1. **Problem Statement:** Determine the quadrant in which angle $\theta$ lies given the inequalities: $$\sin(\theta) < 0$$ $$\sin(\theta) \cos(\theta) > 0$$ 2. **Recall the signs of sine and cosine in each quadrant:** - Quadrant I: $\sin(\theta) > 0$, $\cos(\theta) > 0$ - Quadrant II: $\sin(\theta) > 0$, $\cos(\theta) < 0$ - Quadrant III: $\sin(\theta) < 0$, $\cos(\theta) < 0$ - Quadrant IV: $\sin(\theta) < 0$, $\cos(\theta) > 0$ 3. **Analyze the first inequality:** $$\sin(\theta) < 0$$ This means $\theta$ must be in either Quadrant III or Quadrant IV. 4. **Analyze the second inequality:** $$\sin(\theta) \cos(\theta) > 0$$ Since the product is positive, both $\sin(\theta)$ and $\cos(\theta)$ must have the same sign. 5. **Combine both conditions:** - From step 3, $\sin(\theta) < 0$ (negative). - For the product to be positive, $\cos(\theta)$ must also be negative. 6. **Conclusion:** The only quadrant where both $\sin(\theta)$ and $\cos(\theta)$ are negative is Quadrant III. **Final answer:** $\boxed{\text{Quadrant III}}$