Question

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

Explanation:

Step1: Recall sign - rules of trig functions

In the unit - circle, $\tan\theta=\frac{\sin\theta}{\cos\theta}$, $\cos\theta$ is the $x$ - coordinate of the point on the unit - circle corresponding to the angle $\theta$, and $\sin\theta$ is the $y$ - coordinate.

Step2: Analyze the condition $\tan\theta>0$

Since $\tan\theta = \frac{\sin\theta}{\cos\theta}>0$, $\sin\theta$ and $\cos\theta$ have the same sign. This occurs in Quadrant I ($\sin\theta>0,\cos\theta>0$) and Quadrant III ($\sin\theta<0,\cos\theta<0$).

Step3: Analyze the condition $\cos\theta < 0$

The cosine function is negative in Quadrant II ($x<0,y > 0$) and Quadrant III ($x<0,y<0$).

Step4: Find the common quadrant

The common quadrant that satisfies both $\tan\theta>0$ and $\cos\theta < 0$ is Quadrant III.

Answer:

A. III