determine the angle in which angle theta is located if cos theta < 0 and tan theta > 0

To determine the quadrant in which angle \(\theta\) is located given that \(\cos \theta < 0\) and \(\tan \theta > 0\), let's analyze the properties of the trigonometric functions in each quadrant:

### Properties of \(\cos \theta\) and \(\tan \theta\) in Different Quadrants:
1. **Quadrant I:** 
   - \(\cos \theta > 0\)
   - \(\tan \theta > 0\)
   
2. **Quadrant II:** 
   - \(\cos \theta < 0\)
   - \(\tan \theta < 0\)
   
3. **Quadrant III:** 
   - \(\cos \theta < 0\)
   - \(\tan \theta > 0\)
   
4. **Quadrant IV:** 
   - \(\cos \theta > 0\)
   - \(\tan \theta < 0\)

### Analyzing the Conditions:
- The condition \(\cos \theta < 0\) tells us that the angle \(\theta\) must be in either **Quadrant II** or **Quadrant III** because cosine is negative in these two quadrants.
- The condition \(\tan \theta > 0\) tells us that the angle \(\theta\) must be in either **Quadrant I** or **Quadrant III** because tangent is positive in these two quadrants.

### Conclusion:
The only quadrant that satisfies both conditions (\(\cos \theta < 0\) and \(\tan \theta > 0\)) is **Quadrant III**.

So, the angle \(\theta\) is located in **Quadrant III**.

Would you like any more details or further clarifications?

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Here are 5 related questions you might find useful:
1. What are the signs of sine and cosine in each quadrant?
2. How does the sign of the tangent function relate to the signs of sine and cosine?
3. How would you determine the quadrant if \(\sin \theta > 0\) and \(\tan \theta < 0\)?
4. What happens to the trigonometric function values if \(\theta\) is shifted by 180 degrees?
5. Can you determine the exact angle \(\theta\) if \(\cos \theta = -\frac{1}{2}\) and \(\sin \theta = -\frac{\sqrt{3}}{2}\)?

**Tip:** Remember that the tangent of an angle is positive where sine and cosine have the same sign, and negative where they have opposite signs.

Explore how to determine the quadrant of an angle \(\theta\) based on given conditions \(\cos \theta < 0\) and \(\tan \theta > 0\). Understand the properties of trigonometric functions in each quadrant and their implications. Suitable for high school students studying trigonometry.

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