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To convert the given rectangular coordinates \((x, y, z) = (3, 3, 4)\) to spherical coordinates \((\rho, \theta, \phi)\), we use the following formulas:

1. **\(\rho\)** (the radial distance):
   \[
   \rho = \sqrt{x^2 + y^2 + z^2}
   \]
   Plugging in \(x = 3\), \(y = 3\), and \(z = 4\):
   \[
   \rho = \sqrt{3^2 + 3^2 + 4^2} = \sqrt{9 + 9 + 16} = \sqrt{34}
   \]

2. **\(\theta\)** (the azimuthal angle):
   \[
   \theta = \tan^{-1}\left(\frac{y}{x}\right)
   \]
   Since \(x = y = 3\):
   \[
   \theta = \tan^{-1}\left(\frac{3}{3}\right) = \tan^{-1}(1) = \frac{\pi}{4} \, \text{radians} \, (\text{or} \, 45^\circ)
   \]

3. **\(\phi\)** (the polar angle):
   \[
   \phi = \cos^{-1}\left(\frac{z}{\rho}\right)
   \]
   Since \(z = 4\) and \(\rho = \sqrt{34}\):
   \[
   \phi = \cos^{-1}\left(\frac{4}{\sqrt{34}}\right)
   \]

Now, let's calculate the exact values for these. I'll compute them now.The spherical coordinates corresponding to the rectangular coordinates \((3, 3, 4)\) are:

1. **\(\rho\)** (radial distance) = \( \sqrt{34} \approx 5.83 \)
2. **\(\theta\)** (azimuthal angle) = \( \frac{\pi}{4} \) radians \( \approx 45^\circ \)
3. **\(\phi\)** (polar angle) = \( \approx 0.815 \) radians \( \approx 46.69^\circ \)

Would you like more details or have any questions?

Here are some related questions:
1. How do you convert spherical coordinates back to rectangular coordinates?
2. What are cylindrical coordinates, and how do they relate to spherical coordinates?
3. How do you interpret angles \(\theta\) and \(\phi\) geometrically in 3D space?
4. Why are spherical coordinates more useful in some physics problems than rectangular coordinates?
5. Can you generalize this transformation to higher dimensions?

**Tip:** Always ensure you're using the correct units (radians or degrees) for angles when converting or interpreting spherical coordinates!

What are the spherical coordinates of the point whose rectangular coordinates are (3, 3, 4)?

Learn how to convert rectangular coordinates (3, 3, 4) to spherical coordinates step-by-step, using formulas for ρ, θ, and φ. Suitable for high school students in Grades 10-12.

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