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}}
| Question | Answer |
|---|
| What is \tanh in terms of \sinh and \cosh? | \tanh = \frac{\sinh}{\cosh} |
| How do you derive the formula for \tanh using exponentials? | \tanh \equiv \frac{e^x - e^{-x}}{2} \div \frac{e^x + e^{-x}}{2}, which simplifies to \tanh \equiv \frac{e^x - e^{-x}}{e^x + e^{-x}} |