Given the point in cylindrical coordinates $(-4,3,4)$, what are its spherical coordinates $(\rho ,\theta ,\phi )$?

• A. $(5,2.418,0.927)$
• B. $(7,2.214,1.107)$
• C. $(5,2.214,1.107)$
• D. $(5,0.927,2.418)$

Answer: (A)

Verified step by step guidance

1. Recall the relationship between cylindrical coordinates \((r, \theta, z)\) and spherical coordinates \((\rho, \phi, \theta)\), where \(\rho\) is the distance from the origin, \(\phi\) is the angle from the positive \(z\)-axis, and \(\theta\) is the same azimuthal angle as in cylindrical coordinates.
2. Identify the given cylindrical coordinates: \(r = -4\), \(\theta = 3\), and \(z = 4\). Note that \(r\) represents the radial distance in the \(xy\)-plane and should be non-negative, so consider the absolute value \(r = 4\) for conversion purposes.
3. Calculate \(\rho\), the distance from the origin to the point, using the formula: \(\rho = \sqrt{r^2 + z^2} = \sqrt{4^2 + 4^2}\)
4. Calculate \(\phi\), the angle from the positive \(z\)-axis, using the formula: \(\phi = \arccos\left(\frac{z}{\rho}\right) = \arccos\left(\frac{4}{\rho}\right)\)
5. The azimuthal angle \(\theta\) in spherical coordinates is the same as in cylindrical coordinates, so keep \(\theta = 3\). Thus, the spherical coordinates are \((\rho, \phi, \theta)\).