Convert the point (3,4,5) from Cartesian to spherical coordinates

To convert the Cartesian coordinates (x, y, z) = (3, 4, 5) to spherical coordinates (ρ, θ, φ), we use the following formulas:

1. ( ρ = sqrt{x^2 + y^2 + z^2} )
2. ( θ = tan^{-1}left(frac{y}{x}right) )
3. ( φ = cos^{-1}left(frac{z}{ρ}right) )

Now plug in the values:

1. Calculate ( ρ ):

[

ρ = sqrt{3^2 + 4^2 + 5^2} = sqrt{9 + 16 + 25} = sqrt{50} = 5sqrt{2}

]

2. Calculate ( θ ):

[

θ = tan^{-1}left(frac{4}{3}right) approx 0.927 text{ radians} quad (text{or } 53.13^circ)

]

3. Calculate ( φ ):

[

φ = cos^{-1}left(frac{5}{5sqrt{2}}right) = cos^{-1}left(frac{1}{sqrt{2}}right) = frac{pi}{4} text{ radians} quad (text{or } 45