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We are given that:

\[
\sin(\theta) = \frac{8}{17}, \quad \sec(\theta) < 0
\]

We are tasked with finding \(\tan(\theta)\).

### Step 1: Determine the correct quadrant
Since \(\sec(\theta) = \frac{1}{\cos(\theta)}\) and we are told that \(\sec(\theta) < 0\), this implies \(\cos(\theta) < 0\). Therefore, \(\theta\) must be in either the second or third quadrant, as \(\cos(\theta)\) is negative in both of these quadrants.

However, \(\sin(\theta) = \frac{8}{17}\) is positive. The sine function is positive in the first and second quadrants. Therefore, the angle \(\theta\) must lie in the **second quadrant**.

### Step 2: Use the Pythagorean identity to find \(\cos(\theta)\)
We use the Pythagorean identity:

\[
\sin^2(\theta) + \cos^2(\theta) = 1
\]

Substituting \(\sin(\theta) = \frac{8}{17}\):

\[
\left(\frac{8}{17}\right)^2 + \cos^2(\theta) = 1
\]

\[
\frac{64}{289} + \cos^2(\theta) = 1
\]

\[
\cos^2(\theta) = 1 - \frac{64}{289} = \frac{289}{289} - \frac{64}{289} = \frac{225}{289}
\]

\[
\cos(\theta) = -\frac{15}{17}
\]

(The cosine is negative because \(\theta\) is in the second quadrant.)

### Step 3: Find \(\tan(\theta)\)
Now, recall that:

\[
\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}
\]

Substituting \(\sin(\theta) = \frac{8}{17}\) and \(\cos(\theta) = -\frac{15}{17}\):

\[
\tan(\theta) = \frac{\frac{8}{17}}{-\frac{15}{17}} = -\frac{8}{15}
\]

Thus, \(\tan(\theta) = -\frac{8}{15}\).

### Conclusion:
The correct answer is \(-\frac{8}{15}\), which corresponds to option 4 on the list.

Would you like further details or have any questions?

Here are 5 related questions to deepen understanding:
1. How do you determine the signs of trigonometric functions in each quadrant?
2. What is the relationship between secant and cosine?
3. How can you derive \(\tan(\theta)\) from \(\sin(\theta)\) and \(\cos(\theta)\)?
4. Why is \(\cos(\theta)\) negative in the second quadrant?
5. What other trigonometric identities can be useful when solving such problems?

**Tip:** Always remember the ASTC rule (All Students Take Calculus) to quickly determine the sign of trigonometric functions in different quadrants.

Given sin(θ) = 8/17 and sec(θ) < 0, find tan(θ).

Learn how to find tan(θ) given sin(θ) = 8/17 and sec(θ) < 0 using trigonometric identities and quadrant analysis. This step-by-step solution covers the use of the Pythagorean identity and the ASTC rule, suitable for high school students.

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