Let's find the measure of the reference angle for [tex]\(\theta = \frac{7\pi}{6}\)[/tex] and the value of [tex]\(\sin \theta\)[/tex].
1. Determine the reference angle:
The angle [tex]\(\theta = \frac{7\pi}{6}\)[/tex] is in the third quadrant because [tex]\(\frac{7\pi}{6}\)[/tex] is greater than [tex]\(\pi\)[/tex] and less than [tex]\(2\pi\)[/tex].
The reference angle for an angle in the third quadrant can be found by using the formula:
[tex]\[
\text{Reference Angle} = \pi - \left(\theta - \pi\right) = \frac{\pi}{6}
\][/tex]
Converting [tex]\(\frac{\pi}{6}\)[/tex] radians to degrees, we have:
[tex]\[
\frac{\pi}{6} \times \frac{180}{\pi} = 30^{\circ}
\][/tex]
2. Determine [tex]\(\sin \theta\)[/tex]:
Since [tex]\(\theta = \frac{7\pi}{6}\)[/tex] is in the third quadrant, the sine function is negative in this quadrant.
The corresponding angle in the first quadrant for [tex]\(\theta\)[/tex] is [tex]\(\frac{\pi}{6}\)[/tex], and [tex]\(\sin \left(\frac{\pi}{6}\right) = \frac{1}{2}\)[/tex]. Hence, for the third quadrant, we have:
[tex]\[
\sin \left(\frac{7\pi}{6}\right) = -\sin \left(\frac{\pi}{6}\right) = -\frac{1}{2}
\][/tex]
Therefore, the measure of the reference angle is [tex]\(30^{\circ}\)[/tex], and [tex]\(\sin \theta = -\frac{1}{2}\)[/tex].
Thus, the measure of its reference angle is [tex]\(30^{\circ}\)[/tex], and [tex]\(\sin \theta\)[/tex] is [tex]\(-\frac{1}{2}\)[/tex].
