If ` cos A sin(A-(pi)/6)` is maximum , when the value of A is equal to `(pi)/(lambda )`, then the value of ` lambda ` is

Answer

To solve the problem, we need to find the value of \( \lambda \) such that \( \cos A \sin\left(A - \frac{\pi}{6}\right) \) is maximized when \( A = \frac{\pi}{\lambda} \).

### Step-by-Step Solution:

1. **Write the expression**:
We start with the expression given in the problem:
\[ y = \cos A \sin\left(A - \frac{\pi}{6}\right) \]

2. **Use the sine subtraction formula**:
We can expand \( \sin\left(A - \frac{\pi}{6}\right) \) using the sine subtraction formula:
\[ \sin\left(A - \frac{\pi}{6}\right) = \sin A \cos\left(\frac{\pi}{6}\right) - \cos A \sin\left(\frac{\pi}{6}\right) \]
Knowing that \( \cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2} \) and \( \sin\left(\frac{\pi}{6}\right) = \frac{1}{2} \), we substitute these values:
\[ \sin\left(A - \frac{\pi}{6}\right) = \sin A \cdot \frac{\sqrt{3}}{2} - \cos A \cdot \frac{1}{2} \]

3. **Substitute back into the expression**:
Now substituting this back into our expression for \( y \):
\[ y = \cos A \left( \sin A \cdot \frac{\sqrt{3}}{2} - \cos A \cdot \frac{1}{2} \right) \]
Simplifying this gives:
\[ y = \frac{\sqrt{3}}{2} \sin A \cos A - \frac{1}{2} \cos^2 A \]

4. **Use double angle identities**:
We know that \( 2 \sin A \cos A = \sin(2A) \) and \( \cos^2 A = \frac{1 + \cos(2A)}{2} \). Thus:
\[ y = \frac{\sqrt{3}}{4} \sin(2A) - \frac{1}{4}(1 + \cos(2A)) \]
This simplifies to:
\[ y = \frac{\sqrt{3}}{4} \sin(2A) - \frac{1}{4} - \frac{1}{4} \cos(2A) \]

5. **Combine terms**:
To combine the terms, we can express it as:
\[ y = \frac{1}{4} \left( \sqrt{3} \sin(2A) - \cos(2A) - 1 \right) \]

6. **Maximize the sine function**:
The maximum value of \( \sin(2A) \) is 1. For \( y \) to be maximum, we need:
\[ \sqrt{3} \sin(2A) - \cos(2A) = 1 \]
This occurs when:
\[ 2A - \frac{\pi}{6} = \frac{\pi}{2} \]
Solving for \( 2A \):
\[ 2A = \frac{\pi}{2} + \frac{\pi}{6} = \frac{3\pi}{6} + \frac{\pi}{6} = \frac{4\pi}{6} = \frac{2\pi}{3} \]
Thus:
\[ A = \frac{\pi}{3} \]

7. **Relate A to lambda**:
We know that \( A = \frac{\pi}{\lambda} \). Setting these equal gives:
\[ \frac{\pi}{\lambda} = \frac{\pi}{3} \]
Cancelling \( \pi \) from both sides, we find:
\[ \lambda = 3 \]

### Final Answer:
The value of \( \lambda \) is \( 3 \).