Final answer:
To find \(tan(\theta)\) given \(cos(\theta) = \frac{8}{9}\) in quadrant IV, we calculate the corresponding sine using the Pythagorean identity and then divide by the given cosine. The correct tangent value is \(-\frac{\sqrt{17}}{8}\), but this doesn't match any of the provided options.
Step-by-step explanation:
The student has asked, "If cos(\(\theta\)) = \(\frac{8}{9}\) and \(\theta\) is an angle in quadrant IV, what is the value of tan(\(\theta\))?" To solve this, we know that in the fourth quadrant, cosine is positive and tangent is negative. Since we have cos(\(\theta\)) = \(\frac{8}{9}\), we use the Pythagorean identity to find sine: sin(\(\theta\)) = -\sqrt{1 - cos^2(\(\theta\))}. Therefore, sin(\(\theta\)) = -\sqrt{1 - (\(\frac{8}{9}\))^2} which simplifies to sin(\(\theta\)) = -\frac{\sqrt{17}}{9}.
Now, to find the tangent, which is the ratio of sine to cosine, we calculate tan(\(\theta\)) = \frac{sin(\(\theta\))}{cos(\(\theta\))}. Thus, tan(\(\theta\)) = -\frac{\sqrt{17}}{9} \div \frac{8}{9} which simplifies to tan(\(\theta\)) = -\frac{\sqrt{17}}{8}.
Since -\(\frac{\sqrt{17}}{8}\) is not an option in the provided choices, it seems there may be a mistake in the question or the answer choices given. Therefore, I am unable to choose from options a, b, or c without more context or corrected answer choices.