In which quadrant does theta lie if the following statements are true: cot theta >...
Check the final answer first, then review the worked steps.
Problem
In which quadrant does theta lie if the following statements are true: cot theta > 0 and sec theta < 0
Answer
Quadrant III
Step-by-step solution
- Analyze the first condition: The problem states that $\cot \theta > 0$. The cotangent function is positive in Quadrant I (where all trigonometric functions are positive) and Quadrant III (where tangent and cotangent are positive). So, $\theta$ must be in Quadrant I or Quadrant III.
- Analyze the second condition: The problem states that $\sec \theta < 0$. The secant function is the reciprocal of the cosine function, so $\sec \theta$ has the same sign as $\cos \theta$. The cosine function is negative in Quadrant II and Quadrant III. So, $\theta$ must be in Quadrant II or Quadrant III.
- Determine the quadrant: To satisfy both conditions, $\theta$ must be in a quadrant that is common to both sets of possibilities. From step 1, $\theta$ is in Quadrant I or Quadrant III. From step 2, $\theta$ is in Quadrant II or Quadrant III. The only quadrant that satisfies both conditions is Quadrant III.