Question

$\sin h^{-1}\left(2^{\frac{3}{2}}\right)$ is

• A. $\log \left(2+\sqrt{18}\right)$
• B. $\log \left(3+\sqrt{8}\right)$
• C. $\log \left(3–\sqrt{8}\right)$
• D. $\log \left(\sqrt{8}+\sqrt{27}\right)$

Answer 1

The correct option is B

$\log \left(3+\sqrt{8}\right)$

Explanation for the correct answer:

Solving the given expression by using inverse trigonometric identities,

$$\begin{gathered} \sin h^{-1}\left(2^{\frac{3}{2}}\right)=\log \left(2^{\frac{3}{2}}+\sqrt{{\left(2^{\frac{3}{2}}\right)}^2+1}\right)\left[\because \sin h^{-1}x=\log \left(x+\sqrt{x^2+1}\right)\right] \\ =\log \left(2^{\frac{3}{2}}+\sqrt{\left(2^3\right)+1}\right) \\ =\log \left(2^{3.\frac{1}{2}}+\sqrt{8+1}\right) \\ =\log \left(\sqrt{2^3}+\sqrt{8+1}\right) \\ =\log \left(\sqrt{8}+\sqrt{9}\right) \\ =\log \left(\sqrt{8}+3\right) \end{gathered}$$

Thus, $\sin h^{-1}\left(2^{\frac{3}{2}}\right)=$$\log \left(3+\sqrt{8}\right)$

Hence, option (B) is the correct answer.