Single Variable Calculus, Chapter 7, 7.7, Section 7.7, Problem 6

Determine the numerical value of a.) $\sin h 1$ and b.) $\sin h^{-1} 1$

a.) $\sin h 1$

Using Hyperbolic Function


$
\begin{equation}
\begin{aligned}

\sin h x =& \frac{e^x - e^{-x}}{2}
\\
\\
\sin h 1 =& \frac{e^1 - e^{-1}}{2}
\\
\\
\sin h 1 =& \frac{\displaystyle e - \frac{1}{e}}{2}
\\
\\
\sin h 1 =& \frac{e^2 - 1}{2e}
\\
\\
\sin h 1 =& 1.1752

\end{aligned}
\end{equation}
$


b.) $\sin h^{-1} 1$

Using Inverse Hyperbolic Function


$
\begin{equation}
\begin{aligned}

\sin h^{-1} x =& \ln (x + \sqrt{x^2 + 1})
\\
\\
\sin h^{-1} 1 =& \ln (1 + \sqrt{(1)^2 + 1})
\\
\\
\sin h^{-1} 1 =& \ln (1 + \sqrt{2})
\\
\\
\sin h^{-1} 1 =& 0.8814

\end{aligned}
\end{equation}
$