The secant function, sec(t), is defined as the reciprocal of the cosine function, that is, sec(t) = 1/cos(t).
To determine whether sec(t) is positive or negative in each quadrant, we need to consider the sign of cos(t) in those quadrants:
- Quadrant I: In this quadrant, both sine and cosine are positive. Therefore, sec(t) is also positive because it is the reciprocal of a positive value (cos(t)).
- Quadrant II: Here, sine is positive, but cosine is negative. Since sec(t) is the reciprocal of cos(t), sec(t) is negative in this quadrant.
- Quadrant III: In this quadrant, both sine and cosine are negative. As a result, sec(t) is also negative, because the reciprocal of a negative number is still negative.
- Quadrant IV: In the fourth quadrant, sine is negative while cosine is positive. Thus, sec(t) is positive here, since it is the reciprocal of a positive value (cos(t)).
In summary, sec(t) is:
- Positive in Quadrant I
- Negative in Quadrant II
- Negative in Quadrant III
- Positive in Quadrant IV