Find the quadrant in which θ lies if tan θ > 0 and csc θ < 0.

Answer 1

To determine the quadrant in which the angle \( \theta \) lies based on the given conditions \( \tan \theta > 0 \) and \( \csc \theta < 0 \), we analyze these conditions in relation to the quadrants of the unit circle.

1. Analyzing \( \tan \theta > 0 \):

   • The tangent function \( \tan \theta \) is positive in the first and third quadrants.
     • First Quadrant: \( \tan \theta > 0 \)
     • Third Quadrant: \( \tan \theta > 0 \)

2. Analyzing \( \csc \theta < 0 \):

   • The cosecant function \( \csc \theta \) is negative wherever the sine function \( \sin \theta \) is negative, which occurs in the third and fourth quadrants.
     • Third Quadrant: \( \csc \theta < 0 \) (since \( \sin \theta < 0 \))
     • Fourth Quadrant: \( \csc \theta < 0 \) (since \( \sin \theta < 0 \))

3. Finding the Intersection:

   • The only quadrant where both conditions are satisfied (i.e., where \( \tan \theta > 0 \) and \( \csc \theta < 0 \)) is the third quadrant.

Thus, we conclude that the angle \( \theta \) lies in the third quadrant.