The problem involves converting cylindrical coordinates $(r, \theta, z)$ into rectangular (Cartesian) coordinates.

The given cylindrical coordinates are: $r = 8, \quad \theta = \frac{2\pi}{5}, \quad z = 8$

The formulas to convert cylindrical coordinates to rectangular coordinates are: $x = r \cdot \cos(\theta)$ $y = r \cdot \sin(\theta)$ $z = z$

Let’s calculate the rectangular coordinates.The rectangular (Cartesian) coordinates corresponding to the cylindrical coordinates $(r = 8, \theta = \frac{2\pi}{5}, z = 8)$ are approximately: $x \approx 2.47, \quad y \approx 7.61, \quad z = 8$

Do you want more details or have any questions about the conversion?

Here are 5 related questions to explore further:

1. How do you convert from spherical coordinates to Cartesian coordinates?
2. What are the differences between cylindrical and spherical coordinate systems?
3. How does the magnitude of the position vector change in different coordinate systems?
4. Can cylindrical coordinates represent points in 3D space with negative $r$?
5. How does the angle $\theta$ affect the $x$- and $y$-coordinates in cylindrical coordinates?

Tip: Cylindrical coordinates are particularly useful when solving problems with circular symmetry.