Question

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

• A. $\theta +2\varphi =90°$
• B. $\theta +2\varphi =60°$
• C. $\theta +2\varphi =30°$
• D. $\theta +2\varphi =45°$

Answer 1

The correct option is D

$\theta +2\varphi =45°$

Explanation for the correct option.

Step 1. Find the value of $\tan \varphi$

Using the identity ${\sin }^{2}A+{\cos }^{2}A=1,$ for $\sin \varphi =\frac{1}{\sqrt{10}}$, $\cos \varphi$ can be found as:

$$\begin{gathered} {\left(\frac{1}{\sqrt{10}}\right)}^2+{\cos }^2\varphi =1 \\ \Rightarrow {\cos }^2\varphi =1-\frac{1}{10} \\ \Rightarrow {\cos }^2\varphi =\frac{9}{10} \\ \Rightarrow \cos \varphi =\frac{3}{\sqrt{10}} \end{gathered}$$

So, the value of $\tan \varphi$ is

$\begin{array}{rcl}\tan \varphi & =& \frac{\sin \varphi }{\cos \varphi }\\ & =& \frac{1}{3}\end{array}$

Step 2. Find the value of $\tan 2\varphi$.

Using the identity $\tan 2A=\frac{2\tan A}{1-{\tan }^{2}A}$ and $\tan \varphi =\frac{1}{3}$, the value of $\tan 2\varphi$ is found as:

$\begin{array}{rcl}\tan 2\varphi & =& \frac{2\tan \varphi }{1-{\tan }^2\varphi }\\ & =& \frac{2\times {\displaystyle \frac{1}{3}}}{1-{\displaystyle \frac{1}{9}}}\\ & =& \frac{{\displaystyle \frac{2}{3}}}{{\displaystyle \frac{8}{9}}}\\ & =& \frac{2\times 9}{3\times 8}\\ & =& \frac{3}{4}\end{array}$

Step 3: Find the value of $\theta +2\varphi$.

Using the identity $\tan \left(A+B\right)=\frac{\tan A+\tan B}{1-\tan A\tan B}$, $\tan \theta =\frac{1}{7}$ and $\tan 2\varphi =\frac{3}{4}$ the value of $\tan \left(\theta +2\varphi \right)$

$\begin{array}{rcl}\tan \left(\theta +2\varphi \right)& =& \frac{\tan \theta +\tan \left(2\varphi \right)}{1-\tan \theta \tan \left(2\varphi \right)}\\ & =& \frac{{\displaystyle \frac{1}{7}}+{\displaystyle \frac{3}{4}}}{1-{\displaystyle \frac{1}{7}}\times {\displaystyle \frac{3}{4}}}\\ & =& \frac{{\displaystyle \frac{25}{28}}}{{\displaystyle \frac{25}{28}}}\\ & =& 1\end{array}$

For $45°$, then $\tan 45°=1$.

Therefore, $\theta +2\varphi =45°$.

Hence, the correct option is D.