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## Multivariable calculus

# Polar coordinates

Introduction to polar coordinates.

## Video transcript

Everything we've done up to now
has dealt with Cartesian coordinates, even though you
might have not realized that they were Cartesian
coordinates, because I never called it that before just now. So what is a Cartesian
coordinate? Let me draw the axis
that you're hopefully familiar with by now. If you're not, review
those videos. Let me just draw the
y-axis and the x-axis. And so, Cartesian coordinates--
and actually, they can apply to more than just 2 dimensions. But 2 dimensions is what
we tend to deal with. Then if I want to specify any
point in 2 dimensional space, I just tell you how far in the x
direction I have to go, and how far in the y direction. And I can specify that point. Like, let's see, if I said,
well, let me give the Cartesian coordinates for that
point right there. And I might say, well, to get
to that point right there, I have to go to the
right 3 steps. So this is 3, right here. And then I have to
go up 4 steps. 1, 2, 3, 4. And you end up there. And by convention, we call this
the x-coordinate, and we call this the y-coordinate, and
we call this 3 comma 4. And this is just a convention
where we say the first coordinate is how far in the x
direction we go, and the second coordinate is how far in the up
and down, or the y direction, we go. So this is 3 comma 4. And we could have gone 3 to the
right and 4 up, or we could have gone 4 up and
3 to the right. And we'd have gotten to
the exact same point. This is just one way to
really specify a point in 2 dimensions. Another option, and you might
have done this in everyday life, is to say, hey, let me
just point you in a direction, and then you go a certain
distance in that direction. So I could have said,
you know what? Let me just point you
in this direction. Let me just point you in
that direction, and you need to go this far. You need to go this far
in that direction. That would also get you
to the same point. So how do I specify
the direction? Well, how about I
call this 0 degrees? How about I call
this the x-axis? How about I call
that 0 degrees? And we'll deal with degrees. But hopefully, you know
how to convert between degrees and radians. So you could just as easily
convert to radians. But you could specify the
direction if you call this 0, saying, ok, this is theta
degrees, and I want you-- I'm going to point you in a degree
in-- you know, you could always imagine someone
who's blindfolded. You point them in this
direction and you say, walk r units. And then they'll end up
at that same point. And so you could also specify
this point-- instead of specifying, this is the
Cartesian coordinates, this is x comma y-- you could also
specify it-- maybe, if we can figure out a way to do it. And I'll do it in magenta. You can also specify
it as r comma theta. Which essentially says,
you know, walk r units in the theta direction. Well, let's figure
out what that is. Because I think it's kind of
abstract now, and might be a little difficult to understand. So we're going to break out
a little trigonometry and actually just a little
Pythagorean theorem. Can we figure out r and theta? Well, r, hopefully, is
the easier one here. Because if you think about it,
this is a right triangle. This is a right triangle. That has distance 3. This is distance 4. This is a right triangle. So what is r here? Pythagorean Theorem. 3 squared plus 4 squared is
equal to the hypotenuse squared, or r squared. So we could say, 3 squared
plus 4 squared is equal to r squared. That's 9 plus 16 is
equal to r squared. 25 is equal to r squared. We don't want to deal with
negative distances, so 5 is equal to r, r is equal to 5. Fair enough. So we already figured out
that r is equal to 5. How do we figure out theta? So what do we know here? We know-- well, we're trying
to figure out theta. And we know it's
opposite side, right? This is just going
back to SOHCAHTOA. Let me write down SOHCAHTOA. If this is completely
unfamiliar to you, you might want to watch the basic
trigonometry video. So we know the opposite
side of theta, right? That's 4. And we also know
the adjacent side. That's 3. So which of the trig
functions uses the opposite and the adjacent? Well, the TOA. Opposite and adjacent. So tangent. So tangent of theta is equal to
the opposite-- which is really the y-coordinate, which is
equal to 4-- over the adjacent. Which is the x-coordinate,
which is 3. So tangent of theta
is equal to 4/3. And to solve this, you
essentially just take the inverse tangent of
both sides of this. So this is the same thing. And depending on your
calculator, or the convention you use, you might write this. When you take the inverse
