the
point. One 2/3 of Π. So we take
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the 2/3. That's going all the
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way around. To there.
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And we draw out through there,
and we want a distance of one
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along there, which roughly
called the scale we're using is
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about there, and so that's the
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.1. 2/3 of Π.
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Notice that we've taken theater
first to establish in which
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direction were actually facing.
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OK, let's now have a
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look. Having got used to
plotting points, let's now have
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a look in polar coordinates.
These points 2.
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Minus Π by 2.
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1.
3/4 of
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Π. And 2
- π by three. Now these
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are all in Pohlers.
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What I want to do is convert
them into cartesian coordinates.
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So first a picture whereabouts
are they? And I'll do them one
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at a time. So let's take this
one 2 - π by two initial point
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poll. Oh, and initial line.
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Minus Π by two? Well, that's
coming down here.
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To there. So that's minus
π by two, and we've
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come a distance to to
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there. Well, we don't need to do
much calculation. I don't think
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to find this. If again we take
our origin for our cartesians as
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being the pole, and we align the
X axis with our initial line.
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And there's our X. There's RY
and we can see straight away the
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point in Pohlers that's 2 - π
by two in fact, goes to the
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point. In Cartesians, That's 0 -
2 because it's this point here
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on the Y axis, and it's 2
units below the X axis, so it's
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0 - 2.
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Notice how plotting the point
actually saved as having to do
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any of the calculations. So
let's take the next point now,
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which was one 3/4 of Π.
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1 3/4 of π. So
again, let's plot where it
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is. Take our initial.
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.0 our poll and our initial line
3/4 of π going round. It's
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positive so it drought there be
somewhere out along that.
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Direction there's our angle of
3/4 of Π, where somewhere out
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here at a distance one unit.
So again, let's take our X&Y
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axes, our X axis.
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To be along the initial line.
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And now why access to be
vertical and through the pole?
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Oh So that the polo becomes our
origin of, and it's this point.
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But where after?
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Now how we going to work
this out that remember the
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formula that we had was X
equals our cause theater.
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Let's have a look at that.
Are is one an we've got
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cause of 3/4 of Π and
the cosine of 3/4 of Π
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is minus one over Route 2,
so that's minus one over Route
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2. Why is our sign Theta?
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And so this is one times the
sign of 3/4 of Π and the sign
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of 3/4 of Π is just one over
Route 2, and so we have one over
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Route 2 for RY coordinate. And
notice that these answers agree
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with where the point is in this
particular quadrant. Negative X
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and positive Y, negative X and
positive Y so.
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Even if I've got the calculation
wrong in the sense that I, even
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if I've done the arithmetic
wrong, have no, I've got the
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point in the right quadrant.
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Let's have a look at the
last one of these two.
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Minus Π by
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3. So again, our poll.
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Our initial line.
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Minus Π by three is around here.
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So we've come around there minus
π by three, and we're out a
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distance, two along there.
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Take our X axis to coincide with
the initial line.
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And now origin.
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Coincide with the pole.
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Let's write down our equations
that tell us X is
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our cause theater, which is
2 times the cosine of
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minus π by three, which
is equal to 2.
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Times Now we want the cause
of minus π by three and
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the cosine of minus π by
three is 1/2, and so that
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gives US1.
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Why is equal to our
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sign theater? Which is 2 times
the sign of minus π by three,
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which is 2 Times Now we want the
sign of my minus Pi π three, and
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that is minus Route 3 over 2.
The two is cancelled to give us
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minus Route 3.
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And so again, notice we know
that we've got it in the right
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quadrant. 'cause when we drew
the diagram, we have positive X
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and negative Y, and that's how
we've ended up here.
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What do we do about going back
the other way?
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Well, let's have a look at
some examples that will do that
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for us. What I'm going to look
at as these points, which are
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cartesians. The .22 point
minus 3 four.
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The point minus 2 -
2 Route 3.
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And the .1 - 1 now these
are all points in Cartesian's.
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So let's begin with this one.
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Show where it is.
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To begin with, on the cartesian
axes so it's at 2 for X and two
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for Y. So it's there.
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So again.
