To find the sine value of the angle [tex]\(\frac{7\pi}{6}\)[/tex], we can break down the problem step-by-step.
### Step-by-Step Solution:
1. Identify the Angle in Radians:
The angle given is [tex]\(\frac{7\pi}{6}\)[/tex] radians.
2. Convert the Angle to Degrees (Optional, for Better Understanding):
Although we don't need to convert radians to degrees to solve this, it might help understand the reference angle:
[tex]\[
\frac{7\pi}{6} \times \frac{180^\circ}{\pi} = 210^\circ
\][/tex]
So, [tex]\(\frac{7\pi}{6}\)[/tex] radians is 210 degrees.
3. Determine the Reference Angle:
The reference angle for 210 degrees is:
[tex]\[
210^\circ - 180^\circ = 30^\circ \implies \text{Reference angle is } 30^\circ
\][/tex]
4. Sine Value of the Reference Angle:
We know from trigonometric values that [tex]\(\sin(30^\circ) = \frac{1}{2}\)[/tex].
5. Determine the Sign Based on the Quadrant:
The angle 210 degrees (or [tex]\(\frac{7\pi}{6}\)[/tex] radians) is in the third quadrant, where sine is negative.
6. Calculate the Sine Value:
Given the reference angle and that sine is negative in the third quadrant, the sine of [tex]\(\frac{7\pi}{6}\)[/tex] is:
[tex]\[
\sin\left(\frac{7\pi}{6}\right) = -\frac{1}{2}
\][/tex]
Therefore, the answer is:
[tex]\[
\sin\left(\frac{7\pi}{6}\right) = -\frac{1}{2}
\][/tex]
The correct option is:
[tex]\[
\boxed{D}
\][/tex]
