To find the exact value of [tex]\(\sin \left( \frac{-7\pi}{6} \right)\)[/tex], let's proceed step-by-step.
1. Understanding the Angle:
- First, let's convert the angle [tex]\(\frac{-7\pi}{6}\)[/tex] into a standard position within the range [tex]\([0, 2\pi)\)[/tex] to make it easier to analyze. Since this angle is negative, we need to add [tex]\(2\pi\)[/tex] to it until it lies within the standard range:
[tex]\[
\frac{-7\pi}{6} + 2\pi = \frac{-7\pi}{6} + \frac{12\pi}{6} = \frac{5\pi}{6}
\][/tex]
- So, [tex]\(\frac{-7\pi}{6}\)[/tex] is coterminal with [tex]\(\frac{5\pi}{6}\)[/tex].
2. Using the Identitiy for Negative Angles:
- The sine function has the property that [tex]\(\sin(-\theta) = -\sin(\theta)\)[/tex]. Hence, we can rewrite [tex]\(\sin \left( \frac{-7\pi}{6} \right)\)[/tex] as:
[tex]\[
\sin \left( \frac{-7\pi}{6} \right) = -\sin \left( \frac{7\pi}{6} \right)
\][/tex]
3. Simplifying [tex]\(\sin \left( \frac{7\pi}{6} \right)\)[/tex]:
- Notice that [tex]\(\frac{7\pi}{6}\)[/tex] is in the third quadrant, because [tex]\(\frac{7\pi}{6} = \pi + \frac{\pi}{6}\)[/tex], and the sine function is negative in the third quadrant.
- Also, in trigonometry, we know that:
[tex]\[
\sin \left( \pi + \theta \right) = -\sin(\theta)
\][/tex]
So,
[tex]\[
\sin \left( \frac{7\pi}{6} \right) = \sin \left( \pi + \frac{\pi}{6} \right) = -\sin \left( \frac{\pi}{6} \right)
\][/tex]
4. Finding [tex]\(\sin \left( \frac{\pi}{6} \right)\)[/tex]:
- From the unit circle or common trigonometric values, we know:
[tex]\[
\sin \left( \frac{\pi}{6} \right) = \frac{1}{2}
\][/tex]
- Therefore,
[tex]\[
\sin \left( \frac{7\pi}{6} \right) = -\frac{1}{2}
\][/tex]
5. Combining the Results:
- Recall from step 2 that [tex]\(\sin \left( \frac{-7\pi}{6} \right)\)[/tex] is the negative of [tex]\(\sin \left( \frac{7\pi}{6} \right)\)[/tex]:
[tex]\[
\sin \left( \frac{-7\pi}{6} \right) = -\left( -\frac{1}{2} \right) = \frac{1}{2}
\][/tex]
Hence, the exact value of [tex]\(\sin \left( \frac{-7\pi}{6} \right)\)[/tex] is:
[tex]\[
\boxed{\frac{1}{2}}
\][/tex]
