To solve [tex]\(\sin\left(\frac{7\pi}{6}\right)\)[/tex], let's follow a detailed step-by-step process:
1. Understanding the Angle:
- The angle [tex]\(\frac{7\pi}{6}\)[/tex] is in radians. We need to convert this to degrees to better understand its position on the unit circle:
[tex]\[
\frac{7\pi}{6} \times \frac{180^\circ}{\pi} = \frac{7 \times 180^\circ}{6} = 210^\circ
\][/tex]
So, [tex]\(\frac{7\pi}{6}\)[/tex] radians is equivalent to 210 degrees.
2. Unit Circle and Quadrants:
- The angle [tex]\(210^\circ\)[/tex] is in the third quadrant. In the unit circle, angles between [tex]\(180^\circ\)[/tex] and [tex]\(270^\circ\)[/tex] lie in the third quadrant.
- In the third quadrant, sine values are negative.
3. Reference Angle:
- To find the sine of [tex]\(210^\circ\)[/tex], we first determine the reference angle. The reference angle is how far the given angle is from the nearest x-axis:
[tex]\[
210^\circ - 180^\circ = 30^\circ
\][/tex]
Thus, the reference angle for [tex]\(210^\circ\)[/tex] is [tex]\(30^\circ\)[/tex].
4. Sine of the Reference Angle:
- We know the sine of [tex]\(30^\circ\)[/tex]:
[tex]\[
\sin(30^\circ) = \frac{1}{2}
\][/tex]
5. Sine in the Third Quadrant:
- Since [tex]\(210^\circ\)[/tex] is in the third quadrant, and sine values are negative in the third quadrant, we have:
[tex]\[
\sin(210^\circ) = -\sin(30^\circ) = -\frac{1}{2}
\][/tex]
6. Conclusion:
- Thus, [tex]\(\sin\left(\frac{7\pi}{6}\right) = \sin(210^\circ) = -\frac{1}{2}\)[/tex].
Hence, the solution to the question is:
[tex]\[
\boxed{-\frac{1}{2}}
\][/tex]
So, the correct answer is D. [tex]\(-\frac{1}{2}\)[/tex].
