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# Polar coordinates 3

Converting between Cartesian and Polar functions. Created by Sal Khan.

Video transcript

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. 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.