Final answer:
Cosine and cotangent are both negative in Quadrant II, where the x-coordinate (cosine) is negative and sine is positive, making the ratio of cosine to sine (cotangent) negative as well. Therefore, the answer to the question is c) Quadrant III.
Step-by-step explanation:
The question asks in which quadrant the cosine and cotangent functions are negative. Recalling the unit circle, we know that cosine represents the x-coordinate of a point on the unit circle and cotangent is the ratio of cosine to sine. In Quadrant II, the x-coordinate (cosine) is negative because we are to the left of the y-axis, and sine is positive because we are above the x-axis.
Thus, the cotangent, which depends on both sine and cosine (cotangent = cosine/sine), will also be negative when cosine is negative and sine is positive. Similarly, in Quadrant III, both cosine and sine are negative, but because cotangent is the ratio of these (cosine/sine), it will be positive. So, cosine and cotangent are both negative in Quadrant II (b).
n mathematics, the cosine function represents the ratio of the length of the adjacent side to the hypotenuse in a right triangle. The cotangent function represents the ratio of the length of the adjacent side to the opposite side in a right triangle.
In quadrant II, the cosine function is negative because the x-coordinates are negative. Similarly, in quadrant III, both the cosine and cotangent functions are negative because both the x-coordinates and y-coordinates are negative. Therefore, the answer to the question is c) Quadrant III.