To determine which expression is equivalent to [tex]\(\sin \frac{7 \pi}{6}\)[/tex], let's explore each sine function provided:
1. [tex]\(\sin \frac{\pi}{6}\)[/tex]
2. [tex]\(\sin \frac{5 \pi}{6}\)[/tex]
3. [tex]\(\sin \frac{5 \pi}{3}\)[/tex]
4. [tex]\(\sin \frac{11 \pi}{6}\)[/tex]
Firstly, recall the value and properties of the sine function at key angles, specifically those related to [tex]\(\pi\)[/tex].
### Step-by-step Solution:
1. Evaluating [tex]\(\sin \frac{\pi}{6}\)[/tex]:
[tex]\[
\sin \frac{\pi}{6} = \frac{1}{2}
\][/tex]
2. Evaluating [tex]\(\sin \frac{5 \pi}{6}\)[/tex]:
Knowing that sine is positive in the second quadrant:
[tex]\[
\sin \frac{5 \pi}{6} = \sin \left(\pi - \frac{\pi}{6}\right) = \sin \frac{\pi}{6} = \frac{1}{2}
\][/tex]
3. Evaluating [tex]\(\sin \frac{5 \pi}{3}\)[/tex]:
To find the value of [tex]\(\sin \frac{5 \pi}{3}\)[/tex], note that:
[tex]\[
\frac{5 \pi}{3} = 2\pi - \frac{\pi}{3}
\][/tex]
Hence, it lies in the fourth quadrant where sine is negative:
[tex]\[
\sin \frac{5 \pi}{3} = \sin \left(2\pi - \frac{\pi}{3}\right) = -\sin \frac{\pi}{3} = -\frac{\sqrt{3}}{2}
\][/tex]
4. Evaluating [tex]\(\sin \frac{11 \pi}{6}\)[/tex]:
To find the value of [tex]\(\sin \frac{11 \pi}{6}\)[/tex], note that:
[tex]\[
\frac{11 \pi}{6} = 2\pi - \frac{\pi}{6}
\][/tex]
Hence, it also lies in the fourth quadrant:
[tex]\[
\sin \frac{11 \pi}{6} = \sin \left(2\pi - \frac{\pi}{6}\right) = -\sin \frac{\pi}{6} = -\frac{1}{2}
\][/tex]
Now, compare these values with [tex]\(\sin \frac{7 \pi}{6}\)[/tex].
Evaluating [tex]\(\sin \frac{7 \pi}{6}\)[/tex]:
[tex]\[
\frac{7 \pi}{6} = \pi + \frac{\pi}{6}
\][/tex]
Since this angle lies in the third quadrant where sine is negative:
[tex]\[
\sin \frac{7 \pi}{6} = -\sin \frac{\pi}{6} = -\frac{1}{2}
\][/tex]
From the above calculations, we see that:
- [tex]\(\sin \frac{7 \pi}{6} = -\frac{1}{2}\)[/tex]
- [tex]\(\sin \frac{11 \pi}{6} = -\frac{1}{2}\)[/tex]
Therefore, the expression equivalent to [tex]\(\sin \frac{7 \pi}{6}\)[/tex] is:
[tex]\[
\boxed{\sin \frac{11 \pi}{6}}
\][/tex]
So the answer to the question is the fourth option, [tex]\(\sin \frac{11 \pi}{6}\)[/tex].
