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Hi everyone! Welcome back to integralcalc.com. Today we’re going to be doing an example
about how to convert spherical coordinates and in this case we’re going to be converting
them to cylindrical coordinates. The spherical coordinates that were given are three, pie
over two, and pie, for our spherical coordinate, and we’re going to take that and convert
it to a cylindrical coordinate. So, the formulas we’re going to need to
do that are the following: First of all, we have to recognize that when we have spherical
coordinates, they are given to us in this form here, in the form… instead of like…
rectangular or Cartesian coordinates are x, y, z, spherical coordinates are rho, phi or
phi, (I still don’t know how to say that, if you know please send me the email because
I don’t know), but rho, we’ll go with the phi, and theta, and then, we’re going
to convert that to cylindrical coordinates which are in the form r, theta, and z. So,
this is the conversion we’re going to be using and the formulas we need to find r,
theta and z are these three here. So, notice that the formulas we’re using, r, theta
and z, will give us our cylindrical coordinates here and the formulas are in the form or contain
the variables that are over here in our cylindrical coordinates. So, the left hand side here are
the cylindrical coordinates that we’re working toward, the right hand side here are the spherical
coordinates which we’re going to get from our original spherical coordinate, in our
case, three, pie over two, and pie. So, the first thing we’ll do is write out
our three equations here. So, we’ll take rho, in our case, three, and plug it in over
here for… in our formula for r and then we’re going to take… we’re going to
take pie for theta and plug it in here for theta into our equation for r. Theta, in terms
of cylindrical coordinates, is going to be equal to phi over here in our spherical coordinates
which we’ll grab from our original coordinate, pie over two, so we set up that equation.
And then, instead of z, we say z is equal to three cosine of pie because we grab rho
from our original coordinate as three and theta from our original coordinate as pie.
So we just put those variables into these formulas, we set these three up like this,
and, now it’s just a matter of simplifying these formulas and then collecting our theta
and z into one coordinate to get your cylindrical coordinate. So, the only simplification required
is to simplify sine of pie and cosine of pie. We can either use the unit circle to simplify,
or you can just plug this into your calculator, whichever easiest for you. Sine of pie on
the unit circle is zero, the value at the angle pie on the unit circle, for the y coordinate,
which is sine is equal to zero. So we plug in zero for sine of pie and we get three times
zero which is going to be equal to zero. Theta is still equal to pie over two, nothing we
can do to simplify that one. And, for z equals three cosine of pie, we will find cosine of
pie on the unit circle, we’re looking for the angle pie, and then… and then, we’re
taking the x coordinate at that angle which is cosine of that angle, and cosine of pie
ends up being negative one. So, we replace negative one, or we put negative one in place
of cosine of pie, and we get three times negative one which is going to give us a negative three.
So, now that we’ve simplified as much as we can, we put these three values together
in the order, r, theta, z, and that will give us our cylindrical coordinate. So our final
answer ends up being zero, pie over two, negative three, and this is… this is in the form
r, theta, z, which is what we need for cylindrical coordinates. So we’ve successfully converted
three, pie over two, pie, in spherical coordinate form, to zero, pie over two, negative three,
in cylindrical coordinate form. So, I hope that video helped you guys and
I’ll see you in the next one. Bye!