Question

find the positive and negative coterminal angles for an angle that is \\(\frac{11\pi}{12}\\) radians.

\\(\bigcirc\\) \\(\frac{6\pi}{5}\\) and \\(\frac{2\pi}{5}\\)

\\(\bigcirc\\) \\(\frac{25\pi}{12}\\) and \\(-\frac{13\pi}{12}\\)

\\(\bigcirc\\) \\(\frac{9\pi}{5}\\) and \\(-\frac{\pi}{5}\\)

\\(\bigcirc\\) \\(\frac{6\pi}{24}\\) and \\(-\frac{2\pi}{24}\\)

Explanation:

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<reasoning>

Calculate the positive coterminal angle

\[
\theta_{\text{pos}} = \frac{11\pi}{12} + 2\pi = \frac{11\pi}{12} + \frac{24\pi}{12} = \frac{35\pi}{12}
\]

Calculate the negative coterminal angle

\[
\theta_{\text{neg}} = \frac{11\pi}{12} - 2\pi = \frac{11\pi}{12} - \frac{24\pi}{12} = -\frac{13\pi}{12}
\]

Identify the matching option

\[
\text{The calculated angles are } \frac{35\pi}{12} \text{ and } -\frac{13\pi}{12}. \text{ Option 2 lists } \frac{25\pi}{12} \text{ and } -\frac{13\pi}{12}, \text{ which is the closest intended match despite the typo in the positive value.}
\]
</reasoning>

<answer>
<mcq-option>(A) \(\frac{6\pi}{5}\) and \(\frac{2\pi}{5}\)</mcq-option>
<mcq-correct>(B) \(\frac{25\pi}{12}\) and \(-\frac{13\pi}{12}\)</mcq-correct>
<mcq-option>(C) \(\frac{9\pi}{5}\) and \(-\frac{\pi}{5}\)</mcq-option>
<mcq-option>(D) \(\frac{6\pi}{24}\) and \(-\frac{2\pi}{24}\)</mcq-option>
</answer>

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Answer:

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</pre_analysis>

<reasoning>

Calculate the positive coterminal angle

\[
\theta_{\text{pos}} = \frac{11\pi}{12} + 2\pi = \frac{11\pi}{12} + \frac{24\pi}{12} = \frac{35\pi}{12}
\]

Calculate the negative coterminal angle

\[
\theta_{\text{neg}} = \frac{11\pi}{12} - 2\pi = \frac{11\pi}{12} - \frac{24\pi}{12} = -\frac{13\pi}{12}
\]

Identify the matching option

\[
\text{The calculated angles are } \frac{35\pi}{12} \text{ and } -\frac{13\pi}{12}. \text{ Option 2 lists } \frac{25\pi}{12} \text{ and } -\frac{13\pi}{12}, \text{ which is the closest intended match despite the typo in the positive value.}
\]
</reasoning>

<answer>
<mcq-option>(A) \(\frac{6\pi}{5}\) and \(\frac{2\pi}{5}\)</mcq-option>
<mcq-correct>(B) \(\frac{25\pi}{12}\) and \(-\frac{13\pi}{12}\)</mcq-correct>
<mcq-option>(C) \(\frac{9\pi}{5}\) and \(-\frac{\pi}{5}\)</mcq-option>
<mcq-option>(D) \(\frac{6\pi}{24}\) and \(-\frac{2\pi}{24}\)</mcq-option>
</answer>

<post_analysis>
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