If `theta in [(pi)/(2),3(pi)/(2)] then sin^(-1)(sin theta)` equals

Options

A. `theta`
B. `pi - theta`
C. `2pi - theta`
D. `-pi + theta`

Explanation

To solve the problem, we need to evaluate \( \sin^{-1}(\sin \theta) \) given that \( \theta \) is in the interval \( \left[\frac{\pi}{2}, \frac{3\pi}{2}\right] \).

### Step-by-Step Solution:

1. **Understanding the Interval**:
- The interval \( \left[\frac{\pi}{2}, \frac{3\pi}{2}\right] \) includes angles where the sine function takes values from 1 (at \( \frac{\pi}{2} \)) to -1 (at \( \frac{3\pi}{2} \)).
- In this interval, the sine function is negative for angles between \( \pi \) and \( \frac{3\pi}{2} \).

2. **Using the Property of Sine**:
- We know that \( \sin(\pi - \theta) = \sin \theta \). This property will help us relate \( \theta \) to an angle in the first or fourth quadrant where the sine function is positive.

3. **Finding the Equivalent Angle**:
- If we take \( -\theta \), we can find an equivalent angle in the range of \( \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \).
- For \( \theta \in \left[\frac{\pi}{2}, \frac{3\pi}{2}\right] \), we can express \( \theta \) as \( \theta = \pi + x \) where \( x \) is in the range \( \left[-\frac{\pi}{2}, 0\right] \).

4. **Applying the Inverse Sine Function**:
- Therefore, \( \sin^{-1}(\sin \theta) \) can be expressed as:
\[ \sin^{-1}(\sin \theta) = \sin^{-1}(\sin(\pi - \theta)) = \pi - \theta \]
- Since \( \theta \) is in the interval \( \left[\frac{\pi}{2}, \frac{3\pi}{2}\right] \), we can conclude:
\[ \sin^{-1}(\sin \theta) = \pi - \theta \]

5. **Final Result**:
- Thus, the final answer is:
\[ \sin^{-1}(\sin \theta) = \pi - \theta \]