Convert the following point from rectangular to spherical coordinates:
(1, −√3, 2√3).
I found the answer to be (4, -pi/3, 0.5235987755983), but my online homework is stating that my middle coordinate (-pi/3) is incorrect. I put this into an online calculator and it's giving me the same answer. Am I missing something?
going from (x,y,z) to (r,Ø,θ),
r^2 = x^2+y^2+z^2 = 1+3+12 = 16, so r=4
tanØ = y/x = -√3, so Ø = -π/3 or 2π/3
cosθ = z/r = √3/2, so θ = ±π/6
(1, −√3) is in QIV, so Ø = -π/3
z > 0, so θ = π/6
(r,Ø,θ) = (4,-π/3,π/6)
Looks good to me.
To convert from rectangular coordinates to spherical coordinates, you need to use the following formulas:
r = √(x^2 + y^2 + z^2) where (x, y, z) are the rectangular coordinates
θ = arccos(z / r)
ϕ = arctan(y / x)
Let's calculate the spherical coordinates for the point (1, -√3, 2√3):
First, calculate r:
r = √(1^2 + (-√3)^2 + (2√3)^2)
= √(1 + 3 + 12)
= √16
= 4
Next, calculate θ:
θ = arccos(2√3 / 4)
= arccos(√3 / 2)
= π / 3
Lastly, calculate ϕ:
ϕ = arctan((-√3) / 1)
= arctan(-√3)
= -π / 3
So, the spherical coordinates for the point (1, -√3, 2√3) are (4, π/3, -π/3).
Based on the calculations, it seems like your answer of (4, -π/3, 0.5235987755983) is correct. However, if your online homework system is not accepting it, there might be an issue with their system or a mistake in their answer key. You can reach out to your instructor or the technical support of the online homework platform to seek clarification on the discrepancy.