Determine the quadrant for an angle with the following characteristics: cotangent theta greater than 0and cosecant theta greater than 0.

Question

(1 point)
Responses

Quadrant I
Quadrant I

Quadrant II
Quadrant II

Quadrant III
Quadrant III

Quadrant IV

Answer 1

To determine the quadrant for the angle θ based on the conditions given:

1. The cotangent of an angle (cot θ) is positive when both the sine (sin θ) and cosine (cos θ) are either both positive or both negative.
2. The cosecant of an angle (csc θ) is positive when the sine (sin θ) is positive.

Now, let's analyze the quadrants:

• Quadrant I: Both sine and cosine are positive. Therefore, cotangent is positive and cosecant is positive.
• Quadrant II: Sine is positive, but cosine is negative. Therefore, cotangent is negative (since it is sin/cos) and cosecant is positive.
• Quadrant III: Both sine and cosine are negative. Therefore, cotangent is positive and cosecant is negative.
• Quadrant IV: Sine is negative, but cosine is positive. Therefore, cotangent is negative and cosecant is negative.

Given the conditions that cotangent θ > 0 and cosecant θ > 0, the only quadrant where both these conditions are satisfied is:

Quadrant I.