For the angle \( \theta \) in standard position, determine the quadrant where the terminal side of \( \theta \) lies. \( \csc \theta>0 \) and \( \cot \theta>0 \) Quadrant:

To determine the quadrant where the terminal side of the angle \( \theta \) lies based on the conditions \( \csc \theta > 0 \) and \( \cot \theta > 0 \), we need to analyze the signs of the trigonometric functions involved. 1. **Understanding the conditions:** - The cosecant function \( \csc \theta \) is the reciprocal of the sine function: \[ \csc \theta = \frac{1}{\sin \theta} \] Therefore, \( \csc \theta > 0 \) implies that \( \sin \theta > 0 \). - The cotangent function \( \cot \theta \) is the ratio of the cosine function to the sine function: \[ \cot \theta = \frac{\cos \theta}{\sin \theta} \] Thus, \( \cot \theta > 0 \) implies that \( \cos \theta \) and \( \sin \theta \) must have the same sign. 2. **Analyzing the quadrants:** - **Quadrant I:** Both \( \sin \theta > 0 \) and \( \cos \theta > 0 \) (both are positive). - **Quadrant II:** \( \sin \theta > 0 \) and \( \cos \theta < 0 \) (sine is positive, cosine is negative). - **Quadrant III:** Both \( \sin \theta < 0 \) and \( \cos \theta < 0 \) (both are negative). - **Quadrant IV:** \( \sin \theta < 0 \) and \( \cos \theta > 0 \) (sine is negative, cosine is positive). 3. **Conclusion:** Since \( \sin \theta > 0 \) and \( \cot \theta > 0 \) (which means \( \cos \theta > 0 \)), the only quadrant where both conditions are satisfied is Quadrant I. Thus, the terminal side of \( \theta \) lies in **Quadrant I**.