All right, let's keep converting Cartesian functions
to polar coordinates.
The next one I have here is 3y minus 7x is equal to 10.
I cut and pasted our tool kit here, and let's
see what we can do.
We want to convert this to a function of r and theta.
So the simplest thing, we have a y and an x,
we just substitute.
We know that y is equal to r sine of theta and x is equal
to r cosine of theta, so let's just substitute.
So 3 times y, well, y is the same thing as r sine of
theta minus 7 times x.
Well, that's 7 times r cosine theta is equal to-- now,
remember, these just came from SOHCAHTOA, nothing
fancier than that.
Let's see if we can simplify this or write it explicitly
in terms of theta.
So we could factor out an r, so you get r times 3 sine theta
minus 7 cosine theta, right?
We just factored out an r, and that's equal to 10.
And now divide both sides by this big expression, and you're
left with r is equal to 10, 3 sine of theta-- 10 divided by 3
sine of theta minus 7 cosine of theta.
And we could also write this, you know, we could say, well,
this r is a function of theta.
We could write r of theta, right? r is a
function of theta.
I just felt like doing that so that you could show that the
function notation even works in polar coordinates.
Let's do another one.
Let me cut and paste our tool kit.
Look at that.
All right, next one.
So it says y is equal to 2x minus 3.
Well, we can do the same thing.
We know what y and x are in terms of r and theta. y is
equal to r sine of theta, so let's write that.
r sine of theta is equal to 2 times x.
Well, x is r cosine of theta minus 3.
Let's see if we can separate the r's and thetas, so let's
subtract this from both sides.
So we get r sine of theta minus 2r cosine of theta
is equal to minus 3.
Just like the last problem, we can factor an r out, so we get
r times sine of theta minus 2 cosine of theta is
equal to minus 3.
Now, divide both sides by this expression, so you're left with
r is equal to minus 3 divided by sine of theta minus
2 cosine of theta.
There you go.
Now they want us to do some problems.
Let me clear everything.
I want to do some problems.
We're going to convert the other way.
Let's write our tool kit down.
It's proved to be useful.
All right, now they want to convert from polar to Cartesian
coordinates, and so they give us the polar function r is
equal to 4 sine of theta.
So how do we convert this into a function of x and y?
So let's see here.
There's no obvious-- let's think about it a little bit.
So we know we have this equation y is equal
to r sine of that.
So can we write sine of theta?
Well, let me do a little side here.
So y is equal to r sine of theta.
If we divide both sides by r, what do we get?
We get y/r is equal to sine of theta, right?
Well, that seems to be helpful.
We have a sine of theta there, and then we have something that
at least it gets sort of the theta out of the equation.
We're still left with an r, but it makes it a
lot easier to look at.
So let's do that.
Sine of theta is equal to y/r, so let's substitute that there.
So we get r is equal to 4 times sine of theta,
which is y/r, so 4y/r.
We can multiply both sides of the equation by r, and you're
left with r squared is equal to 4y.
And we know what r squared is equal to.
It tells us right there. r squared is equal to x
squared plus y squared.
So you get x squared plus y squared is equal to 4y.
And there we have it.
We have it at least in an implicit form.
We don't have an explicit equation, but I think that's
good enough for now.
So you see, it's really just a lot of algebra and I guess a
little bit of trigonometry, but really a lot of algebra and
just kind of figuring out how to use this tool kit.
Let's do a couple more.
So let's say-- let me paste the tool kit.
Let's say that we have the polar coordinates r is equal
to sine of theta plus cosine of theta.
And just so you don't lose the big picture of what we're
doing, we could graph this.
You could put your graphing calculator on the polar
coordinates, and I'll do future videos where we do graph it,
and it'll produce some graph.
As theta changes, as we go around the circle, the radius
will change, and it'll produce something.
I don't know what it might look like, like flower
petals or something.
I don't know I'll have to-- I don't have the intuition of
exactly what this'll look like, but what we're saying is when
we convert to x and y, if you actually graphed in the
Cartesian coordinates, it would look the exact same way.
That's all we're doing.
How do you express the same relationship in
terms of x and y's?
So how do we do this?
Well, in the last one, we can-- these two, we can rewrite.
We can divide both sides of these equations by r.
This is the same thing as sine of theta is
equal to y/r, right?
I just divide both sides by r.
And this is the same thing as cosine of
theta is equal to x/r.
Just divided both sides of this equation by
r, and you get this.
So now we can use this to substitute back here, so r
is equal to sine of theta.
Well, we know that's equal to y/r plus cosine of theta.
Well, cosine of theta is equal to x/r.
And now we can multiply both sides of this equation by r,
and we're left with r squared is equal to y plus x.
And we know what r squared is now.
That should be second nature, hopefully.
So it's x squared plus y squared, right?
That's r squared is equal to x-- sorry,
is equal to y plus x.
There you go.
Let's do one more.
Let's do it without our tool kit, and I'll do it
in a different color.
Let's say we had r is equal to a squared, so they're
leaving it abstract.
But a is some constant, so we hopefully know our intuition
from a couple of videos ago that this should be
a circle, right?
If a was 3, then we'd say, oh, r is equal to 9, right?
3 squared, and then there'd just be a circle with a
constant radius of 9.
So how would you write this in Cartesian coordinates?
Well, we know that x squared plus y squared
is equal to r squared.
So one thing we could do, maybe just square both
sides of this equation.
So you get r squared is equal to-- what's a squared squared?
Well, it's a to the fourth, right?
It's a to the 2 times 2, so a to the fourth.
And we know what r squared is.
It's x squared plus y squared, so you get x squared plus y
squared is equal to a to the fourth.
Not too bad.
And that's really-- I mean, that'll get you pretty far in
converting between Cartesian and polar coordinates.
In the next video, we'll maybe explore a couple of these
graphs to hopefully give you a little intuition of really how
the relationships between r and theta really works.
See you in the next video.