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To convert the point given in spherical coordinates \((\rho, \theta, \phi)\) to Cartesian coordinates \((x, y, z)\), we use the following formulas:

\[
x = \rho \sin \phi \cos \theta
\]
\[
y = \rho \sin \phi \sin \theta
\]
\[
z = \rho \cos \phi
\]

Where:
- \(\rho = 6\)
- \(\theta = 2\pi\)
- \(\phi = \frac{3\pi}{4}\)

### Let's apply the formulas:

1. **For \(x\):**
   \[
   x = 6 \cdot \sin\left(\frac{3\pi}{4}\right) \cdot \cos(2\pi)
   \]

2. **For \(y\):**
   \[
   y = 6 \cdot \sin\left(\frac{3\pi}{4}\right) \cdot \sin(2\pi)
   \]

3. **For \(z\):**
   \[
   z = 6 \cdot \cos\left(\frac{3\pi}{4}\right)
   \]

I'll calculate these now.The Cartesian coordinates corresponding to the spherical coordinates \((\rho, \theta, \phi) = \left(6, 2\pi, \frac{3\pi}{4}\right)\) are approximately:

\[
(x, y, z) = (4.2, 0.0, -4.2)
\]

Would you like more details on how this conversion was done, or have any questions?

Here are some related questions to explore further:
1. What are the spherical coordinate formulas for Cartesian to spherical conversion?
2. How does the value of \(\theta\) affect the Cartesian coordinates in spherical to Cartesian conversion?
3. Can spherical coordinates represent points at the origin of the coordinate system?
4. How would a point with \(\phi = \frac{\pi}{2}\) (90 degrees) affect the Cartesian coordinates?
5. What practical applications use spherical coordinate systems?

**Tip:** When converting coordinates, always pay attention to the angle units (radians or degrees) to avoid confusion.

Convert the point (ρ, θ, ϕ) = (6, 2π, 3π/4) to Cartesian coordinates. Give answers either as expressions, or decimals to at least one decimal place.

Learn how to convert spherical coordinates (ρ, θ, ϕ) = (6, 2π, 3π/4) to Cartesian coordinates with step-by-step explanations. This tutorial covers the use of trigonometric functions and coordinate transformation equations.

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