tanh

\tanh is the hyperbolic tangent. It doesn’t have a definition directly in terms of e, but just like regular \tan, we know we can find \tanh with the formula:

\tanh=\frac{\sinh}{\cosh}

Finding a formula

Because \tanh=\frac{\sinh}{\cosh}, we know that:

\tanh\equiv\frac{e^x-e^{-x}}2\div\frac{e^x+e^{-x}}2

We can simplify that to

\tanh\equiv\frac{e^x-e^{-x}}{e^x+e^{-x}}