To convert polar coordinates $(r, \theta)$ to Cartesian coordinates $(x, y)$, we use the formulas:

$x = r \cos(\theta)$ $y = r \sin(\theta)$

Here, $r = 8$ and $\theta = \frac{\pi}{4}$. Using these:

1. $x = 8 \cos\left(\frac{\pi}{4}\right)$

   • $\cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}$, so: $x = 8 \cdot \frac{\sqrt{2}}{2} = 4\sqrt{2}$

2. $y = 8 \sin\left(\frac{\pi}{4}\right)$

   • $\sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}$, so: $y = 8 \cdot \frac{\sqrt{2}}{2} = 4\sqrt{2}$

Final Answer:

$x = 4\sqrt{2}, \quad y = 4\sqrt{2}$

Would you like a detailed explanation of the formulas or their derivations? Here are 5 related questions to expand your knowledge:

1. What is the geometric interpretation of converting between polar and Cartesian coordinates?
2. How do sine and cosine values relate to angles in a unit circle?
3. What happens to the Cartesian coordinates if $r$ is negative in polar coordinates?
4. Can you derive the reverse formulas to convert Cartesian coordinates to polar coordinates?
5. Why are exact values like $\sqrt{2}$ used instead of decimal approximations in mathematics?

Tip: Always visualize polar and Cartesian coordinates on a graph to better understand the conversion process!