On the unit circle, in which quadrant are both $\cos$ and $\cot$ negative?

• A. Quadrant $\mathrm{II}$
• B. Quadrant $\mathrm{I}$
• C. Quadrant $\mathrm{III}$
• D. Quadrant $\mathrm{IV}$

Answer: (C)

Verified step by step guidance

1. Recall the signs of the trigonometric functions in each quadrant of the unit circle: In Quadrant I, all functions are positive; in Quadrant II, sine is positive while cosine and tangent are negative; in Quadrant III, tangent is positive while sine and cosine are negative; and in Quadrant IV, cosine is positive while sine and tangent are negative.
2. Understand that cotangent is the reciprocal of tangent, so cotangent has the same sign as tangent in each quadrant.
3. Since cotangent and tangent share the same sign, identify the quadrants where cotangent is negative. From the sign chart, cotangent is negative in Quadrants II and IV.
4. Next, identify the quadrants where cosine is negative. Cosine is negative in Quadrants II and III.
5. Find the quadrant where both cosine and cotangent are negative by looking for the intersection of the two sets: cosine negative (II, III) and cotangent negative (II, IV). The common quadrant is Quadrant II.