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For what values of [tex]\theta \, (0 \leq \theta \leq 2 \pi)[/tex] do maximum [tex]r[/tex]-values occur on the graph of the polar equation [tex]r = 2 \sin 4 \theta[/tex]? Note that the maximum [tex]r[/tex]-value occurs at a point that is the maximum distance from the pole.

a. [tex]\frac{\pi}{8}, \frac{3 \pi}{8}, \frac{5 \pi}{8}, \frac{7 \pi}{8}, \frac{9 \pi}{8}, \frac{11 \pi}{8}, \frac{13 \pi}{8}, \frac{15 \pi}{8}[/tex]

b. [tex]0, \frac{\pi}{4}, \frac{\pi}{2}, \frac{3 \pi}{4}, \pi, \frac{5 \pi}{4}, \frac{3 \pi}{2}, \frac{7 \pi}{4}[/tex]

c. [tex]\frac{\pi}{8}, \frac{5 \pi}{8}, \frac{9 \pi}{8}, \frac{13 \pi}{8}[/tex]

d. [tex]0, \frac{\pi}{2}, \pi, \frac{3 \pi}{2}[/tex]

Please select the best answer from the choices provided:

A

B

C

D



Answer :

GinnyAnswer
To determine the values of [tex]\(\theta\)[/tex] that lead to the maximum [tex]\(r\)[/tex]-values for the given polar equation [tex]\( r = 2 \sin(4 \theta) \)[/tex], we need to analyze when [tex]\( r = 2 \)[/tex] since the sine function has a maximum value of 1. Specifically, we want to find when [tex]\( \sin(4 \theta) = 1 \)[/tex].

The equation [tex]\(\sin(4 \theta) = 1\)[/tex] holds true when:
[tex]\[ 4 \theta = \frac{\pi}{2} + 2 k \pi \][/tex]
where [tex]\(k\)[/tex] is any integer.

Solving for [tex]\(\theta\)[/tex], we have:
[tex]\[ \theta = \frac{\pi}{8} + \frac{k \pi}{2} \][/tex]

Since [tex]\(\theta\)[/tex] must be between [tex]\(0\)[/tex] and [tex]\(2\pi\)[/tex], we consider different values of [tex]\(k\)[/tex]:

1. For [tex]\( k = 0 \)[/tex]:
[tex]\[ \theta = \frac{\pi}{8} \][/tex]

2. For [tex]\( k = 1 \)[/tex]:
[tex]\[ \theta = \frac{\pi}{8} + \frac{\pi}{2} = \frac{\pi}{8} + \frac{4\pi}{8} = \frac{5\pi}{8} \][/tex]

3. For [tex]\( k = 2 \)[/tex]:
[tex]\[ \theta = \frac{\pi}{8} + \pi = \frac{\pi}{8} + \frac{8\pi}{8} = \frac{9\pi}{8} \][/tex]

4. For [tex]\( k = 3 \)[/tex]:
[tex]\[ \theta = \frac{\pi}{8} + \frac{3\pi}{2} = \frac{\pi}{8} + \frac{12\pi}{8} = \frac{13\pi}{8} \][/tex]

Thus, the values of [tex]\(\theta\)[/tex] that satisfy the conditions within the interval [tex]\([0, 2\pi]\)[/tex] are:
[tex]\[ \frac{\pi}{8}, \frac{5\pi}{8}, \frac{9\pi}{8}, \frac{13\pi}{8} \][/tex]

Among the provided choices, these values match with option C.

Therefore, the best answer is:
C

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