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)

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