Question

If $\theta$ and $\varphi$ are angles in the 1st quadrant such that $\tan \theta =\frac{1}{7}$ and $\sin \varphi =\frac{1}{\sqrt{10}}$ .

• A. $\theta +2\varphi ={90}^{\circ }$
• B. $\theta +2\varphi ={60}^{\circ }$
• C. $\theta +2\varphi ={30}^{\circ }$
• D. $\theta +2\varphi ={45}^{\circ }$

Answer 1

The correct option is D $\theta +2\varphi ={45}^{\circ }$

$\tan \theta =\frac{1}{7}$

$\sin \varphi =\frac{1}{\sqrt{10}}\Rightarrow {\sin }^2\varphi =\frac{1}{10}\Rightarrow cose{c}^2\varphi =10$

We know, ${\cot }^2\varphi =cose{c}^2\varphi -1=10-1=9\Rightarrow \tan \varphi =\frac{1}{3}$

Now, $\tan 2\varphi =\frac{2\tan \varphi }{1-{\tan }^2\varphi }=\frac{3}{4}$

Now, $\tan (\theta +2\varphi )=\frac{\tan \theta +\tan 2\varphi }{1-\tan \theta \tan 2\varphi }=\frac{\frac{1}{7}+\frac{3}{4}}{1-\frac{1}{7}.\frac{3}{4}}=1\Rightarrow \theta +2\varphi ={45}^o$