If `sin theta = (5)/(13), 0 lt theta lt (pi)/(2)` then find the value of following.
`sin 3 theta`

Answer

To find the value of \( \sin 3\theta \) given that \( \sin \theta = \frac{5}{13} \) and \( 0 < \theta < \frac{\pi}{2} \), we can use the trigonometric identity for \( \sin 3\theta \):

\[ \sin 3\theta = 3\sin \theta - 4\sin^3 \theta \]

### Step-by-Step Solution:

1. **Identify the value of \( \sin \theta \)**:
\[ \sin \theta = \frac{5}{13} \]

2. **Calculate \( \sin^3 \theta \)**:
\[ \sin^3 \theta = \left( \frac{5}{13} \right)^3 = \frac{5^3}{13^3} = \frac{125}{2197} \]

3. **Substitute \( \sin \theta \) and \( \sin^3 \theta \) into the identity**:
\[ \sin 3\theta = 3\left(\frac{5}{13}\right) - 4\left(\frac{125}{2197}\right) \]

4. **Calculate \( 3\sin \theta \)**:
\[ 3\sin \theta = 3 \times \frac{5}{13} = \frac{15}{13} \]

5. **Calculate \( 4\sin^3 \theta \)**:
\[ 4\sin^3 \theta = 4 \times \frac{125}{2197} = \frac{500}{2197} \]

6. **Combine the two results**:
To combine \( \frac{15}{13} \) and \( \frac{500}{2197} \), we need a common denominator. The common denominator is \( 2197 \).

First, convert \( \frac{15}{13} \) to have a denominator of \( 2197 \):
\[ \frac{15}{13} = \frac{15 \times 169}{13 \times 169} = \frac{2535}{2197} \]

7. **Subtract the two fractions**:
\[ \sin 3\theta = \frac{2535}{2197} - \frac{500}{2197} = \frac{2035}{2197} \]

### Final Answer:
\[ \sin 3\theta = \frac{2035}{2197} \]