To solve this problem, we need to analyze the given conditions:

1. $\cot(\theta) > 0$: The cotangent of an angle is positive when both sine and cosine have the same sign. This occurs in Quadrants I and III.

2. $\sin(\theta) < 0$: The sine of an angle is negative in Quadrants III and IV.

Combining the conditions:

• $\cot(\theta) > 0$: Possible in Quadrants I and III.
• $\sin(\theta) < 0$: Possible in Quadrants III and IV.
• The overlap is Quadrant III.

Final Answer:

The angle $\theta$ lies in Quadrant III.

Let me know if you'd like a detailed explanation or have further questions! Here are some related questions for practice:

1. How do you determine the signs of trigonometric functions in each quadrant?
2. Why is $\cot(\theta) > 0$ when sine and cosine have the same sign?
3. How do you use the ASTC rule (All Students Take Calculus) to determine signs of trigonometric functions?
4. What are the values of other trigonometric functions in Quadrant III?
5. How can you find $\theta$ if $\tan(\theta) = -1$ and $\sin(\theta) < 0$?

Tip: Remember the ASTC rule to quickly determine the signs of trigonometric functions in each quadrant!