If theta is in between 0 and 180 is cos theta over 2 negative or positive

If \(\theta\) is in the range \(0^\circ < \theta < 180^\circ\), we need to determine whether \(\cos\left(\frac{\theta}{2}\right)\) is negative or positive.

### Step-by-Step Analysis:
1. **Understanding the range of \(\theta\):**
   - \(\theta\) is between \(0^\circ\) and \(180^\circ\).
   - This implies that \(\frac{\theta}{2}\) is between \(\frac{0^\circ}{2}\) and \(\frac{180^\circ}{2}\), i.e., \(0^\circ < \frac{\theta}{2} < 90^\circ\).

2. **Evaluating \(\cos\left(\frac{\theta}{2}\right)\):**
   - The cosine function \(\cos(x)\) is positive in the first quadrant, i.e., when \(0^\circ < x < 90^\circ\).

Since \(\frac{\theta}{2}\) is in the first quadrant (\(0^\circ < \frac{\theta}{2} < 90^\circ\)), \(\cos\left(\frac{\theta}{2}\right)\) will be positive.

### Conclusion:
\[
\cos\left(\frac{\theta}{2}\right) > 0
\]

Thus, \(\cos\left(\frac{\theta}{2}\right)\) is positive when \(\theta\) is between \(0^\circ\) and \(180^\circ\).

Would you like more details, or do you have any other questions?

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Here are 5 related questions:
1. What is the sign of \(\cos\left(\frac{\theta}{2}\right)\) if \(\theta\) is between \(180^\circ\) and \(360^\circ\)?
2. How does the sign of \(\sin\left(\frac{\theta}{2}\right)\) change in different quadrants?
3. What happens to \(\cos\left(\frac{\theta}{2}\right)\) if \(\theta\) is in the third quadrant?
4. How would the range of \(\theta\) affect \(\sin\left(\frac{\theta}{2}\right)\)?
5. How does the value of \(\cos(\theta)\) compare to \(\cos\left(\frac{\theta}{2}\right)\) for different ranges of \(\theta\)?

**Tip:** Remember that cosine is positive in the first quadrant (0° to 90°) and negative in the second quadrant (90° to 180°).

Explore the behavior of \( \cos\left(\frac{\theta}{2}\right) \) when \( \theta \) is between 0° and 180°. Learn how the cosine function behaves in the first quadrant and apply it to halved angles in trigonometric calculations. Suitable for Grades 9-12.

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