If `sin 2 A = 4/5,` find the value of `tan A, (0^(@) le A le (pi)/(3))`

Answer

To solve the problem, we need to find the value of \( \tan A \) given that \( \sin 2A = \frac{4}{5} \) and \( 0 \leq A \leq \frac{\pi}{3} \).

### Step-by-Step Solution:

1. **Use the double angle identity for sine:**
\[ \sin 2A = 2 \sin A \cos A \]
We also know that:
\[ \sin 2A = \frac{2 \tan A}{1 + \tan^2 A} \]
Setting these equal gives:
\[ \frac{2 \tan A}{1 + \tan^2 A} = \frac{4}{5} \]

2. **Cross-multiply to eliminate the fraction:**
\[ 2 \tan A \cdot 5 = 4(1 + \tan^2 A) \]
This simplifies to:
\[ 10 \tan A = 4 + 4 \tan^2 A \]

3. **Rearrange the equation:**
\[ 4 \tan^2 A - 10 \tan A + 4 = 0 \]

4. **This is a quadratic equation in terms of \( \tan A \). We can use the quadratic formula:**
\[ \tan A = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 4 \), \( b = -10 \), and \( c = 4 \).

5. **Calculate the discriminant:**
\[ b^2 - 4ac = (-10)^2 - 4 \cdot 4 \cdot 4 = 100 - 64 = 36 \]

6. **Substituting into the quadratic formula:**
\[ \tan A = \frac{10 \pm \sqrt{36}}{2 \cdot 4} \]
\[ \tan A = \frac{10 \pm 6}{8} \]

7. **Calculate the two possible values for \( \tan A \):**
- First value:
\[ \tan A = \frac{10 + 6}{8} = \frac{16}{8} = 2 \]
- Second value:
\[ \tan A = \frac{10 - 6}{8} = \frac{4}{8} = \frac{1}{2} \]

8. **Determine which value is valid given the constraint \( 0 \leq A \leq \frac{\pi}{3} \):**
- Since \( \tan A = 2 \) corresponds to an angle greater than \( \frac{\pi}{3} \), we discard this solution.
- Therefore, the valid solution is:
\[ \tan A = \frac{1}{2} \]

### Final Answer:
\[ \tan A = \frac{1}{2} \]