If $\tan \theta =\frac{1}{7}$ and $\sin \varphi =\frac{1}{\sqrt{10}}$, where $\theta$ and $\varphi$ are acute angles, then find the value of $\theta +2\varphi$.

Solution

1. Given $\tan \theta =\frac{1}{7}$ and $\sin \varphi =\frac{1}{\sqrt{10}}$, we have $\tan \varphi =\frac{1}{3}$.
2. Calculate $$\begin{array}{rl}\tan (2\varphi )& =\frac{2\tan \varphi }{1-{\tan }^2\varphi }\\ & =\frac{2/3}{1-1/9}\\ & =\frac{2/3}{8/9}\\ & =\frac{3}{4}\end{array}$$.
3. Calculate $$\begin{array}{rl}\tan (\theta +2\varphi )& =\frac{\tan \theta +\tan 2\varphi }{1-\tan \theta \tan 2\varphi }\\ & =\frac{1/7+3/4}{1-(1/7)(3/4)}\\ & =\frac{25/28}{25/28}\\ & =1\end{array}$$.
4. Since $\theta$ and $\varphi$ are acute, $\theta +2\varphi =\frac{\pi }{4}$.
5. Hence the answer is (A).