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In which quadrant does
theta lie if the following statements are true:

tan theta < 0" and "sin theta < 0

In which quadrant does $\theta$ lie if the following statements are true: $\newline$ $\tan \theta<0 \text { and } \sin \theta<0$

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Full solution

Q. In which quadrant does

\theta

lie if the following statements are true:

\newline

\tan \theta<0 \text { and } \sin \theta<0

Identify Signs in Quadrants: Identify the signs of the trigonometric functions in each quadrant. $\newline$ In Quadrant $I$ , all trigonometric functions are positive. $\newline$ In Quadrant $II$ , $\sin$ is positive, but $\cos$ and $\tan$ are negative. $\newline$ In Quadrant $III$ , $\tan$ is positive, but $\sin$ and $\cos$ are negative. $\newline$ In Quadrant $IV$ , $\cos$ is positive, but $\sin$ and $\tan$ are negative.
Determine Quadrant Conditions: Determine the quadrant where both conditions are met. $\newline$ Since $\tan(\theta) < 0$ and $\sin(\theta) < 0$ , we are looking for a quadrant where sine is negative and tangent is also negative. $\newline$ From the previous step, we know that sine is negative in Quadrants $\text{III}$ and $\text{IV}$ , and tangent is negative in Quadrants $\text{II}$ and $\text{IV}$ .
Find Common Quadrant: Find the common quadrant where both conditions are satisfied. $\newline$ The only quadrant where both sine and tangent are negative is Quadrant $IV$ .

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