If `theta` lies in second quadrant, then the quadrant in which `-theta/(2)` lies is

Options

A. I quadrant
B. II quadrant
C. III quadrant
D. IV quadrant

Answer

To determine the quadrant in which \(-\frac{\theta}{2}\) lies, given that \(\theta\) is in the second quadrant, we can follow these steps:

### Step 1: Understand the range of \(\theta\)
Since \(\theta\) lies in the second quadrant, we know that:
\[ \frac{\pi}{2} < \theta < \pi \]

### Step 2: Calculate \(-\frac{\theta}{2}\)
To find \(-\frac{\theta}{2}\), we will first find the bounds for this expression using the range of \(\theta\).

1. **Find the lower bound**:
\[ \text{If } \theta = \frac{\pi}{2}, \text{ then } -\frac{\theta}{2} = -\frac{\pi}{4} \]

2. **Find the upper bound**:
\[ \text{If } \theta = \pi, \text{ then } -\frac{\theta}{2} = -\frac{\pi}{2} \]

### Step 3: Determine the range of \(-\frac{\theta}{2}\)
From the calculations above, we can conclude that:
\[ -\frac{\pi}{4} < -\frac{\theta}{2} < -\frac{\pi}{2} \]

### Step 4: Identify the quadrant
Now we need to determine in which quadrant the angle \(-\frac{\theta}{2}\) lies.

- The angle \(-\frac{\pi}{4}\) is in the fourth quadrant.
- The angle \(-\frac{\pi}{2}\) is exactly on the negative y-axis, which is also considered part of the fourth quadrant.

Since \(-\frac{\theta}{2}\) lies between \(-\frac{\pi}{4}\) and \(-\frac{\pi}{2}\), it is in the fourth quadrant.

### Conclusion
Thus, the quadrant in which \(-\frac{\theta}{2}\) lies is the **fourth quadrant**.

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