To determine the values of [tex]\(\theta\)[/tex] for which the maximum [tex]\(r\)[/tex]-values occur for the polar equation [tex]\(r = 2 \cos 4\theta\)[/tex], we need to find the points where [tex]\(r\)[/tex] is maximized.
1. Polar Equation Analysis:
[tex]\[
r = 2 \cos 4\theta
\][/tex]
2. Maximum [tex]\(r\)[/tex]-values:
Since the maximum value of the cosine function [tex]\(\cos x\)[/tex] is 1, to find where [tex]\(r\)[/tex] is maximum, we need to find the values of [tex]\(\theta\)[/tex] for which [tex]\( \cos 4\theta = 1 \)[/tex].
3. Solving [tex]\( \cos 4\theta = 1 \)[/tex]:
The cosine function achieves a value of 1 at multiple angles which are integer multiples of [tex]\(2\pi\)[/tex]. Hence,
[tex]\[
4\theta = 2k\pi \quad (k \in \mathbb{Z})
\][/tex]
Dividing both sides by 4:
[tex]\[
\theta = \frac{k\pi}{2}
\][/tex]
Now, we need to find the values of [tex]\(\theta\)[/tex] within the interval [tex]\(0 \leq \theta \leq 2\pi\)[/tex]:
[tex]\[
\theta = 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}, 2\pi
\][/tex]
However, we discard [tex]\(2\pi\)[/tex] because it's equivalent to [tex]\(0\)[/tex] in this context and already accounted for. Therefore, the relevant [tex]\(\theta\)[/tex] values are:
[tex]\[
\theta = 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}
\][/tex]
4. Conclusion:
Therefore, the correct answer is:
[tex]\[
\boxed{A}
\][/tex]
This corresponds to the values where the maximum [tex]\(r\)[/tex]-values occur for the given polar equation.
