The problem asks to convert the given spherical coordinates to Cartesian coordinates. (a) The spherical coordinates are $(8, \pi/4, \pi/6)$. (b) The spherical coordinates are $(4, \pi/3, 3\pi/4)$.

1. Problem Description

The problem asks to convert the given spherical coordinates to Cartesian coordinates.

(a) The spherical coordinates are $(8, \pi/4, \pi/6)$.

(b) The spherical coordinates are $(4, \pi/3, 3\pi/4)$.

2. Solution Steps

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

$x = \rho \sin(\phi) \cos(\theta)$

$y = \rho \sin(\phi) \sin(\theta)$

$z = \rho \cos(\phi)$

(a) Given $(\rho, \theta, \phi) = (8, \pi/4, \pi/6)$:

$x = 8 \sin(\pi/6) \cos(\pi/4) = 8 * (1/2) * (\sqrt{2}/2) = 8 * \frac{\sqrt{2}}{4} = 2\sqrt{2}$

$y = 8 \sin(\pi/6) \sin(\pi/4) = 8 * (1/2) * (\sqrt{2}/2) = 8 * \frac{\sqrt{2}}{4} = 2\sqrt{2}$

$z = 8 \cos(\pi/6) = 8 * (\sqrt{3}/2) = 4\sqrt{3}$

So, the Cartesian coordinates are $(2\sqrt{2}, 2\sqrt{2}, 4\sqrt{3})$.

(b) Given $(\rho, \theta, \phi) = (4, \pi/3, 3\pi/4)$:

$x = 4 \sin(3\pi/4) \cos(\pi/3) = 4 * (\sqrt{2}/2) * (1/2) = 4 * \frac{\sqrt{2}}{4} = \sqrt{2}$

$y = 4 \sin(3\pi/4) \sin(\pi/3) = 4 * (\sqrt{2}/2) * (\sqrt{3}/2) = 4 * \frac{\sqrt{6}}{4} = \sqrt{6}$

$z = 4 \cos(3\pi/4) = 4 * (-\sqrt{2}/2) = -2\sqrt{2}$

So, the Cartesian coordinates are $(\sqrt{2}, \sqrt{6}, -2\sqrt{2})$.

3. Final Answer

(a) The Cartesian coordinates are $(2\sqrt{2}, 2\sqrt{2}, 4\sqrt{3})$.

(b) The Cartesian coordinates are $(\sqrt{2}, \sqrt{6}, -2\sqrt{2})$.