If `0 lt A lt pi//2` and `cos A =4//5`, find the value of sin 2A.

Answer

To solve the problem, we need to find the value of \( \sin 2A \) given that \( \cos A = \frac{4}{5} \) and \( 0 < A < \frac{\pi}{2} \).

### Step-by-Step Solution:

1. **Identify the relationship for \( \sin 2A \)**:
\[ \sin 2A = 2 \sin A \cos A \]

2. **Use the given value of \( \cos A \)**:
\[ \cos A = \frac{4}{5} \]

3. **Find \( \sin A \)**:
Since we know \( \cos A \), we can use the Pythagorean identity:
\[ \sin^2 A + \cos^2 A = 1 \]
Substituting the value of \( \cos A \):
\[ \sin^2 A + \left(\frac{4}{5}\right)^2 = 1 \]
\[ \sin^2 A + \frac{16}{25} = 1 \]
\[ \sin^2 A = 1 - \frac{16}{25} \]
\[ \sin^2 A = \frac{25}{25} - \frac{16}{25} = \frac{9}{25} \]
Taking the square root:
\[ \sin A = \sqrt{\frac{9}{25}} = \frac{3}{5} \]
(Since \( A \) is in the first quadrant, \( \sin A \) is positive.)

4. **Substitute \( \sin A \) and \( \cos A \) into the formula for \( \sin 2A \)**:
\[ \sin 2A = 2 \sin A \cos A = 2 \left(\frac{3}{5}\right) \left(\frac{4}{5}\right) \]

5. **Calculate \( \sin 2A \)**:
\[ \sin 2A = 2 \cdot \frac{3 \cdot 4}{5 \cdot 5} = \frac{24}{25} \]

### Final Answer:
\[ \sin 2A = \frac{24}{25} \]