Question

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

Explanation:

Step1: Recall reciprocal - trigonometric identities

We know that $\csc\theta=\frac{1}{\sin\theta}$. Given $\csc\theta < 0$, then $\frac{1}{\sin\theta}<0$. Since the numerator 1 is positive, $\sin\theta<0$. In the unit - circle, $\sin\theta$ represents the $y$ - coordinate of the point on the unit - circle corresponding to the angle $\theta$. So, $\theta$ lies in either Quadrant III or Quadrant IV where $y<0$.

Step2: Recall the definition of cotangent

We know that $\cot\theta=\frac{\cos\theta}{\sin\theta}$. Given $\cot\theta > 0$, and we already know $\sin\theta<0$. For $\frac{\cos\theta}{\sin\theta}>0$, since $\sin\theta<0$, then $\cos\theta<0$ (because a negative divided by a negative is positive). In the unit - circle, $\cos\theta$ represents the $x$ - coordinate of the point on the unit - circle corresponding to the angle $\theta$. When $\cos\theta < 0$, $\theta$ lies in either Quadrant II or Quadrant III where $x<0$.

Step3: Determine the quadrant

Combining the two conditions: $\sin\theta<0$ and $\cos\theta<0$. The angle $\theta$ that satisfies both $\sin\theta < 0$ and $\cos\theta<0$ is in Quadrant III.

Answer:

III