sin − 1 ( sin θ ) > π 2 − sin − 1 ( sin θ ) find the range of θ

Options

A. `((pi)/(4), pi)`
B. `(pi, (3pi)/(2))`
C. `((pi)/(4), (3pi)/(4))`
D. `((3pi)/(4), 2pi)`

Explanation

To solve the inequality \( \sin^{-1}(\sin \theta) > \frac{\pi}{2} - \sin^{-1}(\sin \theta) \), we will follow these steps:

### Step 1: Rewrite the Inequality
We start with the given inequality:
\[ \sin^{-1}(\sin \theta) > \frac{\pi}{2} - \sin^{-1}(\sin \theta) \]
We can add \( \sin^{-1}(\sin \theta) \) to both sides:
\[ 2 \sin^{-1}(\sin \theta) > \frac{\pi}{2} \]

### Step 2: Divide by 2
Now, we divide both sides by 2:
\[ \sin^{-1}(\sin \theta) > \frac{\pi}{4} \]

### Step 3: Analyze the Function \( \sin^{-1}(\sin \theta) \)
The function \( \sin^{-1}(\sin \theta) \) gives us:
- \( \sin^{-1}(\sin \theta) = \theta \) when \( \theta \) is in the range \( \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \).
- For \( \theta \) in the range \( \left[\frac{\pi}{2}, \frac{3\pi}{2}\right] \), \( \sin^{-1}(\sin \theta) = \pi - \theta \).

### Step 4: Solve the Inequality
1. **Case 1:** If \( \theta \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \):
\[ \theta > \frac{\pi}{4} \]
Thus, \( \theta \) must be in the range \( \left(\frac{\pi}{4}, \frac{\pi}{2}\right] \).

2. **Case 2:** If \( \theta \in \left[\frac{\pi}{2}, \frac{3\pi}{2}\right] \):
\[ \pi - \theta > \frac{\pi}{4} \]
This simplifies to:
\[ \pi - \frac{\pi}{4} > \theta \implies \frac{3\pi}{4} > \theta \]
Thus, \( \theta \) must be in the range \( \left[\frac{\pi}{2}, \frac{3\pi}{4}\right) \).

### Step 5: Combine the Ranges
Combining both cases, we find the overall range of \( \theta \):
\[ \theta \in \left(\frac{\pi}{4}, \frac{3\pi}{4}\right) \]

### Final Answer
The range of \( \theta \) is:
\[ \left(\frac{\pi}{4}, \frac{3\pi}{4}\right) \]