tangent of both sides, you get-- well, I'll
just write it out. You could write it as the
arctan of the tangent of theta is equal to the arctan of 4/3. And this, of course-- the
arctan of the tangent. This is the same thing
as the inverse tan of the tangent of theta. This is just equal to theta. That just simplifies to theta. Is equal to arctan of 4/3. And then another way to write
arctan is often-- they'll often write it-- I'll write it. So the same exact statement
could be written as tan inverse to the negative 1 power. They sometimes say it. Although it's a little
misleading sometimes, when they write it like this. Because you don't know. Oh, am I taking the whole
tangent of theta to the negative power or something? But sometimes I'll write arctan
as 10 to the negative 1 power. But either way. So we can figure out theta by
taking the arctangent of 4/3. And most people do not have the
arctangent of 4/3 memorized. And I don't know an easy way
to figure it out, without getting my TI-85 out. So let's get it out. All right. So how do we know the arctan--
so we want to do this inverse-- let me get the calculator up a
little higher, so you can see. We want to get this
inverse tan right here. So you press second inverse
tangent of 4 divided by 3. And I already set my
calculator to degree mode. It gives you 53.13 degrees. So theta is equal
to 53.13 degrees. So there you have it. We already knew that we could
specify this point in the 2 dimensional plane by the point
x is equal to 3, y is equal to 4. We can also specify it by r
is equal to 5, and theta is equal to 53 degrees. So I'll write that. And polar coordinates, it can
be specified as r is equal to 5, and theta is 53.13 degrees. So all that says is, OK, orient
yourself 53.13 degrees counterclockwise from the
x-axis, and then walk 5 units. And you'll get to the
exact same point. And that's all polar
coordinates are telling you. Let's do another one. Actually, before that, let's
see if we can come up with something general here. Because once you have the
general, then you can always solve for the particular. So let me just write
an arbitrary point. Last time I gave a
specific point. The point-- was it 3 comma 4? I think it was. So let's say I have
the point x comma y. It's right there. So its x-coordinate is there,
its y-coordinate is there. So how do we convert this to
r, to polar coordinates? r, theta. Well, let's do the same thing
we did in the last video. If this length is r, and
this angle is theta. So just going into what we
did in the last video. We use Pythagorean Theorem. We say that x squared plus--
because this distance here is also y, right? This distance here is x. Pythagorean Theorem, you get
x squared plus y squared is equal to r squared. That's just the
Pythagorean Theorem. And then you have the
tangent of theta. The tangent of this angle. SOHCAHTOA. TOA. Opposite over adjacent. Tangent of theta is equal to
the opposite-- which is y-- over the adjacent-- which is x. And this is really
all you need. And if you wanted to go one
step further, and this is actually something you
should write at the side of your paper. How can we express y as
a function if we wanted to go the other way? If we were given r and
theta, how could we get y? Well, let's think about it. y-- if we want to know-- let's
see, if we are given r and y and we want to figure out
theta-- what deals with the r and the y, with respect to
theta? r is the hypotenuse and y is the opposite. Let me write SOHCAHTOA down. So what deals with the
opposite and the hypotenuse? Sine. So the sine of theta is equal
to the opposite-- which is the y side, which is equal to y--
over the hypotenuse, which is r. Equal to r. And so if you multiply both
sides by r, you get r sine theta is equal to y. And then let's do
it the other way. What if we wanted to have an
equation for theta, r, and x? x is adjacent to the angle. r is still the hypotenuse. So what deals with
adjacent and hypotenuse? Well, that's CAH. Cosine. So we have the cosine of theta
is equal to the adjacent, is equal to x, over the
hypotenuse, which is r. Multiply both sides by r
and you get x is equal to r cosine theta. And really, when you're
equipped with this formula-- which really just comes from
our trig identities-- this formula, which you know, is
similar, just comes from SOHCAHTOA. This, which comes from
SOHCAHTOA-- and really is, you know-- if you're
given these first. If you're given these two, you
can easily derive that one. And this one, which is just
the Pythagorean Theorem, you're actually ready. You're fully equipped to
convert between polar and rectangular coordinates. And we'll do that in the
next video, because I just realized I'm out of time.