Question

identify the quadrant(s) of an angle θ that satisfies the given conditions
tan θ > 0 and cot θ > 0

choose the correct answer below.

a. iii and iv
b. ii and iv
c. ii and iii
d. i and iii

Explanation:

Step1: Recall sign of tanθ

Tangent function \( \tan\theta=\frac{\sin\theta}{\cos\theta} \). \( \tan\theta>0 \) when \( \sin\theta \) and \( \cos\theta \) have the same sign (both positive or both negative). This occurs in Quadrant I (both \( \sin\theta, \cos\theta>0 \)) and Quadrant III (both \( \sin\theta, \cos\theta<0 \)).

Step2: Recall sign of cotθ

Cotangent function \( \cot\theta = \frac{\cos\theta}{\sin\theta} \). \( \cot\theta>0 \) when \( \cos\theta \) and \( \sin\theta \) have the same sign (both positive or both negative). This also occurs in Quadrant I (both \( \sin\theta, \cos\theta>0 \)) and Quadrant III (both \( \sin\theta, \cos\theta<0 \)).

Step3: Find common quadrants

The quadrants where both \( \tan\theta>0 \) and \( \cot\theta>0 \) are the quadrants common to both conditions, which are Quadrant I and Quadrant III.

Answer:

D. I and III