INSTRUCTOR: --and cylindrical coordinates. Spherical coordinates tell us that our radius, or our rho, is equal to 2. Theta is equal to negative 5 pi over 6, which that is our angle with the positive x-axis. And then phi is pi over 6, which is the angle with respect to the positive z-axis.

So we now have a radius of 2. So we can kind of imagine it sticking out from the origin, an arm of length 2. Now if we go from the x-axis and go to a negative 5 pi over 6, that will take us over into the area where we have negative x values, negative y values, and, well, we're not worried about z values yet. So we're in that octant over there.

And then we want to rotate away from the positive z-axis pi over 6. So that puts us in an area where x and y are negative but z is positive. That's where that point would be.

Now let's see if those coordinates, when we put them in rectangular and cylindrical, if they give us the same location. They should. So to go from rho, theta, phi to points that are of the form x, y, z in Cartesian coordinates, there are a few equations we could use.

We have x equals rho sine phi cosine theta. We have y equals rho sine phi sine theta, and z equals rho cosine phi or "ph-eye," depending on how you pronounce that.

So if we replace those values in here, that means our x-coordinate will be 2 sine pi over 6 multiplied by cosine of negative 5 pi over 6. Our y component will be 2 sine of pi over 6 multiplied by sine of negative 5 pi over 6.

And our z component will be 2 multiplied by cosine of pi over 6.

So our point in x, y, z, once we evaluate all of this, should be negative root 3 over 2 comma negative 1/2 comma root 3, which is in the same location-- roughly the same location as we predicted with the spherical coordinates.

Now if we need to convert from points that are of the form rho, theta, phi to points that are of the form in cylindrical coordinates that are r theta z, we can use another set of equations. These will be useful to us at the moment.

We have that r equals rho sine phi, theta equals theta-- those are actually the same interpretations so the value doesn't change-- and z equals rho cosine phi.

So if we fit our values here, rho is 2. 2 times sine of phi, which is pi over 6. So theta is going to be equal to the same theta value, so negative 5 pi over 6.

And z will actually be the same value in the last one, so that it would be 2 cosine phi pi over 6.

So we have these values. When we evaluate those, we have a value of 1, negative 5 pi over 6, and square root of 3. So that means our cylindrical coordinate point will be 1 comma negative 5 pi over 6, and squared root of 3.

Thinking about where that is located, we have a radius of 1, so we're 1 away from the origin. We go in the negative 5 pi over 6 rotating from the positive x-axis, so that's now over in where x and y are negative. And then we go up a value of square root of 3, which, again, puts us in the same location as in rectangular or spherical coordinates.