If `sin3A=1 and 0 le A le 90^(@)`, find :
`cos 2A`

Answer

To solve the problem, we need to find the value of \( \cos 2A \) given that \( \sin 3A = 1 \) and \( 0 \leq A \leq 90^\circ \).

### Step-by-Step Solution:

1. **Understand the Given Equation**:
We start with the equation:
\[ \sin 3A = 1 \]
The sine function equals 1 at specific angles.

2. **Identify the Angle**:
The sine function reaches its maximum value of 1 at:
\[ 3A = 90^\circ \]
This is because \( \sin 90^\circ = 1 \).

3. **Solve for A**:
To find \( A \), we divide both sides of the equation by 3:
\[ A = \frac{90^\circ}{3} = 30^\circ \]

4. **Find \( \cos 2A \)**:
Now we need to calculate \( \cos 2A \):
\[ 2A = 2 \times 30^\circ = 60^\circ \]
Therefore, we need to find \( \cos 60^\circ \).

5. **Use the Known Value**:
We know from trigonometric ratios that:
\[ \cos 60^\circ = \frac{1}{2} \]

6. **Final Answer**:
Thus, the value of \( \cos 2A \) is:
\[ \cos 2A = \frac{1}{2} \]

### Summary of the Solution:
- We found \( A = 30^\circ \) from \( \sin 3A = 1 \).
- We calculated \( \cos 2A \) as \( \cos 60^\circ \).
- The final answer is \( \frac{1}{2} \).