What are hyperbolic functions?

Hyperbolic functions are functions defined as follows

\begin{align} \sinh{x}&=\frac{e^{x}-e^{-x}}{2}\notag\\[+10pt] \cosh{x}&=\frac{e^{x}+e^{-x}}{2}\notag\\[+10pt] \tanh{x}&=\frac{e^{x}-e^{-x}}{e^{x}+e^{-x}}\notag \end{align}

The following three equations are the basic formulas for hyperbolic functions.

\begin{align} &\cosh^{2}{x}-\sinh^{2}{x}=1\notag\\[+10pt] &\tanh{x}=\frac{\sinh{x}}{\cosh{x}}\notag\\[+10pt] &1-\tanh^{2}{x}=\frac{1}{\cosh^{2}{x}}\notag \end{align}

Example

Example 1

Calculate \( \sinh{0} \).

Solution

\begin{align} \sinh{0}&=\frac{e^{0}-e^{0}}{2}\notag\\[+10pt] &=0\notag \end{align}

Example 2

Calculate \( \cosh{0} \).

Solution

\begin{align} \cosh{0}&=\frac{e^{0}+e^{0}}{2}\notag\\[+10pt] &=1\notag \end{align}

Example 3

Calculate \( \tanh{0} \).

Solution

\begin{align} \tanh{0}&=\frac{e^{0}-e^{0}}{e^{0}+e^{0}}\notag\\[+10pt] &=0\notag \end{align}