In which quadrant does θ lie if the following statements are true:secant, theta, is greater than, 0, and , cosecant, theta, is less than, 0secθ>0 and cscθ<0

1. Break Down the Problem

To determine the quadrant in which the angle $\theta$ lies, we analyze the given trigonometric functions:

• $\sec \theta > 0$ indicates that the cosine function must be positive.
• $\csc \theta < 0$ indicates that the sine function must be negative.

2. Relevant Concepts

The secant function is the reciprocal of the cosine function: $\sec \theta = \frac{1}{\cos \theta}$ The cosecant function is the reciprocal of the sine function: $\csc \theta = \frac{1}{\sin \theta}$

3. Analysis and Detail

Considering the conditions:

• $\sec \theta > 0$ means $\cos \theta > 0$. This is true in the first quadrant (where $\sin$ and $\cos$ are both positive) and the fourth quadrant (where $\sin$ is negative and $\cos$ is positive).
• $\csc \theta < 0$ means $\sin \theta < 0$. This condition is satisfied in the third quadrant (where both $\sin$ and $\cos$ are negative) and fourth quadrant (where $\sin$ is negative and $\cos$ is positive).

4. Verify and Summarize

The angle $\theta$ must satisfy both conditions simultaneously:

• Fourth Quadrant: $\sec \theta > 0$ (positive cosine) and $\csc \theta < 0$ (negative sine).

Thus, the only quadrant that satisfies both conditions is the fourth quadrant.

Final Answer

Therefore, the angle $\theta$ lies in the fourth quadrant.