if csc of theta is greater than zero
and sec of theta is less than zero
what quadrant is theta in?
if csc of theta is greater than zero
THAT MEANS SIN IS POSITIVE (ABOVE Y AXIS)
and sec of theta is less than zero
THAT MEANS COS IS NEGATIVE (LEFT OF Y AXIS)
SO IT IS IN THE UPPER LEFT, QUAD 2
To determine the quadrant in which theta lies, we need to understand the signs of the trigonometric functions in each quadrant. Let's consider the two given conditions:
1. csc(theta) > 0: The cosecant function is positive in the I and II quadrants. This means that theta could be in either quadrant I or quadrant II.
2. sec(theta) < 0: The secant function is negative in the II and III quadrants. This means that theta could be in either quadrant II or quadrant III.
Now we can determine the common quadrant that satisfies both conditions: theta lies in quadrant II. This is because the value of csc(theta) is positive in quadrant II, but sec(theta) is negative in quadrant II.
If the cosecant (csc) of theta is greater than zero, it means that the sine of theta is positive. And if the secant (sec) of theta is less than zero, it means that the cosine of theta is negative.
In which quadrant is theta located?
We know that in the first quadrant, both sine and cosine are positive.
In the second quadrant, sine is positive, but cosine is negative.
In the third quadrant, both sine and cosine are negative.
In the fourth quadrant, sine is negative, but cosine is positive.
Considering that the sine is positive (csc > 0) and the cosine is negative (sec < 0), we can conclude that theta is in the second quadrant.