Question

$\sin h^{-1}\left(\frac{1}{2}\right)=?$

• A. $\tan h^{-1}\left(\sqrt{5}\right)$
• B. $\tan h^{-1}\left(\frac{1}{\sqrt{5}}\right)$
• C. $\tan h^{-1}\left(\sqrt{3}\right)$
• D. $\tan h^{-1}\left(\frac{2}{\sqrt{5}}\right)$

Answer 1

The correct option is B

$\tan h^{-1}\left(\frac{1}{\sqrt{5}}\right)$

Explanation for the correct answer:

Solving the given expression by using inverse trigonometric identities:

$$\begin{gathered} \sin h^{-1}\left(\frac{1}{2}\right)=\tan h^{-1}\left[\frac{{\displaystyle \frac{1}{2}}}{\sqrt{1+{\left({\displaystyle \frac{1}{2}}\right)}^2}}\right]\left[\because {\sin }^{-1}x={\tan }^{-1}\left(\frac{x}{\sqrt{1+x^2}}\right)\right] \\ =\tan h^{-1}\left[\frac{{\displaystyle \frac{1}{2}}}{\sqrt{1+{\displaystyle \frac{1}{4}}}}\right] \\ =\tan h^{-1}\left[\frac{{\displaystyle \frac{1}{2}}}{{\displaystyle \frac{\sqrt{5}}{2}}}\right] \\ =\tan h^{-1}\left(\frac{1}{\sqrt{5}}\right) \end{gathered}$$

Thus, $\sin h^{-1}\left(\frac{1}{2}\right)=$$\tan h^{-1}\left(\frac{1}{\sqrt{5}}\right)$

Therefore, option (B) is the correct answer.