To determine in which quadrant $\theta$ lies given the conditions:

1. $\sin \theta > 0$
2. $\tan \theta < 0$

Analyzing the conditions:

1. $\sin \theta > 0$: This indicates that $\theta$ must be in either Quadrant I or Quadrant II, as sine is positive in these two quadrants.
2. $\tan \theta < 0$: Tangent is negative in Quadrants II and IV, since tangent is positive when both sine and cosine have the same sign, which only occurs in Quadrants I and III.

Conclusion:

Given both conditions, $\theta$ must be in Quadrant II, where $\sin \theta$ is positive, and $\tan \theta$ is negative.

Answer: Quadrant II

Would you like more details or have any questions about this solution?

Here are some related questions to expand on this concept:

1. What are the signs of $\sin \theta$, $\cos \theta$, and $\tan \theta$ in each quadrant?
2. How would the answer change if $\cos \theta > 0$ and $\sin \theta < 0$?
3. In which quadrant is $\theta$ if both $\sin \theta < 0$ and $\tan \theta > 0$?
4. Can you determine the quadrant if only $\sin \theta > 0$ is known?
5. How would you determine the quadrant if given only the sign of $\cos \theta$?

Tip: Remember that the CAST rule (Cosine, All, Sine, Tangent) can help quickly identify the sign of trigonometric functions in each quadrant.