# Is the second equation a sphere?

• gnome
In summary, the first equation is easily converted to the equation of a sphere in Cartesian coordinates, while the second equation is not a sphere due to the out of phase nature of sin&phi; and cos&theta;. It is likely a quartic equation when simplified.
gnome
This equation of a sphere in spherical coordinate form:
&rho; = 4sin&phi;cos&theta; converts very readily to (x-2)2 + y2 + z2 = 4 with very little effort.

Now this similar equation looks to me like it should also be a sphere, but I can't seem to get anywhere with it:
&rho; = 4sin&theta;cos&phi;

I just end up with a very ugly
x2 + y2 + z2 = 4yz/(&radic;(x2+y2)
and I have no idea what to do with that.

Is this a dead end? Is the second equation not a sphere?

I can only ask what makes you think the second equation would be a sphere. Spherical coordinates are not "symmetric" in &theta; and &phi;.

I guess I can only plead insanity on this one.

When it comes to spherical coordinates I'm an absolute greenhorn. The only reason I thought it might be a sphere is that I thought that was the equation that was given on my calc 3 exam last night, and I guess I was "mis-remembering".

But actually, I did wake up this morning thinking that the 2nd equation probably isn't a sphere; I realized that the sin&phi; and cos&theta; are "out of phase", i.e. &phi; is approaching its max when &theta; is approaching its min, and I was going to post that as a "supplementary" question. So thanks for answering my second question before I even asked it.

And thanks for pointing out so succinctly what characterizes a sphere's equation in spherical coordinates.

So, do you have any idea what "my" equation looks like on a graph?

It's a quartic of some kind. Clear the square roots and fractions and I believe you will have a fourth degree equation.

## 1. What is the formula for converting spherical coordinates to rectangular coordinates?

The formula for converting spherical coordinates (r, θ, φ) to rectangular coordinates (x, y, z) is:
x = r * sin(θ) * cos(φ)
y = r * sin(θ) * sin(φ)
z = r * cos(θ)

## 2. How do you determine the values of r, θ, and φ for a given point in spherical coordinates?

The values of r, θ, and φ can be determined by using the distance formula and trigonometric functions.
r = √(x² + y² + z²)
θ = arctan(y / x)
φ = arccos(z / r)

## 3. Can spherical coordinates be negative?

Yes, spherical coordinates can be negative. The r value can be negative if the point is located in the opposite direction from the origin, while the θ and φ values can be negative if the point is in a different quadrant or hemisphere.

## 4. What is the difference between spherical and rectangular coordinates?

The main difference between spherical and rectangular coordinates is the way they represent a point in 3D space. Spherical coordinates use a distance from the origin (r), an angle from the positive z-axis (θ), and an angle from the positive x-axis (φ), while rectangular coordinates use the Cartesian coordinates (x, y, z) which represent a point's distance from the x, y, and z axes.

## 5. Can you convert rectangular coordinates to spherical coordinates?

Yes, rectangular coordinates can be converted to spherical coordinates using the following formulas:
r = √(x² + y² + z²)
θ = arctan(y / x)
φ = arccos(z / r)

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