The problem in the image is asking:

At how many points in the interval $[-2\pi, 2\pi]$ is $\cos \theta = 1$?

Solution:

We know that $\cos \theta = 1$ at $\theta = 0$, and the cosine function repeats every $2\pi$ because it is periodic.

• The general solution for $\cos \theta = 1$ is $\theta = 2k\pi$, where $k$ is an integer.

In the interval $[-2\pi, 2\pi]$, we need to find how many values of $\theta$ satisfy $\cos \theta = 1$.

1. $\theta = 0$ (this is within the interval).
2. $\theta = 2\pi$ (also within the interval).
3. $\theta = -2\pi$ (also within the interval).

Thus, the points where $\cos \theta = 1$ are at $\theta = -2\pi, 0, 2\pi$.

Conclusion:

There are 3 points where $\cos \theta = 1$ in the interval $[-2\pi, 2\pi]$.

Would you like further details on this? Here are some related questions:

1. Why is the cosine function periodic, and how does that affect its graph?
2. What is the significance of the interval $[-2\pi, 2\pi]$ in trigonometric functions?
3. How would the answer change if the interval was $[0, 4\pi]$?
4. Can you explain how the general solution for $\cos \theta = 1$ is derived?
5. How would the result change if we were asked for $\cos \theta = -1$?

Tip: Cosine and sine functions are periodic, and understanding their periods can help solve many trigonometric equations.