To solve this problem, let's break down the steps required to find the value of [tex]\(\tan(\theta)\)[/tex] given [tex]\(\cos(\theta) = -\frac{7}{8}\)[/tex] and knowing that [tex]\(\theta\)[/tex] is in quadrant III.
1. Understand the given values and quadrant properties:
- Given: [tex]\(\cos(\theta) = -\frac{7}{8}\)[/tex].
- Quadrant III implies both sine and cosine are negative.
2. Find [tex]\(\sin(\theta)\)[/tex]:
- Use the Pythagorean identity: [tex]\(\sin^2(\theta) + \cos^2(\theta) = 1\)[/tex].
3. Calculate [tex]\(\sin(\theta)\)[/tex]:
[tex]\[
\cos^2(\theta) = \left(-\frac{7}{8}\right)^2 = \left(\frac{7}{8}\right)^2 = \frac{49}{64}
\][/tex]
[tex]\[
\sin^2(\theta) = 1 - \cos^2(\theta) = 1 - \frac{49}{64} = \frac{64}{64} - \frac{49}{64} = \frac{15}{64}
\][/tex]
Since [tex]\(\theta\)[/tex] is in quadrant III, [tex]\(\sin(\theta)\)[/tex] must be negative:
[tex]\[
\sin(\theta) = -\sqrt{\frac{15}{64}} = -\frac{\sqrt{15}}{8}
\][/tex]
4. Calculate [tex]\(\tan(\theta)\)[/tex]:
[tex]\[
\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} = \frac{-\frac{\sqrt{15}}{8}}{-\frac{7}{8}} = \frac{\sqrt{15}}{7}
\][/tex]
Therefore, the correct value of [tex]\(\tan(\theta)\)[/tex] is:
[tex]\[
\boxed{\frac{\sqrt{15}}{7}}
\][/tex]
