To find the value of [tex]\(\tan \left(\sin ^{-1}\left(-\frac{1}{2}\right)\right)\)[/tex], we will proceed step-by-step:
1. Identify the angle [tex]\(\theta\)[/tex]:
- We are given [tex]\(\sin ^{-1}\left(-\frac{1}{2}\right)\)[/tex], which represents the angle [tex]\(\theta\)[/tex] for which [tex]\(\sin \theta = -\frac{1}{2}\)[/tex].
2. Determine [tex]\(\theta\)[/tex]:
- The sine function takes negative values in the third and fourth quadrants. However, [tex]\(\sin^{-1}\)[/tex] or [tex]\(\arcsin\)[/tex] function has a range of [tex]\([-\frac{\pi}{2}, \frac{\pi}{2}]\)[/tex], so we must consider angles in this range.
- The angle [tex]\(\theta\)[/tex] for which [tex]\(\sin \theta = -\frac{1}{2}\)[/tex] within this range is [tex]\(\theta = -\frac{\pi}{6}\)[/tex].
3. Find [tex]\(\tan \theta\)[/tex]:
- We now need to find [tex]\(\tan\left(-\frac{\pi}{6}\right)\)[/tex].
- Recall that [tex]\(\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}\)[/tex].
- For [tex]\(\theta = -\frac{\pi}{6}\)[/tex], [tex]\(\sin\left(-\frac{\pi}{6}\right) = -\frac{1}{2}\)[/tex].
- The corresponding cosine value is [tex]\(\cos\left(-\frac{\pi}{6}\right)\)[/tex], and since cosine is an even function, [tex]\(\cos\left(-\frac{\pi}{6}\right) = \cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2}\)[/tex].
4. Calculate [tex]\(\tan \left(-\frac{\pi}{6}\right)\)[/tex]:
- Using the values from above, we get:
[tex]\[
\tan \left(-\frac{\pi}{6}\right) = \frac{\sin\left(-\frac{\pi}{6}\right)}{\cos\left(-\frac{\pi}{6}\right)} = \frac{-\frac{1}{2}}{\frac{\sqrt{3}}{2}} = -\frac{1}{\sqrt{3}} = -\frac{\sqrt{3}}{3}.
\][/tex]
Thus, the value of [tex]\(\tan \left(\sin ^{-1}\left(-\frac{1}{2}\right)\right)\)[/tex] is [tex]\(-\frac{\sqrt{3}}{3}\)[/tex]. The correct answer is:
[tex]\[
\boxed{-\frac{\sqrt{3}}{3}}
\][/tex]
