Question

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

Explanation:

Step1: Recall tangent and secant properties

Tangent $\tan\theta=\frac{\sin\theta}{\cos\theta}$, and $\sec\theta = \frac{1}{\cos\theta}$.

Step2: Analyze $\tan\theta>0$

$\tan\theta>0$ when $\sin\theta$ and $\cos\theta$ have the same sign. This occurs in quadrants I and III.

Step3: Analyze $\sec\theta < 0$

Since $\sec\theta=\frac{1}{\cos\theta}<0$, then $\cos\theta<0$. This occurs in quadrants II and III.

Step4: Find the common quadrant

The common quadrant for $\tan\theta>0$ and $\sec\theta < 0$ is quadrant III.

Answer:

III