Derive Formulas (6) and (7) for 3 -space....

Prove Formula 6 of Theorem 3.

So if we have X equals r co sign Pi over six, well, coastline of power six is going to be something that's positive. And there's no restrictions on what Arkan be. So X is just going to be some positive thing here. And then why is our times sign hi over six, which is also going to be positive. So we know that we have to live in the first quadrant here because X and Y are both going to be positive. I live in the first quadrant. I mean, when we look at this in two dimensional space, we're in the first quadrant. J. R is positive when this first quadrant, the only restriction is this data here. We have to be at this angle pi over six. All right, so that might not look like an angle of poverty. Six. But that's what I'm going for him. And then in three dimensional space, this line is allowed to extend outward from the board. So then it just becomes a flat plane. So I guess, Ah, plane. It's already implied that plans are flat. And as you can see from the picture, the plane that we get is going to be at some angle

So once we've done all these steps, then we'll have the correct format for the answer.

Okay, so the graph of by is you could have a constant pi over three. Um, it should describe the cone, and we can see that, Um, since we know that, uh, by is just the the angle with the Z axis. So this angle here, um, if we have a non constant, um uh, five in a non constant data, um, larger fi larger row, um, radius, um, extends outward like this. And then when you rotate around x y plane, you should end up with circle all around, so you end up with this, uh, cone.

Okay, so we have the surface, um, ro is equal to three. And, um, what this means is that, um So we have a three D coordinate system and in cylindrical coordinates, we know that really goes three means that the the distance from the origin it must be, um, three. So if we have all the set of all points for a distance, Um, but the origin is you go to three and two d space. That's a circle. And in three D space, that's a spear. So we have, um, in this case, Roe, you know, is ah x squared, plus y squared plus z squared, um, square root of that. And if we know that it's is equal to three, this is just the graph of a sphere with, um, we have a sphere with radius three.