To determine the values of [tex]\(\theta\)[/tex] at which the maximum [tex]\(r\)[/tex]-values occur for the polar equation [tex]\(r = 3 + 2 \sin \theta\)[/tex], let's consider the behavior of the function.
The given polar equation is [tex]\( r = 3 + 2 \sin \theta \)[/tex]. To find the maximum [tex]\(r\)[/tex]-value, we need to find when [tex]\( \sin \theta \)[/tex] is at its maximum value because the term [tex]\(2 \sin \theta\)[/tex] will maximize the value of [tex]\(r\)[/tex].
The sine function, [tex]\(\sin \theta\)[/tex], reaches its maximum value of 1 at:
[tex]\[ \theta = \frac{\pi}{2}, \frac{5\pi}{2}, \text{ and so on.} \][/tex]
However, we are only considering the range [tex]\(0 \leq \theta \leq 2\pi\)[/tex]. Within this interval, the specific value of [tex]\(\theta\)[/tex] at which [tex]\(\sin \theta\)[/tex] is maximum (i.e., 1) is:
[tex]\[ \theta = \frac{\pi}{2} \][/tex]
Thus, the maximum value of [tex]\(r\)[/tex] occurs when:
[tex]\[ r = 3 + 2 \cdot 1 = 5 \][/tex]
Therefore, the [tex]\(\theta\)[/tex] value that gives the maximum [tex]\(r\)[/tex]-value within the given range [tex]\([0, 2\pi]\)[/tex] is:
[tex]\[ \theta = \frac{\pi}{2} \][/tex]
Hence, the correct answer is:
C. [tex]\(\frac{\pi}{2}\)[/tex]
