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