Question

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

• A. $\theta +2\varphi ={90}^o$
• B. $\theta +2\varphi ={30}^o$
• C. $\theta +2\varphi ={75}^o$
• D. $\theta +2\varphi ={45}^o$

Answer 1

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

Since $\tan \theta =\frac{1}{7}$ we have $\sin \theta =\frac{1}{\left(5\sqrt{2}\right)}$ and $\cos \theta =\frac{7}{\left(5\sqrt{2}\right)}$
Also $\sin \varphi =\frac{1}{\sqrt{10}}\Rightarrow \tan \varphi =\frac{1}{3}$
$\therefore \sin 2\varphi =\frac{\left(2\tan \varphi \right)}{(1+{\tan }^2\varphi )}$
$=\frac{\left(\frac{2}{3}\right)}{[1+\left(\frac{1}{9}\right)]}=\frac{3}{5}$
and so $\cos 2\varphi =\frac{4}{5}$
Hence $\sin (\theta +2\varphi )=\sin \theta \cos 2\varphi +\cos \theta \sin 2\varphi$
$=\frac{1}{5\surd \left(2\right)}.\frac{4}{5}+\frac{7}{5\surd \left(2\right)}.\frac{3}{5}=\frac{25}{25\surd \left(2\right)}=\frac{1}{\surd \left(2\right)}$
$\theta +2\varphi ={45}^o$