### Instruction: Convert the point $\left( 8, \frac{\pi}{4}, \sqrt{3} \right)$ in cylindrical coordinates to rectangular coordinates.### Response: I know that cylindrical coordinates are related to rectangular coordinates by the formulas $x = r \cos \theta$, $y = r \sin \theta$, and $z = z$.
So to convert the given point, I just need to plug in the values of $r$, $\theta$, and $z$ into those formulas and simplify.
Let me start with $x$.
I have $x = r \cos \theta = 8 \cos \left( \frac{\pi}{4} \right)$.
I recall that $\cos \left( \frac{\pi}{4} \right) = \frac{\sqrt{2}}{2}$, so I can write $x = 8 \cdot \frac{\sqrt{2}}{2}$.
Simplifying, I get $x = 4 \sqrt{2}$.
Next, I move on to $y$.
I have $y = r \sin \theta = 8 \sin \left( \frac{\pi}{4} \right)$.
I also recall that $\sin \left( \frac{\pi}{4} \right) = \frac{\sqrt{2}}{2}$, so I can write $y = 8 \cdot \frac{\sqrt{2}}{2}$.
Simplifying, I get $y = 4 \sqrt{2}$ as well.
Finally, I look at $z$.
I have $z = z = \sqrt{3}$, so there is nothing to do here.