Hi everyone. Welcome back to integralcalc.com. Today we're going to be talking about how
to convert cylindrical coordinates to rectangular coordinates. The cylindrical coordinates that
we're given are 1, 1/2 pi or pi/2 and 2. The formula that we're going to use to convert
these cylindrical coordinates to rectangular coordinates are the three formulas below.
The only thing outside these formulas that you have to remember is that our original
cylindrical coordinates are given in the form, r, theta and z.
And we’re trying to go into x, y and z. So in order to get r, theta, z to x, y, we
need to use these formulas here. All we do is plugging in what we get here
in r, theta, z into these formulas. Let’s go ahead and look at the x-coordinate first.
The way that we're going to convert the x-coordinate, 1 in the cylindrical coordinate to rectangular
coordinate is by using this r cos theta. So we're going to plug in 1 for r because remember
our cylindrical coordinates are r, theta and z. So we plug in 1 for r and we plug in 1/2
pi for theta and that is going to give us our x-coordinate.
We’re also going to get the equation here for y, plugging in 1 for r and 1/2 pi for
theta. We get the equation z equals 2 because we
just have this z equal’s z formula. We’re going to pull the 2 coordinate straight from
the original coordinate that we're given. So the z equation is already simplified as
much as possible, obviously so all we need to do is simplify these equations for x and
y. The way that we're going to do that is with the Unit Circle. So we have to find where
on the unit circle the angle is equal to 1/2 pi or pi/2.
We can see that when the angle is pi/2 at this point, we're looking for the coordinates
there which in this case, the coordinates are (0, 1). x is equal to 0, y is equal to
1. So when were asked for cos of the angle we're looking at the x-coordinate of that
point which is 0 and when were asked for sin of the angle, we're looking for the y-coordinate
which is 1. So we go back to our formulas here.
cos of pi/2 is 0 so we multiply here by 0. This replaces the entire cos of pi/2 and we
end up with x equals 0. And then for our y equation, this sin of pi
over 2 is going to be replaced with 1, because remember that sin of the angle pi/2 is the
y-coordinate at that angle in the Unit Circle, which in our case is 1. So we end up with
1 times 1 and so we get y equals 1. And now all we have to do is put our three
coordinates together for x, y, and z. We get our final answer, which is 0, 1, 2.
So, converting this cylindrical coordinates 1, pi/2 and 2, to rectangular coordinates
yields a (0, 1, 2). That’s our final answer. I hope that helped and I'll see you guys in
the next problem. Bye.
