Question

given the stated conditions, identify the quadrant in which θ lies. cotθ>0 and sinθ<0 the angle θ must be in quadrant select

Explanation:

Step1: Recall cotangent and sine signs by quadrant

Cotangent $\cot\theta=\frac{\cos\theta}{\sin\theta}$. $\cot\theta> 0$ when $\cos\theta$ and $\sin\theta$ have the same sign (both positive or both negative). $\sin\theta<0$ means the $y -$coordinate of the point on the unit - circle corresponding to the angle $\theta$ is negative.

Step2: Determine the quadrant

In Quadrant I: $\sin\theta>0,\cos\theta>0,\cot\theta>0$. In Quadrant II: $\sin\theta>0,\cos\theta<0,\cot\theta<0$. In Quadrant III: $\sin\theta<0,\cos\theta<0,\cot\theta>0$. In Quadrant IV: $\sin\theta<0,\cos\theta>0,\cot\theta<0$. Since $\cot\theta > 0$ and $\sin\theta<0$, the angle $\theta$ must have $\cos\theta<0$ (because $\cot\theta=\frac{\cos\theta}{\sin\theta}>0$ and $\sin\theta<0$). So, $\theta$ is in Quadrant III.

Answer:

III