Question

If $\tan \alpha =\frac{1}{7},\sin \beta =\frac{1}{\sqrt{10}}$,then the value of $\alpha +2\beta$

• A. $0^0$
• B. ${30}^0$
• C. ${45}^0$
• D. ${60}^0$

Answer 1

The correct option is C ${45}^0$

$\sin \beta =\frac{1}{\sqrt{10}}$
Therefore $\cos \beta =\frac{\sqrt{10-1}}{\sqrt{10}}=\frac{3}{\sqrt{10}}$
Hence $\tan \beta =\frac{\sin \beta }{\cos \beta }=\frac{1}{3}$ ...(i)
$\tan 2\beta =\frac{2\tan \beta }{1-{\tan }^2\beta }$
Substituting the value of $\tan \beta$ from (i) we get
$\tan 2\beta =\frac{3}{4}$ ...(ii)
$\tan \alpha =\frac{1}{7}$ ...(iii)
Now
$\tan (\alpha +2\beta )=\frac{\tan \alpha +\tan 2\beta }{1-\tan \alpha +\tan 2\beta }$
Substituting the value of $\tan \alpha$ and $\tan 2\beta$ from iii and ii and by simplifying we get
$\tan (\alpha +2\beta )=\frac{4+21}{28-3}$$=1$
$\tan (\alpha +2\beta )=1$
$\alpha +2\beta ={45}^0$
Hence answer is C