Convert the point \( (x, y, z)=(-4,-1,5) \) to spherical coordinates. Give answers as positive values, either as expressions, or decimals to one decimal place. \( (\rho, \theta, \phi)=\square \)

To convert the point \((x, y, z)=(-4,-1,5)\) to spherical coordinates, we use the formulas: \[ \rho = \sqrt{x^2 + y^2 + z^2} \] \[ \theta = \tan^{-1} \left( \frac{y}{x} \right) \] \[ \phi = \cos^{-1} \left( \frac{z}{\rho} \right) \] First, calculate \(\rho\): \[ \rho = \sqrt{(-4)^2 + (-1)^2 + 5^2} = \sqrt{16 + 1 + 25} = \sqrt{42} \approx 6.5 \] Next, calculate \(\theta\): \[ \theta = \tan^{-1} \left( \frac{-1}{-4} \right) = \tan^{-1} \left( \frac{1}{4} \right) \approx 0.245 \text{ radians} \] Since the point \((-4, -1)\) is in the third quadrant, we add \(\pi\) to adjust: \[ \theta \approx 0.245 + \pi \approx 0.245 + 3.142 \approx 3.387 \text{ radians} \] Finally, calculate \(\phi\): \[ \phi = \cos^{-1} \left( \frac{5}{\sqrt{42}} \right) \approx \cos^{-1}(0.768) \approx 0.665 \text{ radians} \] Combining these results, the spherical coordinates are: \[ (\rho, \theta, \phi) \approx \left(\sqrt{42}, 3.4, 0.7\right) \]