We'll associate the origin in
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Cartesians with the pole in
polar's, and the X axis, with
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the initial line and what we
want to calculate is what's that
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angle there an what's that
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radius there? Well.
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All squared is equal to X
squared plus Y squared.
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So that's 2 squared +2 squared,
keeps us 8 and so are is
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equal to 2 Route 2.
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When we take the square root of
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8. What about theater? Well,
tan Theta is equal to
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Y over X.
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In this case it's two over
2, which is one, and so
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theater is π by 4, and
so therefore the polar
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coordinates of this point are
two route 2π over 4.
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Let's have a look at this one
now, minus 3 four.
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Let's
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begin.
By establishing whereabouts it
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is on our cartesian
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axes. Minus 3 means it's back
here somewhere, so there's
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minus three and the four on
the Y. It's up there, so I'll
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point is there.
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Join it up to the origin as our
point P and we are after. Now
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the polar coordinates for this
point. So again we associate the
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pole with the origin and the
initial line with the X axis,
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and so there's the value of
theater that we're after. And
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this opie is the length are that
were after, so R-squared is
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equal to X squared.
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Plus Y squared, which in this
case is minus 3 squared, +4
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squared. That's 9 + 16, gives us
25, and so R is the square
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root of 25, which is just five.
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What about finding
theater now well?
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Tan Theta is.
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Y over X.
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Which gives us.
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4 over minus three.
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Now when you put that into your
Calculator, you will get.
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A slightly odd answers. It will
actually give you a negative
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answer. That might be
difficult for you to
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interpret. It sits actually
telling you this angle out
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here.
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And we want to be all the way
around there now the way that I
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think these are best done is
actually to look at a right
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angle triangle like this and
call that angle Alpha. Now let's
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have a look at what an Alpha is.
Tan Alpha is 4 over 3 and when
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you put that into your
Calculator it will tell you that
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Alpha is nought .9.
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Three radians. Remember, theater
has to be in radians and
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therefore. Theater here is
equal to π minus
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Alpha, and so that's
π - 4.9. Three,
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which gives us 2.2
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one radians. And that's
theater so you can see that
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the calculation of our is
always going to be relatively
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straightforward, but the
calculation this angle theater
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is going to be quite tricky,
and that's one of the reasons
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why it's best to plot these
points before you try and
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workout what theater is.
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Now the next example was the
point minus 2.
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Minus 2 Route 3.
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So again.
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Let's have a look where it is in
the cartesian plane. These are
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its cartesian coordinates, so
we've minus two for X. So we
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somewhere back here and minus 2
route 3 four Y. So where
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somewhere down here? So I'll
point is here.
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Join it up to our origin.
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Marking our point P.
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Again, we'll take the origin to
be the pole and the X axis to be
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the initial line, and we can see
that the theater were looking
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for is around there.
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That's our theater, and here's
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our. So again, let's
calculate R-squared that's X
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squared plus Y squared.
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Is equal to. Well, in this
case we've got minus two all
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squared plus minus 2 route 3
all squared, which gives us 4
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+ 12. 16 and so R is equal
to the square root of 16, which
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is just 4.
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Now, what about this? We can
see that theater should be
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negative, so let's just
calculate this angle as an
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angle in a right angle
triangle. So tan Alpha
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equals, well, it's going to
be the opposite, which is
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this side here.
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2 Route 3 in length over the
adjacent, which is just two
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which gives us Route 3. So Alpha
just calculated as an angle is π
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by three. So if that's pie by
three this angle in size is 2π
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by three, but of course we must
measure theater negatively when
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we come clockwise from the
initial line, and so Theta.
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Is minus 2π by three the 2π
three giving us the size the
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minus sign giving us the
direction so we can see that the
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point we've got described as
minus 2 - 2 route 3 in
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Cartesians is the .4 - 2π over 3
or minus 2/3 of Π in Pollas.
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Now we've taken a point in this
quadrant. A point in this
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quadrant appointing this
quadrant. Let's have a look at a
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point in the fourth quadrant
just to finish off this set of
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examples and the point we chose
was 1 - 1.
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So again.
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Let's have a look at where it is
in our cartesian system.
