Question

A possible value of $\tan \left(\frac{1}{4}{\sin }^{-1}\left(\frac{\sqrt{63}}{8}\right)\right)$ is:

• A. $\frac{1}{2\sqrt{2}}$
• B. $\frac{1}{\sqrt{7}}$
• C. $\sqrt{7}-1$
• D. $2\sqrt{2}-1$

Answer 1

The correct option is B

$\frac{1}{\sqrt{7}}$

Explanation for the correct option:

Step 1: Find the $\cos$ value of the given angle:

Use ${\sin }^{-1}\left(\frac{\sqrt{63}}{8}\right)$ to find $\cos$ value.

$\begin{array}{rcl}{\sin }^{-1}\left(\frac{\sqrt{63}}{8}\right)& =& \theta \\ \sin \theta & =& \left(\frac{\sqrt{63}}{8}\right)\\ \cos \theta & =& \frac{1}{8}\end{array}$

Step 2: Express $\cos$ in terms of $\tan$:

Use trignometric identities to get the required value.

$\begin{array}{rcl}2{\cos }^2\frac{\theta }{2}-1& =& \frac{1}{8}\\ \cos \frac{\theta }{2}& =& \frac{3}{4}\\ & \Rightarrow & \frac{\left(1-{\tan }^2\frac{\theta }{4}\right)}{\left(1+{\tan }^2\frac{\theta }{4}\right)}=\frac{3}{4}\\ \therefore \tan \frac{\theta }{4}& =& \frac{1}{\sqrt{7}}\end{array}$

Hence, option (B) is the correct answer.