Questions: In which quadrant does θ lie if the following statements are true: sin θ<0 and (sin θ)(tan θ)>0

Transcript text: In which quadrant does $\theta$ lie if the following statements are true: \[ \sin \theta<0 \text { and }(\sin \theta)(\tan \theta)>0 \]

Solution

Solution Steps

To determine the quadrant in which $\theta$ lies, we need to analyze the given conditions: $\sin \theta < 0$ and $(\sin \theta)(\tan \theta) > 0$. The sine function is negative in the third and fourth quadrants. The tangent function is positive in the first and third quadrants. Therefore, $\theta$ must be in the third quadrant where both conditions are satisfied.

Step 1: Analyze the Conditions

We are given two conditions:

$ \sin \theta < 0 $
$ (\sin \theta)(\tan \theta) > 0 $

Step 2: Determine the Implications of Each Condition

The first condition, $ \sin \theta < 0 $, indicates that $\theta$ is in either the third quadrant or the fourth quadrant, since sine is negative in these quadrants.

The second condition, $ (\sin \theta)(\tan \theta) > 0 $, implies that both $\sin \theta$ and $\tan \theta$ must have the same sign. Since $\sin \theta < 0$, for the product to be positive, $\tan \theta$ must also be negative. The tangent function is negative in the second and fourth quadrants.

Step 3: Combine the Conditions

From the analysis:

The first condition restricts $\theta$ to the third or fourth quadrant.
The second condition restricts $\theta$ to the third quadrant (since $\tan \theta$ is negative in the fourth quadrant, but $\sin \theta$ is negative there as well).

Thus, the only quadrant that satisfies both conditions is the third quadrant.