Question

in which quadrant does theta lie if the following statements are true tan theta<0 and cos theta >0

Answer 1

To determine the quadrant in which \(\theta\) lies given the conditions \( \tan \theta < 0 \) and \( \cos \theta > 0 \), we can analyze the signs of the trigonometric functions in each quadrant.

1. **Quadrant I**:
- \(\tan \theta > 0\) (sine and cosine are positive)
- \(\cos \theta > 0\)

2. **Quadrant II**:
- \(\tan \theta < 0\) (sine is positive, cosine is negative)
- \(\cos \theta < 0\)

3. **Quadrant III**:
- \(\tan \theta > 0\) (sine and cosine are negative)
- \(\cos \theta < 0\)

4. **Quadrant IV**:
- \(\tan \theta < 0\) (sine is negative, cosine is positive)
- \(\cos \theta > 0\)

Given:
- \(\tan \theta < 0\) means that \(\theta\) must be in either Quadrant II or Quadrant IV.
- \(\cos \theta > 0\) means that \(\theta\) cannot be in Quadrant II since cosine is negative there.

Therefore, the only quadrant that satisfies both conditions is **Quadrant IV**.