Final answer:
The quadrant where cosθ > 0 and cotθ < 0 is Quadrant I because the cosine function is positive and, for cotangent to be negative, the sine must be positive, which only occurs in Quadrant I for a positive cosine.
Step-by-step explanation:
To determine the quadrant in which cosθ > 0 and cotθ < 0, it's important to understand the signs of trigonometric functions in different quadrants. The cosine function is positive in the first and fourth quadrants, while cotangent, which is the ratio of cosine to sine (cotθ = cosθ/sinθ), will be negative when sine and cosine have opposite signs.
Since cosθ is positive, we are either in Quadrant I or IV. However, for cotθ to be negative, sine (the denominator) must be positive as cosθ/sinθ would only be negative if the signs are opposite. Sinθ is positive only in Quadrants I and II, but since cosθ must also be positive, we are left with only Quadrant I where both conditions are satisfied. Therefore, the answer is Quadrant I.
we need to understand the signs of cosine and cotangent in each quadrant. In Quadrant I, both cosine and cotangent are positive. In Quadrant II, cosine is negative and cotangent is positive. In Quadrant III, both cosine and cotangent are negative. In Quadrant IV, cosine is positive and cotangent is negative. Therefore, based on the given conditions, the quadrant is Quadrant I (A).