If `sin^(3)theta+sin^(3)(theta+(2pi)/(3))+sin^(3)(theta+(4pi)/(3))=a sinb theta`. Find the value of `|(b)/(a)|`.

Answer

To solve the equation \( \sin^3 \theta + \sin^3 \left( \theta + \frac{2\pi}{3} \right) + \sin^3 \left( \theta + \frac{4\pi}{3} \right) = a \sin b \theta \), we will use the identity for \( \sin^3 \theta \).

### Step-by-step Solution:

1. **Use the identity for \( \sin^3 \theta \)**:
The identity for \( \sin^3 \theta \) is:
\[ \sin^3 \theta = \frac{3 \sin \theta - \sin 3\theta}{4} \]
We will apply this identity to each term in the equation.

2. **Apply the identity**:
We can rewrite each term:
\[ \sin^3 \theta = \frac{3 \sin \theta - \sin 3\theta}{4} \]
\[ \sin^3 \left( \theta + \frac{2\pi}{3} \right) = \frac{3 \sin \left( \theta + \frac{2\pi}{3} \right) - \sin \left( 3\theta + 2\pi \right)}{4} = \frac{3 \sin \left( \theta + \frac{2\pi}{3} \right) - \sin 3\theta}{4} \]
\[ \sin^3 \left( \theta + \frac{4\pi}{3} \right) = \frac{3 \sin \left( \theta + \frac{4\pi}{3} \right) - \sin \left( 3\theta + 4\pi \right)}{4} = \frac{3 \sin \left( \theta + \frac{4\pi}{3} \right) - \sin 3\theta}{4} \]

3. **Combine the terms**:
Now we can combine these terms:
\[ \sin^3 \theta + \sin^3 \left( \theta + \frac{2\pi}{3} \right) + \sin^3 \left( \theta + \frac{4\pi}{3} \right) = \frac{1}{4} \left( 3 \sin \theta + 3 \sin \left( \theta + \frac{2\pi}{3} \right) + 3 \sin \left( \theta + \frac{4\pi}{3} \right) - 3 \sin 3\theta \right) \]

4. **Evaluate the sine terms**:
The sum \( \sin \theta + \sin \left( \theta + \frac{2\pi}{3} \right) + \sin \left( \theta + \frac{4\pi}{3} \right) \) is known to equal 0. Thus, we have:
\[ 3 \left( \sin \theta + \sin \left( \theta + \frac{2\pi}{3} \right) + \sin \left( \theta + \frac{4\pi}{3} \right) \right) = 0 \]

5. **Final expression**:
Therefore, the equation simplifies to:
\[ \sin^3 \theta + \sin^3 \left( \theta + \frac{2\pi}{3} \right) + \sin^3 \left( \theta + \frac{4\pi}{3} \right) = -\frac{3}{4} \sin 3\theta \]
This means we can express it as:
\[ -\frac{3}{4} \sin 3\theta = a \sin b \theta \]
Here, we identify \( a = -\frac{3}{4} \) and \( b = 3 \).

6. **Calculate \( \left| \frac{b}{a} \right| \)**:
Now we need to find \( \left| \frac{b}{a} \right| \):
\[ \left| \frac{b}{a} \right| = \left| \frac{3}{-\frac{3}{4}} \right| = \left| -4 \right| = 4 \]

### Final Answer:
The value of \( \left| \frac{b}{a} \right| \) is \( 4 \).