Question

If $\tan \, A = \frac{1}{3} $ and $\tan \, B = \frac{1}{7}$ , then what is the value of 2A + B?

• A. $30^{\circ}$
• B. $45^{\circ}$
• C. $60^{\circ}$
• D. $135^{\circ}$

Answer 1

Answer: (B)

$\tan\left(2A+B\right) = \frac{\tan2A + \tan B}{1- \tan2A \tan B} $
$\tan2A = \frac{2 \tan A}{1+\tan^{2}A} = \frac{2\left(\frac{1}{3}\right)}{1-\left(\frac{1}{3}\right)^{2}} $
$= \frac{\frac{2}{3}}{1- \frac{1}{9}} = \frac{2/3}{8/9} = \frac{3}{4}$
Hence, $ \tan\left(2A+B\right) = \frac{\tan2A +\tan B}{1-\tan2A \tan B} $
$= \frac{\frac{3}{4} + \frac{1}{7}}{1- \frac{3}{4} .\frac{1}{7}} = \frac{\frac{21+4}{28}}{\frac{28-3}{28}} = 1 $
$\Rightarrow \, \tan (2A + B) = \tan \, 45^{\circ}$
$\Rightarrow \, 2A + B = 45^{\circ}$