Find the spherical coordinates

Question

The rectangular coordinates (x, y, z) of a point are given. Find the spherical coordinates (𝜌, 𝜃, φ) of the point. Express the measure of the angles in degrees rounded to the nearest integer. (−1, 2, 1)

Answer 1

To convert the rectangular coordinates (x,y,z)=(−1,2,1) to spherical coordinates (ρ,θ,ϕ), we use the following formulas:

1. ρ=sqrt(x²+y²+z²)
2. θ=arctan⁡(y/x)
3. ϕ=arccos⁡(z/ρ)

First, let's find ρ:

ρ=sqrt((−1)²+2²+1²)=sqrt(6)​

Next, we find θ. Note that θ is the azimuthal angle in the xy-plane, measured from the positive x-axis:

θ=arctan⁡(y/x)=arctan⁡(2/−1)=arctan⁡(−2)

Since the point (−1,2) is in the second quadrant, we add 180° to the result:

θ=arctan⁡(−2)+180°

Using a calculator to find arctan⁡(−2):

arctan⁡(−2)≈−63.4°

Thus,

θ≈−63.4^∘+180^∘≈116.6^∘

Finally, we find ϕ, which is the polar angle measured from the positive z-axis:

ϕ=arccos⁡(z/ρ)=arccos⁡(1/sqrt(6))

Using a calculator to find arccos⁡(1/sqrt(6)):

arccos⁡(1/sqrt(6))≈65.9^∘

So, the spherical coordinates (ρ,θ,ϕ) of the point (−1,2,1) are:

ρ≈2.45,θ≈117^∘,ϕ≈66^∘ or (2.45,117^∘,66^∘)

Hope this was helpful.