If `tanA=(1)/(3)andtanB=(2)/(5)` , what is the value of tan (2A+B) ?

Options

A. `(8)/(15)`
B. `(6)/(13)`
C. `(37)/(115)`
D. `(23)/(14)`

Answer

To find the value of \( \tan(2A + B) \) given that \( \tan A = \frac{1}{3} \) and \( \tan B = \frac{2}{5} \), we can use the formula for the tangent of a sum:

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

### Step 1: Calculate \( \tan(2A) \)

Using the double angle formula for tangent:

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

Substituting \( \tan A = \frac{1}{3} \):

\[ \tan(2A) = \frac{2 \cdot \frac{1}{3}}{1 - \left(\frac{1}{3}\right)^2} \]

Calculating \( \tan(2A) \):

\[ = \frac{\frac{2}{3}}{1 - \frac{1}{9}} = \frac{\frac{2}{3}}{\frac{8}{9}} = \frac{2}{3} \cdot \frac{9}{8} = \frac{6}{8} = \frac{3}{4} \]

### Step 2: Substitute \( \tan(2A) \) and \( \tan B \) into the formula

Now we substitute \( \tan(2A) = \frac{3}{4} \) and \( \tan B = \frac{2}{5} \) into the formula for \( \tan(2A + B) \):

\[ \tan(2A + B) = \frac{\frac{3}{4} + \frac{2}{5}}{1 - \left(\frac{3}{4}\right) \left(\frac{2}{5}\right)} \]

### Step 3: Calculate the numerator

Finding a common denominator for the numerator:

\[ \frac{3}{4} + \frac{2}{5} = \frac{15}{20} + \frac{8}{20} = \frac{23}{20} \]

### Step 4: Calculate the denominator

Calculating the denominator:

\[ 1 - \left(\frac{3}{4} \cdot \frac{2}{5}\right) = 1 - \frac{6}{20} = 1 - \frac{3}{10} = \frac{10}{10} - \frac{3}{10} = \frac{7}{10} \]

### Step 5: Combine the results

Now we can substitute back into the formula:

\[ \tan(2A + B) = \frac{\frac{23}{20}}{\frac{7}{10}} = \frac{23}{20} \cdot \frac{10}{7} = \frac{230}{140} = \frac{23}{14} \]

### Final Answer

Thus, the value of \( \tan(2A + B) \) is:

\[ \boxed{\frac{23}{14}} \]