To find the period of the function $f(x) = \cos(8x)$, we use the formula for the period of cosine functions:

$\text{Period} = \frac{2\pi}{|b|}$

Here, $b$ is the coefficient of $x$ in the argument of the cosine function. In this case, $b = 8$.

$\text{Period} = \frac{2\pi}{8} = \frac{\pi}{4}$

Thus, the period of $f(x) = \cos(8x)$ is:

$\boxed{\frac{\pi}{4}}$

Let me know if you'd like further clarification or have any related questions! Here are some additional questions for exploration:

1. What is the amplitude of $f(x) = \cos(8x)$?
2. How would the period change if $f(x) = \cos(kx)$ where $k = 12$?
3. What is the phase shift of $f(x) = \cos(8x + \pi/3)$?
4. How does the graph of $f(x) = \cos(8x)$ compare to $f(x) = \cos(x)$?
5. What is the frequency of $f(x) = \cos(8x)$?

Tip: For trigonometric functions, always identify the amplitude, period, and phase shift to fully describe the function.