Question

identify all possible quadrants of an angle θ that satisfies the given conditions.
cos θ > 0 and tan θ < 0

select all possible quadrants below.
a. quadrant iv
b. quadrant i
c. quadrant iii
d. quadrant ii

Explanation:

Step1: Recall cosine sign by quadrant

In Quadrant I: $\cos\theta>0$; Quadrant IV: $\cos\theta>0$; Quadrants II, III: $\cos\theta<0$. So $\cos\theta>0$ implies $\theta$ in I or IV.

Step2: Recall tangent sign by quadrant

$\tan\theta=\frac{\sin\theta}{\cos\theta}$. In Quadrant I: $\sin\theta>0,\cos\theta>0\Rightarrow\tan\theta>0$; Quadrant II: $\sin\theta>0,\cos\theta<0\Rightarrow\tan\theta<0$; Quadrant III: $\sin\theta<0,\cos\theta<0\Rightarrow\tan\theta>0$; Quadrant IV: $\sin\theta<0,\cos\theta>0\Rightarrow\tan\theta<0$. So $\tan\theta<0$ implies $\theta$ in II or IV.

Step3: Find intersection of quadrants

From $\cos\theta>0$ (I, IV) and $\tan\theta<0$ (II, IV), intersection is Quadrant IV.

Answer:

A. Quadrant IV