Answer:
1. Quadrant IV
2. Quadrant IV
3. Quadrant III
Step-by-step explanation:
The quadrants in a Cartesian coordinate system are the four sections formed by the intersection of the x-axis and y-axis, numbered counterclockwise beginning with quadrant I in the top-right section, where both x and y are positive.
The unit circle is a circle centered at the origin (0, 0) with a radius of 1 unit. The x-coordinate of each point on the unit circle represents the cosine of the associated angle, and the y-coordinate represents the sine of the same angle. Angles are measured counterclockwise from the positive x-axis.
In quadrant I, cos θ is positive and sin θ is positive.
In quadrant II, cos θ is negative and sin θ is positive.
In quadrant III, cos θ is negative and sin θ is negative.
In quadrant IV, cos θ is positive and sin θ is negative.
sin θ < 0 implies that the y-coordinate is negative.
cos θ > 0 implies that the x-coordinate is positive.
This corresponds to quadrant IV.
sin θ < 0 implies that the y-coordinate is negative.
tan θ < 0 implies that the ratio of sin θ / cos θ is negative. Since sin θ < 0, it means that cos θ must be positive, so the x-coordinate is positive.
This corresponds to quadrant IV.
Since sec θ is the reciprocal of cos θ, then sec θ < 0 implies that cos θ < 0, so the x-coordinate is negative.
cot θ > 0 implies that the ratio of cos θ / sin θ is positive. Since cos θ < 0, it means that sin θ must also be negative, so the y-coordinate is negative.
This corresponds to quadrant III.
