Hyperbolic Functions Reference Cheat Sheet

Key Facts

• The main definitions are $\sinh x = \frac{e^x - e^{-x}}{2}$, $\cosh x = \frac{e^x + e^{-x}}{2}$, and $\tanh x = \frac{\sinh x}{\cosh x}$.
• The fundamental hyperbolic identity is $\cosh^2 x - \sinh^2 x = 1$.
• The reciprocal hyperbolic functions are $\operatorname{sech} x = \frac{1}{\cosh x}$, $\operatorname{csch} x = \frac{1}{\sinh x}$, and $\coth x = \frac{\cosh x}{\sinh x}$.
• The basic derivatives are $\frac{d}{dx}\sinh x = \cosh x$, $\frac{d}{dx}\cosh x = \sinh x$, and $\frac{d}{dx}\tanh x = \operatorname{sech}^2 x$.
• The reciprocal derivatives include $\frac{d}{dx}\operatorname{sech} x = -\operatorname{sech} x\tanh x$, $\frac{d}{dx}\operatorname{csch} x = -\operatorname{csch} x\coth x$, and $\frac{d}{dx}\coth x = -\operatorname{csch}^2 x$.
• Useful integrals include $\int \sinh x\,dx = \cosh x + C$, $\int \cosh x\,dx = \sinh x + C$, and $\int \operatorname{sech}^2 x\,dx = \tanh x + C$.
• The inverse forms include $\operatorname{arsinh} x = \ln\left(x + \sqrt{x^2 + 1}\right)$ and $\operatorname{arcosh} x = \ln\left(x + \sqrt{x^2 - 1}\right)$ for $x \ge 1$.
• Hyperbolic functions are not periodic, and $\cosh x$ is even while $\sinh x$ and $\tanh x$ are odd.

Vocabulary

Hyperbolic function

A function defined using exponential expressions that is related to the geometry of a hyperbola.

Hyperbolic sine

The function $\sinh x = \frac{e^x - e^{-x}}{2}$, which is odd and has derivative $\cosh x$.

Hyperbolic cosine

The function $\cosh x = \frac{e^x + e^{-x}}{2}$, which is even and has derivative $\sinh x$.

Hyperbolic tangent

The function $\tanh x = \frac{\sinh x}{\cosh x}$, which has horizontal asymptotes at $y = 1$ and $y = -1$.

Inverse hyperbolic function

A function such as $\operatorname{arsinh} x$ or $\operatorname{arcosh} x$ that reverses a hyperbolic function on an appropriate domain.

Fundamental identity

The identity $\cosh^2 x - \sinh^2 x = 1$, which is the hyperbolic counterpart of a trigonometric Pythagorean identity.