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

Answer

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Answer

Final Answer: $\theta$ lies in the $\boxed{\text{third quadrant}}$.

Steps

Step 1 ：The cosecant function, $\csc \theta$, is negative in the third and fourth quadrants.

Step 2 ：The cosine function, $\cos \theta$, is negative in the second and third quadrants.

Step 3 ：Therefore, if the product of $\csc \theta$ and $\cos \theta$ is negative, then one of the functions must be positive and the other must be negative.

Step 4 ：This can only occur in the third quadrant, where $\csc \theta$ is negative and $\cos \theta$ is positive.

Step 5 ：Final Answer: $\theta$ lies in the $\boxed{\text{third quadrant}}$.

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

Answer

Popular Problems

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