Question

name the quadrant in which the angle \\( \theta \\) lies. \\( \sin \theta<0, \cot \theta>0 \\) choose the correct answer below. quadrant ii quadrant iii quadrant iv quadrant i

Explanation:

Step1: Analyze the sign of \(\sin\theta\)

Recall that \(\sin\theta=\frac{y}{r}\), where \(r = \sqrt{x^{2}+y^{2}}>0\). If \(\sin\theta<0\), then \(y < 0\). So the angle \(\theta\) lies in either Quadrant III or Quadrant IV.

Step2: Analyze the sign of \(\cot\theta\)

Recall that \(\cot\theta=\frac{\cos\theta}{\sin\theta}=\frac{x}{y}\). Since \(\cot\theta>0\), then \(x\) and \(y\) have the same sign. From Step 1, we know \(y < 0\), so \(x<0\) as well.

Step3: Determine the quadrant

When \(x < 0\) and \(y < 0\), the angle \(\theta\) lies in Quadrant III.

Answer:

Quadrant III