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 i o iv o ii o iii

Explanation:

Step1: Recall the sign - rules of trig functions

In the first quadrant ($0\lt\theta\lt\frac{\pi}{2}$), $\tan\theta> 0$ and $\sec\theta=\frac{1}{\cos\theta}>0$. In the second quadrant ($\frac{\pi}{2}\lt\theta\lt\pi$), $\tan\theta < 0$ and $\sec\theta<0$ since $\cos\theta<0$. In the third quadrant ($\pi\lt\theta\lt\frac{3\pi}{2}$), $\tan\theta>0$ and $\sec\theta < 0$. In the fourth quadrant ($\frac{3\pi}{2}\lt\theta\lt2\pi$), $\tan\theta<0$ and $\sec\theta>0$.

Step2: Analyze the given conditions

We are given that $\tan\theta < 0$ and $\sec\theta<0$. The only quadrant where $\tan\theta < 0$ (tangent is negative) and $\sec\theta<0$ (cosine is negative) is the second quadrant.

Answer:

C. II