To determine the quadrant where $\theta$ is located, let’s analyze the given information step by step:

1. Given information:

   • $\sec \theta = \sqrt{6}$ (positive value).
   • $\csc \theta < 0$ (negative value).

2. Recap of quadrants and signs:

   • In Quadrant I, all trig functions are positive.
   • In Quadrant II, only sine and cosecant ($\sin \theta$, $\csc \theta$) are positive.
   • In Quadrant III, only tangent and cotangent ($\tan \theta$, $\cot \theta$) are positive.
   • In Quadrant IV, only cosine and secant ($\cos \theta$, $\sec \theta$) are positive.

3. Step 1: Analyze $\sec \theta$ being positive:

   • $\sec \theta$ is the reciprocal of $\cos \theta$, so $\sec \theta > 0$ means $\cos \theta > 0$.
   • $\cos \theta > 0$ occurs in Quadrant I and Quadrant IV.

4. Step 2: Analyze $\csc \theta < 0$:

   • $\csc \theta$ is the reciprocal of $\sin \theta$, so $\csc \theta < 0$ means $\sin \theta < 0$.
   • $\sin \theta < 0$ occurs in Quadrant III and Quadrant IV.

5. Combine the information:

   • $\sec \theta > 0$: Quadrants I and IV.
   • $\csc \theta < 0$: Quadrants III and IV.
   • The only quadrant that satisfies both conditions is Quadrant IV.

Final Answer:

$\theta$ is in Quadrant IV.

Let me know if you'd like a detailed explanation about the reference triangle or solving the rest of the problem!

5 Related Questions:

1. What are the steps to find the reference angle in Quadrant IV?
2. How is $\sec \theta = \sqrt{6}$ used to find $\cos \theta$?
3. Why is $\csc \theta$ negative in Quadrant IV?
4. Can you explain how to calculate all six trigonometric functions for $\theta$?
5. What are the formulas to convert reference angles into actual angles in Quadrants II, III, and IV?

Tip: Always remember that the signs of trig functions depend on both their definitions and the quadrant the angle is in!