Value(s) of `theta` satisfying the equation
`sin 7 theta = sin 4 theta - sin theta`, where `0 lt theta lt (pi//2)`, are

Options

A. `pi//9, pi//4`
B. `pi//3, pi//9`
C. `pi//6, pi//9`
D. `pi//3, pi//4`

Answer

To solve the equation \( \sin 7\theta = \sin 4\theta - \sin \theta \) for \( 0 < \theta < \frac{\pi}{2} \), we can follow these steps:

### Step 1: Rewrite the equation
We start with the given equation:
\[ \sin 7\theta = \sin 4\theta - \sin \theta \]
We can rearrange this to:
\[ \sin 7\theta + \sin \theta = \sin 4\theta \]

### Step 2: Use the sine addition formula
Using the sine addition formula, we know that:
\[ \sin A + \sin B = 2 \sin\left(\frac{A + B}{2}\right) \cos\left(\frac{A - B}{2}\right) \]
Let \( A = 7\theta \) and \( B = \theta \). Then:
\[ \sin 7\theta + \sin \theta = 2 \sin\left(\frac{7\theta + \theta}{2}\right) \cos\left(\frac{7\theta - \theta}{2}\right) = 2 \sin(4\theta) \cos(3\theta) \]
Thus, we can rewrite our equation as:
\[ 2 \sin(4\theta) \cos(3\theta) = \sin 4\theta \]

### Step 3: Factor the equation
We can factor out \( \sin 4\theta \):
\[ \sin 4\theta (2 \cos 3\theta - 1) = 0 \]

### Step 4: Solve the factors
This gives us two cases to consider:
1. \( \sin 4\theta = 0 \)
2. \( 2 \cos 3\theta - 1 = 0 \)

#### Case 1: \( \sin 4\theta = 0 \)
The solutions for \( \sin 4\theta = 0 \) are:
\[ 4\theta = n\pi \quad \text{for } n \in \mathbb{Z} \]
Thus,
\[ \theta = \frac{n\pi}{4} \]
Considering \( 0 < \theta < \frac{\pi}{2} \), we have:
- For \( n = 1 \): \( \theta = \frac{\pi}{4} \)

#### Case 2: \( 2 \cos 3\theta - 1 = 0 \)
Solving for \( \cos 3\theta \):
\[ \cos 3\theta = \frac{1}{2} \]
The solutions for \( \cos 3\theta = \frac{1}{2} \) are:
\[ 3\theta = \frac{\pi}{3} + 2k\pi \quad \text{or} \quad 3\theta = \frac{5\pi}{3} + 2k\pi \quad \text{for } k \in \mathbb{Z} \]
Thus,
\[ \theta = \frac{\pi}{9} + \frac{2k\pi}{3} \]
Considering \( 0 < \theta < \frac{\pi}{2} \):
- For \( k = 0 \): \( \theta = \frac{\pi}{9} \)

### Step 5: Collect the solutions
The values of \( \theta \) that satisfy the original equation in the interval \( 0 < \theta < \frac{\pi}{2} \) are:
\[ \theta = \frac{\pi}{4} \quad \text{and} \quad \theta = \frac{\pi}{9} \]

### Final Answer
The values of \( \theta \) satisfying the equation are:
\[ \theta = \frac{\pi}{4}, \frac{\pi}{9} \]