If tan A=`1/2` and tan B=`1/3` , then the value of A+B is

Options

A. `pi/6`
B. `pi`
C. zero
D. `pi/4`

Answer

To solve the problem where \( \tan A = \frac{1}{2} \) and \( \tan B = \frac{1}{3} \), we need to find the value of \( A + B \).

### Step-by-Step Solution:

1. **Use the formula for \( \tan(A + B) \)**:
The formula for \( \tan(A + B) \) is given by:
\[ \tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} \]

2. **Substitute the values of \( \tan A \) and \( \tan B \)**:
Given \( \tan A = \frac{1}{2} \) and \( \tan B = \frac{1}{3} \), we substitute these values into the formula:
\[ \tan(A + B) = \frac{\frac{1}{2} + \frac{1}{3}}{1 - \left(\frac{1}{2} \cdot \frac{1}{3}\right)} \]

3. **Calculate the numerator**:
To add \( \frac{1}{2} \) and \( \frac{1}{3} \), we need a common denominator. The least common multiple of 2 and 3 is 6:
\[ \frac{1}{2} = \frac{3}{6}, \quad \frac{1}{3} = \frac{2}{6} \]
Therefore, the numerator becomes:
\[ \frac{3}{6} + \frac{2}{6} = \frac{5}{6} \]

4. **Calculate the denominator**:
Now, we calculate \( 1 - \left(\frac{1}{2} \cdot \frac{1}{3}\right) \):
\[ \frac{1}{2} \cdot \frac{1}{3} = \frac{1}{6} \]
Thus, the denominator becomes:
\[ 1 - \frac{1}{6} = \frac{6}{6} - \frac{1}{6} = \frac{5}{6} \]

5. **Combine the results**:
Now substituting back into the formula for \( \tan(A + B) \):
\[ \tan(A + B) = \frac{\frac{5}{6}}{\frac{5}{6}} = 1 \]

6. **Find the angle**:
Since \( \tan(A + B) = 1 \), we know that:
\[ A + B = 45^\circ \]

7. **Convert degrees to radians**:
To convert degrees to radians, we use the conversion factor \( \frac{\pi}{180} \):
\[ A + B = 45^\circ \cdot \frac{\pi}{180} = \frac{\pi}{4} \]

### Final Answer:
Thus, the value of \( A + B \) is:
\[ \boxed{\frac{\pi}{4}} \]