Question

let θ be an angle in standard position. name the quadrant in which θ lies. tan θ < 0, csc θ > 0 the angle θ lies in which quadrant? iv i ii iii

Explanation:

Step1: Recall trig - function signs in quadrants

Tangent function $\tan\theta=\frac{\sin\theta}{\cos\theta}$. A negative tangent means $\sin\theta$ and $\cos\theta$ have opposite signs.

Step2: Recall the sign of cosecant

Cosecant function $\csc\theta=\frac{1}{\sin\theta}$. A positive $\csc\theta$ means $\sin\theta>0$.

Step3: Determine the quadrant

Since $\sin\theta > 0$, the angle is in either quadrant I or II. Since $\tan\theta<0$ and $\tan\theta=\frac{\sin\theta}{\cos\theta}$, and $\sin\theta>0$, then $\cos\theta < 0$. In quadrant II, $\sin\theta>0$ and $\cos\theta < 0$.

Answer:

C. II