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Hi everyone. Welcome back to integralcalc.com. Today were going to be doing another cylindrical
coordinate example. In this example we're given the cylindrical coordinates (2, pi/3,
-5). Those are cylindrical coordinates in the form
r, theta and z as opposed to rectangular coordinates where we always have x, y and z.
With cylindrical coordinates we have r, theta and z.
Given that, we’re going to be using these three formulas here to convert these cylindrical
coordinates to rectangular coordinates. So the first formula is going to give us the
x-coordinate the second formula is going to give us the y-coordinate and the third formula
is going to give us the z-coordinate of our rectangular coordinates, which will be our
final answer. Once we have these formulas, and if we have
them memorized or if we have them in our calculator, all we have to do is plug in our cylindrical
coordinates to find the rectangular coordinates. So remember we said that the cylindrical coordinate
is in the form (r, theta, z), which means we're going to plug 2 for r, pi/3 everywhere
we see theta and negative 5 for z. So you can see in this case, when were solving
for the x-coordinate we're going to be plugging in 2 for r and pi/3 for theta.
So this is the equation that we get for our x-coordinate.
For the y-coordinate, again, I'm going to plug in 2 for r, and pi/3 for theta so the
y-coordinate is 2 sin of pi/3 and of course we're going to be simplifying these.
And then to find the z coordinate, we are just going to plug in negative 5 for z.
So z in our rectangular coordinate is also going to equal negative 5. It’s the same
in both cylindrical and rectangular coordinates. So z is done. We have to simplify the x and
y-coordinates however. The way that we're going to do that is using the unit circle.
We're going to look at the unit circle and find the place in the unit circle where the
angle is equal to pi/3. We head over to our Unit Circle and we're
looking for pi/3 so we move along the angle and we find here pi/3. This is where the angle
is equal to pi/3. For x-coordinate, we're looking for cos of that angle (pi/3). And
cos on the unit circle is the x-coordinate. So we look at the coordinate here, we see
that the x-coordinate is equal to 1/2. Heading back to our equation, we're going
to replace pi/3 with 1/2. Of course when we multiply those two out,
the 2s cancel and we get x equal to 1. Now to simplify the y-coordinate here, again
we're going to be finding pi/3 on the Unit Circle and taking sin of that angle. So if
we head back over here to our unit circle, again we find the angle pi/3 and this time
we need sin of that angle which when you’re looking at the unit circle, sin is the y-coordinate
there. So we're going to be looking at the y-coordinate and we see here that it's the
square root of 3 over 2. So we're going to grab the square root of
3 over 2, head back over to our equation and plug in the square root of 3 over 2 for sin
of pi/3. Of course when we multiply that by 2 here
which we have out in front, the 2s are going to cancel and we'll be left simply with square
root of 3. So y is equal to the square root of 3. So
now we have x, y and z. And to find our rectangular coordinate, we
just put the three of those together. So our rectangular coordinate is going to be equal
to (1, square root of 3, -5). And this is exactly the same place in our
three-dimensional space as this cylindrical coordinate (2, pi/3, -5). It’s just in rectangular
coordinate form. And that's what we have to do to find rectangular coordinates. I hope
that helped you guys and I'll see you in the next problem. Bye.