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So we've a value of one 4X
and the value of minus one
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for Y. So there's our point
P. Join it to the origin.
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And again will associate the
origin in the Cartesian's with
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the pole of the polar
coordinates and the initial line
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will be the X axis, so we're
looking for this angle theater.
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And this length of OP.
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So all squared is equal to X
squared plus Y squared.
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So that's one squared plus minus
one squared, and that's one plus
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one is 2 so far is equal
to Route 2.
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Let's not worry about the
direction here. Let's just
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calculate the magnitude of
theater well. The magnitude of
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theater, in fact, to do that,
I'd rather actually call it
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Alpha, just want to calculate
the magnitude. So tan Alpha is.
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Opposite, which is one over the
adjacent, which is one which is
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just one. So Alpha is in fact
Π by 4. That means that my
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angle theater for the coordinate
coming around this way is minus
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π by 4, and so my polar
coordinates for this point, our
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Route 2 and minus π by 4.
• 28:09 - 28:15
So. We've seen here why it's
so important to plot your points
• 28:15 - 28:17
before you do any calculation.
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Having looked at what happens
with points, let's see if we can
• 28:22 - 28:26
now have a look at what happens
to a collection of points. In
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other words, a curve.
• 28:28 - 28:31
Let's take a very simple curve
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in Cartesians. X squared plus Y
squared equals a squared.
• 28:38 - 28:42
Now this is a circle, a
circle centered on the
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origin of Radius A.
• 28:45 - 28:47
So if we think about that.
• 28:50 - 28:57
Circle centered on the origin of
radius a, so it will go through.
• 28:58 - 29:01
These points on the axis.
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Like so.
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If we think about what that
tells us, it tells us that no
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matter what the angle is.
• 29:22 - 29:25
For any one of our points.
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If we were thinking in Pohlers,
the radius is always a constant.
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So if we were to guess at the
polar equation, it would be our
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equals A and it wouldn't involve
theater at all. Well, it just
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check that we know that X is
equal to our cause theater, and
• 29:51 - 29:55
we know that Y is equal to our
• 29:55 - 30:02
sign Theta. So we can
substitute these in R-squared
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cost, Square theater plus
R-squared. Sine squared Theta is
• 30:08 - 30:14
equal to a squared. We
can take out the R-squared.
• 30:17 - 30:24
And that leaves us with
this factor of Cos squared
• 30:24 - 30:26
plus sign squared.
• 30:26 - 30:31
Now cost squared plus sign
squared is a well known identity
• 30:31 - 30:36
cost squared plus sign squared
at the same angle is always one,
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so this just reduces two
R-squared equals a squared or R
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equals AR is a constant, which
is what we predicted for looking
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at the situation there now.
Another very straightforward
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curve is the straight line Y
equals MX. Let's just have a
• 30:55 - 30:56
look at that.
• 30:57 - 31:01
Y equals MX is a straight line
that goes through the origin.
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Think about it, is it has a
constant gradient and of course
• 31:08 - 31:13
M. The gradient is defined to be
the tangent of the angle that
• 31:13 - 31:17
the line makes with the positive
direction of the X axis.
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So if the gradient is a
constant, the tangent of the
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angle is a constant, and so this
angle theater is a constant. So
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let's just have a look at that.
Why is we know?
• 31:32 - 31:40
All. Sign
Theta equals M times
• 31:40 - 31:44
by our cause theater.
• 31:44 - 31:51
The ask cancel out and so I
have sign theater over Cos Theta
• 31:51 - 31:58
equals M. And so I
have tan Theta equals M and
• 31:58 - 32:02
so theater does equal a
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constant. But and here there is
a big bot for Y equals MX.
• 32:08 - 32:13
That's the picture that we get
if we're working in Cartesians.
• 32:14 - 32:20
But if we're working in Pohlers,
there's our poll. There's our
• 32:20 - 32:26
initial line Theta equals a
constant. There is the angle
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Theta. And remember, we do not
have negative values of R and so
• 32:33 - 32:40
we get a half line. In other
words, we only get this bit of
• 32:40 - 32:45
the line. The half line there.
