To determine which expression is equivalent to [tex]\(\sin \frac{7 \pi}{6}\)[/tex], let's break down the problem step-by-step.
First, we need to understand the value of [tex]\(\sin \frac{7 \pi}{6}\)[/tex].
The angle [tex]\(\frac{7 \pi}{6}\)[/tex] is in the third quadrant of the unit circle. In the third quadrant, the sine function is negative.
The equivalent reference angle for [tex]\(\frac{7 \pi}{6}\)[/tex] is:
[tex]\[
\pi + \frac{\pi}{6} = \frac{7 \pi}{6}
\][/tex]
Since [tex]\(\pi\)[/tex] corresponds to 180 degrees, [tex]\(\frac{7 \pi}{6} = 180^\circ + 30^\circ = 210^\circ\)[/tex].
The sine of 210 degrees (or [tex]\(\frac{7 \pi}{6}\)[/tex]) is:
[tex]\[
\sin(210^\circ) = -\sin(30^\circ)
\][/tex]
Since [tex]\(\sin(30^\circ) = \sin \frac{\pi}{6} = 0.5\)[/tex], we get:
[tex]\[
\sin \frac{7 \pi}{6} = -0.5
\][/tex]
Now, let's evaluate each given option for a match:
- [tex]\(\sin \frac{\pi}{6} = 0.5\)[/tex]
- [tex]\(\sin \frac{5 \pi}{6} = 0.5\)[/tex], since [tex]\(\frac{5 \pi}{6}\)[/tex] is in the second quadrant where sine is positive.
- [tex]\(\sin \frac{5 \pi}{3} = -\sin(\frac{2 \pi}{3}) = -\sin(60^\circ) = -\frac{\sqrt{3}}{2}\)[/tex], which is approximately -0.866.
- [tex]\(\sin \frac{11 \pi}{6} = -\sin(\frac{\pi}{6}) = -0.5\)[/tex], since [tex]\(\frac{11 \pi}{6}\)[/tex] is in the fourth quadrant where sine is negative.
Comparing these values with [tex]\(\sin \frac{7 \pi}{6} = -0.5\)[/tex]:
- [tex]\(\sin \frac{\pi}{6} \neq \sin \frac{7 \pi}{6}\)[/tex]
- [tex]\(\sin \frac{5 \pi}{6} \neq \sin \frac{7 \pi}{6}\)[/tex]
- [tex]\(\sin \frac{5 \pi}{3} \neq \sin \frac{7 \pi}{6}\)[/tex]
- [tex]\(\sin \frac{11 \pi}{6} = \sin \frac{7 \pi}{6}\)[/tex]
Thus, the equivalent expression to [tex]\(\sin \frac{7 \pi}{6}\)[/tex] is [tex]\(\sin \frac{11 \pi}{6}\)[/tex].
So, the correct answer is:
[tex]\[
\boxed{\sin \frac{11 \pi}{6}}
\][/tex]
