If ` tan (A-B) = (1)/(sqrt3) and sin A = (1)/(sqrt2)` find the value of B.

Answer

To solve the problem, we will follow these steps:

### Step-by-Step Solution:

1. **Given Information**: We have two pieces of information:
- \( \tan(A - B) = \frac{1}{\sqrt{3}} \)
- \( \sin A = \frac{1}{\sqrt{2}} \)

2. **Finding Angle A**:
- We know that \( \sin A = \frac{1}{\sqrt{2}} \).
- The angle whose sine is \( \frac{1}{\sqrt{2}} \) is \( 45^\circ \).
- Therefore, \( A = 45^\circ \).

3. **Finding Angle A - B**:
- From the given \( \tan(A - B) = \frac{1}{\sqrt{3}} \).
- The angle whose tangent is \( \frac{1}{\sqrt{3}} \) is \( 30^\circ \).
- Therefore, \( A - B = 30^\circ \).

4. **Substituting A into the Equation**:
- We have \( A = 45^\circ \) and \( A - B = 30^\circ \).
- Substitute \( A \) into the equation:
\[ 45^\circ - B = 30^\circ \]

5. **Solving for B**:
- Rearranging the equation gives:
\[ B = 45^\circ - 30^\circ \]
- Thus, \( B = 15^\circ \).

### Final Answer:
The value of \( B \) is \( 15^\circ \).

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