Determine the quadrant in which $\csc \theta > 0$ and $\sec \theta < 0$.

1. Problem Description

Determine the quadrant in which $\csc \theta > 0$ and $\sec \theta < 0$.

2. Solution Steps

Recall that $\csc \theta = \frac{1}{\sin \theta}$ and $\sec \theta = \frac{1}{\cos \theta}$. Therefore, the conditions $\csc \theta > 0$ and $\sec \theta < 0$ are equivalent to $\sin \theta > 0$ and $\cos \theta < 0$.

We need to find the quadrant where $\sin \theta$ is positive and $\cos \theta$ is negative.

In Quadrant I, both $\sin \theta$ and $\cos \theta$ are positive.

In Quadrant II, $\sin \theta$ is positive and $\cos \theta$ is negative.

In Quadrant III, both $\sin \theta$ and $\cos \theta$ are negative.

In Quadrant IV, $\sin \theta$ is negative and $\cos \theta$ is positive.

Therefore, $\sin \theta > 0$ and $\cos \theta < 0$ in Quadrant II.

3. Final Answer

Quadrant 2.