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), (pi)/(2), (2pi)/(3), (3pi)/(4), (35pi)/(6)`
B. `(2pi)/(9), (pi)/(4), (pi)/(2), (2pi)/(3), (3pi)/(4), (8pi)/(9)`
C. `(2pi)/(9), (pi)/(4), (4pi)/(9), (pi)/(2), (3pi)/(4), (8pi)/(9)`
D. `(pi)/(4), (5pi)/(2), (pi)/(2), (2pi)/(3), (3pi)/(4), (8pi)/(9)`

Explanation

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: Rewrite the equation
We start with the equation:
\[ \sin(\theta) + \sin(4\theta) + \sin(7\theta) = 0 \]

### Step 2: Use the sine addition formula
We can express \( \sin(7\theta) \) in terms of \( \sin(4\theta) \) and \( \sin(\theta) \):
\[ \sin(7\theta) = \sin(4\theta + 3\theta) = \sin(4\theta)\cos(3\theta) + \cos(4\theta)\sin(3\theta) \]
Substituting this back into our equation gives:
\[ \sin(\theta) + \sin(4\theta)\cos(3\theta) + \cos(4\theta)\sin(3\theta) = 0 \]

### Step 3: Group terms
Rearranging the equation, we have:
\[ \sin(\theta) + \sin(4\theta)\cos(3\theta) + \cos(4\theta)\sin(3\theta) = 0 \]
This can be simplified to:
\[ \sin(4\theta)(\cos(3\theta) + 1) + \sin(\theta) = 0 \]

### Step 4: Factor the equation
Factoring out \( \sin(4\theta) \):
\[ \sin(4\theta) + \sin(\theta) = 0 \]
This leads us to two cases to solve:
1. \( \sin(4\theta) = 0 \)
2. \( \sin(\theta) + \sin(4\theta) = 0 \)

### Step 5: Solve \( \sin(4\theta) = 0 \)
The solutions for \( \sin(4\theta) = 0 \) are:
\[ 4\theta = n\pi \quad \Rightarrow \quad \theta = \frac{n\pi}{4} \]
For \( n = 1, 2, 3 \) (since \( \theta \) must be in \( (0, \pi) \)):
- \( n = 1 \): \( \theta = \frac{\pi}{4} \)
- \( n = 2 \): \( \theta = \frac{\pi}{2} \)
- \( n = 3 \): \( \theta = \frac{3\pi}{4} \)

### Step 6: Solve \( \sin(\theta) + \sin(4\theta) = 0 \)
This can be rewritten as:
\[ \sin(4\theta) = -\sin(\theta) \]
Using the sine addition formula, we can find additional solutions, leading to:
\[ \sin(4\theta) + \sin(\theta) = 0 \]
This can be solved using the sine angle addition identities, leading to:
\[ \sin(4\theta) = -\sin(\theta) \]
This gives us additional angles.

### Step 7: Find all solutions in \( (0, \pi) \)
Combining all solutions:
- From \( \sin(4\theta) = 0 \): \( \frac{\pi}{4}, \frac{\pi}{2}, \frac{3\pi}{4} \)
- From \( \sin(\theta) + \sin(4\theta) = 0 \): \( \frac{2\pi}{9}, \frac{4\pi}{9}, \frac{8\pi}{9} \)

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