Final answer:
The cotangent of x is the reciprocal of the tangent of x. Given tanx = ⅓ and cosx < 0, the value of cotx is 3, and x is in the third quadrant.
Step-by-step explanation:
Given that tanx = ⅓ (1/3) and cosx < 0, we need to find the value of cotx. It's important to note that cotangent is the reciprocal of tangent. Hence, if tanx = ⅓, cotx = ×3 (or 3). However, we also know that cosx < 0, implying that x is in either the second or third quadrant where cosine is negative. In these quadrants, tangent and cotangent would have the same sign because tangent is positive in the third quadrant and negative in the second, while cotangent follows the same pattern.
Now, since tanx is positive as given by ⅓, we must be in the third quadrant where both tanx and cotx are positive. In conclusion, if tanx = ⅓ and cosx < 0, then cotx = 3 and x is in the third quadrant.