That simple example should
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warnors that whenever we are
moving between one sort of curve
• 32:50 - 32:54
in cartesians into its
equivalent in polar's, we need
• 32:54 - 32:58
to be very careful about the
results that we get.
• 32:59 - 33:05
So let's just have a look at a
couple more examples. Let's take
• 33:05 - 33:12
X squared plus. Y squared is
equal to 9. We know that X is
• 33:12 - 33:16
our cause theater. And why is
our sign theater?
• 33:17 - 33:23
We can plug those in R-squared,
Cos squared Theta plus
• 33:23 - 33:26
R-squared, sine squared Theta
• 33:26 - 33:29
equals 9. All squared is a
• 33:29 - 33:35
common factor. So we can take it
out and we've got cost squared
• 33:35 - 33:40
Theta plus sign squared. Theta
is equal to 9 cost squared plus
• 33:40 - 33:45
sign squared is an identity cost
squared plus sign squared of the
• 33:45 - 33:51
same angle is always one, and so
R-squared equals 9. R is equal
• 33:51 - 33:54
to three IE a circle of radius
• 33:54 - 33:56
3. Let's
• 33:56 - 34:03
take. The
rectangular hyperbola XY is
• 34:03 - 34:06
equal to 4.
• 34:06 - 34:13
And again, we're going to use
X equals our cause theater and
• 34:13 - 34:19
Y equals R sign theater. So
we're multiplying X&Y together.
• 34:19 - 34:25
So when we do that, we're
going to have our squared.
• 34:25 - 34:33
Sign theater
Cos Theta equals 4.
• 34:34 - 34:42
Now. Sign Theta Cos
Theta will twice sign tita cost
• 34:42 - 34:45
theater would be signed to
• 34:45 - 34:51
theater. But I've taken 2 lots
there, so if I've taken 2 lots
• 34:51 - 34:55
there, it's the equivalent of
multiplying that side by two. So
• 34:55 - 34:57
I've got to multiply that side
• 34:57 - 34:59
by two. So I end up with that.
• 35:01 - 35:02
For my equation.
• 35:03 - 35:07
I still some the other way round
now, but one point to notice
• 35:07 - 35:11
before we do. Knowledge of
trig identity's is very
• 35:11 - 35:14
important. We've used cost
squared plus sign. Squared is
• 35:14 - 35:19
one and we've now used sign to
Theta is equal to two
• 35:19 - 35:22
scientist accosts theater. So
knowledge of those is very
• 35:22 - 35:27
important. So as I said, let's
see if we can turn this around
• 35:27 - 35:31
now and have a look at some
examples going the other way.
• 35:33 - 35:40
First one will take is 2 over,
R is equal to 1 plus cause
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theater. I don't like really the
way it's written, so let's
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multiply up by R so I get R
Plus R cause theater.
• 35:53 - 35:58
Now, because I've done that,
let's just remember that are
• 35:58 - 36:02
squared is equal to X squared
plus Y squared.
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So that means I can replace
this are here by the square
• 36:10 - 36:13
root of X squared plus Y
• 36:13 - 36:20
squared. Our costs theater.
Will, our Cos Theta is equal
• 36:20 - 36:25
to X so I can replace this
bit by X.
• 36:26 - 36:31
Now it looks untidy's got a
square root in it, so naturally
• 36:31 - 36:37
we would want to get rid of that
square root. So let's take X
• 36:37 - 36:38
away from each side.
• 36:38 - 36:44
And then let's Square both
sides. So that gives us X
• 36:44 - 36:51
squared plus Y squared there and
on this side it's 2 minus X
• 36:51 - 36:58
all squared, which will give us
4 - 4 X plus X squared.
• 36:58 - 37:05
So I've got an X squared on each
side that will go out and so I'm
• 37:05 - 37:09
left with Y squared is equal to
4 - 4 X.
• 37:10 - 37:16
And what you should notice there
is that actually a parabola.
• 37:16 - 37:22
So this would seem to be the
way in which we define a
• 37:22 - 37:24
parabola in polar coordinates.
Title:
www.mathcentre.ac.uk/.../Polar%20Co-ordinates.mp4
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