We are given that:

$\cot(\theta) = -\frac{2}{8} = -\frac{1}{4}$

and

$\cos(\theta) < 0.$

To find $\sin(\theta)$, let's proceed step by step:

Step 1: Find $\tan(\theta)$

We know that:

$\cot(\theta) = \frac{1}{\tan(\theta)}$

So,

$\tan(\theta) = -4.$

Step 2: Use the identity involving $\sin(\theta)$ and $\cos(\theta)$

The identity for $\tan(\theta)$ is:

$\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}.$

Thus, we have:

$-4 = \frac{\sin(\theta)}{\cos(\theta)}.$

This gives us the relation:

$\sin(\theta) = -4 \cos(\theta).$

Step 3: Apply the Pythagorean identity

Using the fundamental identity:

$\sin^2(\theta) + \cos^2(\theta) = 1,$

we substitute $\sin(\theta) = -4 \cos(\theta)$ into this equation:

$(-4 \cos(\theta))^2 + \cos^2(\theta) = 1,$

which simplifies to:

$16 \cos^2(\theta) + \cos^2(\theta) = 1,$

$17 \cos^2(\theta) = 1,$

$\cos^2(\theta) = \frac{1}{17}.$

Step 4: Solve for $\cos(\theta)$

Since we are told that $\cos(\theta) < 0$, we take the negative square root:

$\cos(\theta) = -\frac{1}{\sqrt{17}}.$

Step 5: Find $\sin(\theta)$

Now, substitute $\cos(\theta)$ into the equation $\sin(\theta) = -4 \cos(\theta)$:

$\sin(\theta) = -4 \left(-\frac{1}{\sqrt{17}}\right),$

$\sin(\theta) = \frac{4}{\sqrt{17}}.$

Thus, the value of $\sin(\theta)$ is:

$\sin(\theta) = \frac{4}{\sqrt{17}}.$

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Would you like more details on any of these steps or have any questions? Here are some related questions for practice:

1. If $\sin(\theta) = \frac{4}{\sqrt{17}}$, find $\cos(\theta)$.
2. Given $\tan(\theta) = -4$, what is the value of $\sec(\theta)$?
3. If $\sin(\theta) = \frac{4}{\sqrt{17}}$, what is the value of $\csc(\theta)$?
4. Find the exact value of $\sin(2\theta)$ given $\cot(\theta) = -\frac{1}{4}$.
5. What quadrant is $\theta$ located in if $\cos(\theta) < 0$ and $\cot(\theta) < 0$?

Tip: Always determine the correct quadrant based on given information to ensure the correct sign of trigonometric functions.