The question asks in which quadrant the terminal side of the angle $\theta$ lies, given the conditions:

$\sin(\theta) < 0 \quad \text{and} \quad \cos(\theta) < 0$

Explanation:

• $\sin(\theta)$ is negative in Quadrants III and IV.
• $\cos(\theta)$ is negative in Quadrants II and III.

Since both $\sin(\theta)$ and $\cos(\theta)$ are negative, the only quadrant where this happens is Quadrant III.

Answer:

The terminal side of $\theta$ lies in Quadrant III.

Would you like more details or have any other questions?

Here are 5 follow-up questions to extend your understanding:

1. What is the range of angles for each quadrant?
2. How can you determine which trigonometric functions are positive or negative in each quadrant?
3. How do the signs of $\sin(\theta)$ and $\cos(\theta)$ help determine the quadrant of an angle?
4. Can you provide an example of an angle in Quadrant III and its sine and cosine values?
5. What is the relationship between the tangent function and the sine/cosine in Quadrant III?

Tip: Remember the mnemonic "All Students Take Calculus" (ASTC) to recall which trigonometric functions are positive in each quadrant:

• All (Quadrant I: all functions are positive)
• Students (Quadrant II: sine is positive)
• Take (Quadrant III: tangent is positive)
• Calculus (Quadrant IV: cosine is positive).