Identify the quadrant in which θ lies.
sin > 0 and tan < 0
sin positive, quad I and II
Tan negative, Quad II, IV
Quad II
To identify the quadrant in which θ lies based on the given information, we need to consider the signs of two trigonometric functions: sine (sin) and tangent (tan).
Given that sin > 0 and tan < 0, we can determine the quadrant as follows:
1. sin > 0 indicates that the angle θ is either in the first or the second quadrant, because sine is positive in these two quadrants.
2. tan < 0 indicates that the angle θ is in either the second or the fourth quadrant, because tangent is negative in these two quadrants.
Since both conditions indicate that θ lies in the second quadrant, we can conclude that the angle θ lies in the second quadrant.
To determine the quadrant in which θ lies, we need to analyze the signs of the sine and tangent functions.
Given that sin > 0 and tan < 0:
- sin > 0 means that the angle θ is in either the first or second quadrant. Both of these quadrants have positive values for sine.
- tan < 0 means that the angle θ is in the second or fourth quadrant. The tangent function is negative in the second and fourth quadrants.
Therefore, the given information indicates that the angle θ lies in the second quadrant.