if cot of theta is less than zero,
and csc of theta is less than zero what quadrant is theta in?
If the cot of theta is less than zero and the csc of theta is less than zero, we can determine the quadrant of theta as follows:
1. Recall that the cotangent of an angle is the ratio of the adjacent side to the opposite side in a right triangle. When the cot of theta is less than zero, it means that the adjacent side is negative, and the opposite side is positive.
2. The cosecant of an angle is the reciprocal of the sine of that angle. When the csc of theta is less than zero, the sine of theta is negative, which implies that the y-coordinate of the point representing theta is negative.
3. Combining these two conditions, we can conclude that theta lies in the third quadrant.
To determine the quadrant in which θ lies based on the given conditions, we need to recall the signs of trigonometric functions in each quadrant of the standard coordinate plane.
1. Cotangent (cot θ) is negative: Cot θ = cos θ / sin θ. Since cot θ is negative, both cos θ and sin θ must be of opposite signs. In the unit circle, this occurs in Quadrants II and IV.
2. Cosecant (csc θ) is negative: Csc θ = 1 / sin θ. As csc θ is negative, sin θ must also be negative. Sin θ is negative in Quadrants III and IV.
Combining the above information, we conclude that θ must lie in Quadrant IV, as it is the only quadrant where both conditions are satisfied.
COS/SIN IS NEGATIVE
1/SIN IS NEGATIVE
WELL SIN IS - AND THEREFORE COS IS +
SIN NEGATIVE AT BOTTOM
COS POSITIVE AT RIGHT
SO LOWER RIGHT, QUAD 3