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

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.