The possible values of `theta in (0,pi)` such that `sin (theta) + sin (4theta) + sin(7theta) = 0` are (1) `(2pi)/9 , i/4 , (4pi)/9, pi/2, (3pi)/4 , (8pi)/9` (2) `pi/4, (5pi)/12, pi/2 , (2pi)/3, (3pi)/4, (8pi)/9` (3) `(2pi)/9, pi/4 , pi/2 , (2pi)/3 , (3pi)/4 , (35pi)/36` (4) `(2pi)/9, pi/4, pi/2 , (2pi)/3 , (3pi)/4 , (8pi)/9`

Options

A. `(2pi)/(9),(pi)/(4),(4pi)/(9),(pi)/(2),(3pi)/(4),(8pi)/(9)`
B. `(pi)/(4),(5pi)/(12),(pi)/(2),(2pi)/(3),(3pi)/(4),(8pi)/(9)`
C. `(2pi)/(9),(pi)/(4),(pi)/(2),(2pi)/(3),(3pi)/(4),(35pi)/(36)`
D. `(2pi)/(9),(pi)/(4),(pi)/(2),(2pi)/(3),(3pi)/(4),(8pi)/(9)`

Answer

To solve the equation \( \sin(\theta) + \sin(4\theta) + \sin(7\theta) = 0 \) for \( \theta \) in the interval \( (0, \pi) \), we can follow these steps:

### Step 1: Rearranging the Equation
We can rearrange the equation as follows:
\[ \sin(7\theta) + \sin(\theta) + \sin(4\theta) = 0 \]

### Step 2: Using the Sine Addition Formula
Using the sine addition formula, we can combine terms. We know that:
\[ \sin A + \sin B = 2 \sin\left(\frac{A+B}{2}\right) \cos\left(\frac{A-B}{2}\right) \]
We can apply this to \( \sin(7\theta) + \sin(\theta) \):
\[ \sin(7\theta) + \sin(\theta) = 2 \sin\left(\frac{7\theta + \theta}{2}\right) \cos\left(\frac{7\theta - \theta}{2}\right) = 2 \sin(4\theta) \cos(3\theta) \]
Thus, we rewrite the equation as:
\[ 2 \sin(4\theta) \cos(3\theta) + \sin(4\theta) = 0 \]

### Step 3: Factoring the Equation
Factoring out \( \sin(4\theta) \):
\[ \sin(4\theta)(2 \cos(3\theta) + 1) = 0 \]

### Step 4: Solving Each Factor
Now we solve each factor separately.

#### Factor 1: \( \sin(4\theta) = 0 \)
The sine function is zero at integer multiples of \( \pi \):
\[ 4\theta = n\pi \quad \text{for } n = 0, 1, 2, \ldots \]
This gives:
\[ \theta = \frac{n\pi}{4} \]
For \( n = 1, 2, 3, 4 \), we find:
- \( n = 1 \): \( \theta = \frac{\pi}{4} \)
- \( n = 2 \): \( \theta = \frac{\pi}{2} \)
- \( n = 3 \): \( \theta = \frac{3\pi}{4} \)
- \( n = 4 \): \( \theta = \pi \) (not in the interval \( (0, \pi) \))

#### Factor 2: \( 2 \cos(3\theta) + 1 = 0 \)
Solving for \( \cos(3\theta) \):
\[ \cos(3\theta) = -\frac{1}{2} \]
The cosine function is \( -\frac{1}{2} \) at:
\[ 3\theta = \frac{2\pi}{3} + 2k\pi \quad \text{and} \quad 3\theta = \frac{4\pi}{3} + 2k\pi \]
This gives:
\[ \theta = \frac{2\pi}{9} + \frac{2k\pi}{3} \quad \text{and} \quad \theta = \frac{4\pi}{9} + \frac{2k\pi}{3} \]
For \( k = 0 \):
- \( \theta = \frac{2\pi}{9} \)
- \( \theta = \frac{4\pi}{9} \)

For \( k = 1 \):
- \( \theta = \frac{2\pi}{9} + \frac{2\pi}{3} = \frac{2\pi}{9} + \frac{6\pi}{9} = \frac{8\pi}{9} \)

### Step 5: Collecting All Solutions
The solutions in the interval \( (0, \pi) \) are:
- From \( \sin(4\theta) = 0 \): \( \frac{\pi}{4}, \frac{\pi}{2}, \frac{3\pi}{4} \)
- From \( 2\cos(3\theta) + 1 = 0 \): \( \frac{2\pi}{9}, \frac{4\pi}{9}, \frac{8\pi}{9} \)

### Final Step: Listing All Values
Thus, the possible values of \( \theta \) in \( (0, \pi) \) are:
\[ \theta = \frac{2\pi}{9}, \frac{\pi}{4}, \frac{\pi}{2}, \frac{3\pi}{4}, \frac{4\pi}{9}, \frac{8\pi}{9} \]

### Conclusion
The correct option that matches these values is:
**Option 4: \( \frac{2\pi}{9}, \frac{\pi}{4}, \frac{\pi}{2}, \frac{2\pi}{3}, \frac{3\pi}{4}, \frac{8\pi}{9} \)**