Alright, let's work through the problem to determine which expression is equivalent to [tex]\(\sin \frac{7 \pi}{6}\)[/tex].
We need to compare [tex]\(\sin \frac{7 \pi}{6}\)[/tex] with the given angles: [tex]\(\sin \frac{\pi}{6}\)[/tex], [tex]\(\sin \frac{5 \pi}{6}\)[/tex], [tex]\(\sin \frac{5 \pi}{3}\)[/tex], and [tex]\(\sin \frac{11 \pi}{6}\)[/tex].
Given these angles, we'll analyze whether any of them are equivalent to [tex]\(\sin \frac{7 \pi}{6}\)[/tex]:
1. Comparing with [tex]\(\sin \frac{\pi}{6}\)[/tex]:
[tex]\[
\sin \frac{\pi}{6} = \frac{1}{2}
\][/tex]
We need to check if [tex]\(\sin \frac{7 \pi}{6}\)[/tex] equals [tex]\(\frac{1}{2}\)[/tex].
2. Comparing with [tex]\(\sin \frac{5 \pi}{6}\)[/tex]:
[tex]\[
\sin \frac{5 \pi}{6} = \frac{1}{2}
\][/tex]
We need to check if [tex]\(\sin \frac{7 \pi}{6}\)[/tex] equals [tex]\(\frac{1}{2}\)[/tex].
3. Comparing with [tex]\(\sin \frac{5 \pi}{3}\)[/tex]:
[tex]\[
\sin \frac{5 \pi}{3} = -\frac{\sqrt{3}}{2}
\][/tex]
We need to check if [tex]\(\sin \frac{7 \pi}{6}\)[/tex] equals [tex]\(-\frac{\sqrt{3}}{2}\)[/tex].
4. Comparing with [tex]\(\sin \frac{11 \pi}{6}\)[/tex]:
[tex]\[
\sin \frac{11 \pi}{6} = -\frac{1}{2}
\][/tex]
We need to check if [tex]\(\sin \frac{7 \pi}{6}\)[/tex] equals [tex]\(-\frac{1}{2}\)[/tex].
Now we recognize the signs and values of these angles. From the investigation, we observe the results:
[tex]\[
\sin \frac{7 \pi}{6} \neq \sin \frac{\pi}{6}
\][/tex]
[tex]\[
\sin \frac{7 \pi}{6} \neq \sin \frac{5 \pi}{6}
\][/tex]
[tex]\[
\sin \frac{7 \pi}{6} \neq \sin \frac{5 \pi}{3}
\][/tex]
[tex]\[
\sin \frac{7 \pi}{6} \neq \sin \frac{11 \pi}{6}
\][/tex]
Thus, the [tex]\(\sin \frac{7 \pi}{6}\)[/tex] is not equivalent to any of the given expressions. None of the provided angles matches the value of [tex]\(\sin \frac{7 \pi}{6}\)[/tex].
Therefore, the result for the question "Which expression is equivalent to [tex]\(\sin \frac{7 \pi}{6}\)[/tex]?" is:
[tex]\[
\text{None of the above}
\][/tex]
