Convert this rectangular coordinate system point to spherical coordinate system

In summary, the given point is (0, -8, 0) and the conditions for r, θ, and φ are r≥0, 0≤θ≤2∏, and 0≤φ≤∏. Using the equation cos(θ)=y/x, we can determine that θ could be either ∏/4 or 3∏/4. However, we need to plot the point and compare it to a diagram to determine the correct value. As for φ, we can use the equation tan(φ)=y/x, but since y is negative and x is zero, we need to consider the value of arctan(-∞) to determine the
  • #1
skyturnred
118
0

Homework Statement



The point is (0, -8, 0)
r≥0
0≤θ≤2∏
0≤[itex]\varphi[/itex]≤∏

Homework Equations





The Attempt at a Solution



So here is what I've done so far:

I know that r=8 because x and z are 0

I know that θ=∏/4 or 3∏/4, but which one? both of these satisfy the following equation (which is the only one I know for this)

cos(θ)=y/x

Also, I am confused about [itex]\varphi[/itex]. I know that:

tan([itex]\varphi[/itex])=y/x, but y is negative, and x is zero. I know the value of arctan(∞) as well as arctan(-∞) but which one do I use? I am so confused. Thanks!
 
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  • #2
Draw a diagram and plot your point, then go from there by comparing to this.
 

What is a rectangular coordinate system?

A rectangular coordinate system, also known as a Cartesian coordinate system, is a system used to represent points in a two or three-dimensional space using three perpendicular axes (x, y, and z).

What is a spherical coordinate system?

A spherical coordinate system is a system used to represent points in a three-dimensional space using two angles (θ and φ) and a distance (r) from a fixed point called the origin.

How do you convert a point from rectangular to spherical coordinates?

To convert a point from rectangular to spherical coordinates, you can use the following formulas:

r = √(x^2 + y^2 + z^2)

θ = arccos(z/√(x^2 + y^2 + z^2))

φ = arctan(y/x)

What is the purpose of converting coordinates from rectangular to spherical?

Converting coordinates from rectangular to spherical can be useful in various fields, such as physics, engineering, and mathematics, as it allows for a more intuitive representation of points in three-dimensional space.

Are there any limitations or restrictions when converting coordinates from rectangular to spherical?

Yes, there are some limitations and restrictions when converting coordinates from rectangular to spherical, such as the coordinate system being centered at the origin and the distance (r) being a positive value. Additionally, the conversion may not be possible for points on the origin or points with undefined coordinates.

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