To address your question regarding how to find the points $x$ when the $y$-value is $\frac{1}{2}$ after sketching the graph of $f(x) = \cos x$ over the interval $[-2\pi, 2\pi]$, here are the steps:

1. Understand the Function and its Domain:

   • The function given is $f(x) = \cos x$.
   • The domain is restricted to $[-2\pi, 2\pi]$.
   • The cosine function is periodic with a period of $2\pi$, so we expect a repeating pattern within this interval.

2. Identify the Points Where $\cos x = \frac{1}{2}$:

   • We solve the equation $\cos x = \frac{1}{2}$.
   • The general solution for $\cos x = \frac{1}{2}$ in one period is $x = \pm \frac{\pi}{3} + 2k\pi$, where $k$ is any integer representing the periodic nature of the cosine function.

3. Find the Specific Solutions within the Interval:

   • Since the interval is $[-2\pi, 2\pi]$, we need to find the values of $x$ that lie in this interval.
   • Starting with the general solution:
     • For $k = 0$: $x = \frac{\pi}{3}, -\frac{\pi}{3}$
     • For $k = -1$: $x = \frac{\pi}{3} - 2\pi = -\frac{5\pi}{3}$
     • For $k = 1$: $x = \frac{\pi}{3} + 2\pi = \frac{7\pi}{3}$, but this value is outside the given interval, so it doesn't count.

   Therefore, the solutions are $x = -\frac{5\pi}{3}, -\frac{\pi}{3}, \frac{\pi}{3}, \frac{5\pi}{3}$.

4. Sketch the Graph:

   • Draw the cosine wave between $-2\pi$ and $2\pi$. The graph should have a peak at $y = 1$ at $x = 0$ and symmetric troughs at $y = -1$ at $x = \pm \pi$.
   • Mark the points where the graph crosses $y = \frac{1}{2}$ at the calculated $x$-coordinates: $-\frac{5\pi}{3}, -\frac{\pi}{3}, \frac{\pi}{3}, \frac{5\pi}{3}$.

By following these steps, you can find the points where $\cos x = \frac{1}{2}$ after sketching the graph.

Would you like further clarification on any step?

Here are 5 related questions:

1. What is the period of the cosine function, and how does it affect the graph?
2. How would the solutions change if the function were $\cos(2x)$ instead of $\cos x$?
3. What happens to the graph if we shift it by a phase, such as $f(x) = \cos(x - \frac{\pi}{4})$?
4. How does the amplitude affect the points where $y = \frac{1}{2}$?
5. How would you find the points where $\cos x = -\frac{1}{2}$?

Tip: For periodic functions like sine and cosine, understanding the general solution and the behavior of the function across its period is key to finding specific points.