If `sin theta = (5)/(13), 0 lt theta lt (pi)/(2)` then find the value of following.
`sin 4 theta`

Answer

To find the value of \( \sin 4\theta \) given that \( \sin \theta = \frac{5}{13} \) and \( 0 < \theta < \frac{\pi}{2} \), we can follow these steps:

### Step 1: Determine \( \cos \theta \)
Since \( \sin \theta = \frac{5}{13} \), we can use the Pythagorean identity to find \( \cos \theta \).

\[ \cos^2 \theta + \sin^2 \theta = 1 \]

Substituting the value of \( \sin \theta \):

\[ \cos^2 \theta + \left(\frac{5}{13}\right)^2 = 1 \]

Calculating \( \left(\frac{5}{13}\right)^2 \):

\[ \cos^2 \theta + \frac{25}{169} = 1 \]

Subtracting \( \frac{25}{169} \) from both sides:

\[ \cos^2 \theta = 1 - \frac{25}{169} = \frac{169 - 25}{169} = \frac{144}{169} \]

Taking the square root (since \( \theta \) is in the first quadrant, \( \cos \theta \) is positive):

\[ \cos \theta = \frac{12}{13} \]

### Step 2: Use the double angle formula to find \( \sin 2\theta \) and \( \cos 2\theta \)
Using the double angle formulas:

\[ \sin 2\theta = 2 \sin \theta \cos \theta \]
\[ \cos 2\theta = 2 \cos^2 \theta - 1 \]

Calculating \( \sin 2\theta \):

\[ \sin 2\theta = 2 \cdot \frac{5}{13} \cdot \frac{12}{13} = \frac{120}{169} \]

Calculating \( \cos 2\theta \):

\[ \cos 2\theta = 2 \cdot \left(\frac{12}{13}\right)^2 - 1 = 2 \cdot \frac{144}{169} - 1 = \frac{288}{169} - \frac{169}{169} = \frac{119}{169} \]

### Step 3: Use the double angle formula again to find \( \sin 4\theta \)
Now we can find \( \sin 4\theta \) using:

\[ \sin 4\theta = 2 \sin 2\theta \cos 2\theta \]

Substituting the values we found:

\[ \sin 4\theta = 2 \cdot \frac{120}{169} \cdot \frac{119}{169} \]

Calculating this:

\[ \sin 4\theta = \frac{240 \cdot 119}{169^2} = \frac{28560}{28561} \]

### Conclusion
Thus, the value of \( \sin 4\theta \) is:

\[ \sin 4\theta \approx 1 \]