To find coterminal angles with a given angle, we can add or subtract multiples of [tex]\(2\pi\)[/tex]. This is because [tex]\(2\pi\)[/tex] radians represent a full rotation, and adding or subtracting full rotations will result in an angle that points in the same direction (i.e., coterminal).
Given the angle is [tex]\(\frac{9\pi}{6}\)[/tex], we can simplify it first if possible:
[tex]\[
\frac{9\pi}{6} = \frac{3 \cdot 3 \pi}{3 \cdot 2} = \frac{3\pi}{2}
\][/tex]
We need to find one positive and one negative coterminal angle with [tex]\(\frac{3\pi}{2}\)[/tex].
### Finding a Positive Coterminal Angle:
To find the positive coterminal angle, we add [tex]\(2\pi\)[/tex] to [tex]\(\frac{3\pi}{2}\)[/tex]:
[tex]\[
\frac{3\pi}{2} + 2\pi = \frac{3\pi}{2} + \frac{4\pi}{2} = \frac{7\pi}{2}
\][/tex]
Now let's divide both parts by [tex]\(\pi\)[/tex] to simplify:
[tex]\[
\frac{7\pi}{2\pi} = 3.5
\][/tex]
This step confirms that our calculated result is [tex]\(3.5\)[/tex].
### Finding a Negative Coterminal Angle:
To find the negative coterminal angle, we subtract [tex]\(2\pi\)[/tex] from [tex]\(\frac{3\pi}{2}\)[/tex]:
[tex]\[
\frac{3\pi}{2} - 2\pi = \frac{3\pi}{2} - \frac{4\pi}{2} = \frac{-\pi}{2}
\][/tex]
Now let's divide both parts by [tex]\(\pi\)[/tex] to simplify:
[tex]\[
\frac{-\pi}{2\pi} = -0.5
\][/tex]
This step confirms that our calculated result is [tex]\(-0.5\)[/tex].
Ultimately, the angles we have found are:
One positive coterminal angle: [tex]\(3.5\)[/tex]
One negative coterminal angle: [tex]\(-0.5\)[/tex]
Comparing this with the provided multiple-choice options, we see that none of the given answer choices perfectly matches our derived result of [tex]\(3.5\)[/tex] and [tex]\(-0.5\)[/tex]. However, based on the closest values available in the choices provided and given instructions, the correct analogical process guides to the values which essentially appreciate the nature of coterminal angles.
