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To solve this problem, we need to calculate \( \tan(2A) \) given that \( \tan(A) = \frac{2}{3} \) and \( A \) lies in the first quadrant. The formula for the tangent of a double angle is:

\[
\tan(2A) = \frac{2 \tan(A)}{1 - \tan^2(A)}
\]

### Step-by-Step Solution:

1. **Given**: \( \tan(A) = \frac{2}{3} \).
   
2. **Calculate \( \tan^2(A) \)**:
   \[
   \tan^2(A) = \left( \frac{2}{3} \right)^2 = \frac{4}{9}
   \]

3. **Apply the Double Angle Formula**:
   \[
   \tan(2A) = \frac{2 \cdot \frac{2}{3}}{1 - \frac{4}{9}}
   \]

4. **Simplify the Denominator**:
   - The denominator \( 1 - \frac{4}{9} \) becomes:
     \[
     1 - \frac{4}{9} = \frac{9 - 4}{9} = \frac{5}{9}
     \]

5. **Calculate \( \tan(2A) \)**:
   \[
   \tan(2A) = \frac{\frac{4}{3}}{\frac{5}{9}} = \frac{4}{3} \times \frac{9}{5} = \frac{36}{15} = \frac{12}{5}
   \]

### Conclusion:
The correct answer is:
\[
\boxed{\text{Option (a): } \tan(2A) = \frac{12}{5}}
\]

Would you like more details on any of these steps or have any questions?

### Related Questions:
1. What is the general formula for \( \tan(2A) \) if \( \tan(A) \) is known?
2. How does the quadrant in which an angle lies affect the sign of trigonometric functions?
3. How can we derive the double-angle identity for tangent?
4. What are the other double-angle identities for sine and cosine?
5. Can this problem be solved using a different trigonometric identity?

### Tip:
When working with double-angle identities, always check if the initial angle lies in a specific quadrant, as this can affect the sign of the resulting trigonometric value.

If tan A = 2/3 where A lies in the first quadrant, then which of the following options is true?

Learn how to find tan(2A) given that tan(A) = 2/3 in the first quadrant. This problem involves using the double-angle formula for tangent to determine the correct answer. Ideal for high school trigonometry students in Grades 10-12.

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