Hyperbolic Functions

Definitions

The hyperbolic functions are defined as:

$$\begin{array}{rl}\sinh x& =\frac{e^x-e^{-x}}{2}\\ \cosh x& =\frac{e^x+e^{-x}}{2}\\ \tanh x& =\frac{\sinh x}{\cosh x}=\frac{e^x-e^{-x}}{e^x+e^{-x}}\end{array}$$

The graphs of hyperbolic functions are below:

From these graphs, the domain and range of the hyperbolic functions can be found:

• $\sinh x$ has a domain of $x\in \mathbb{R}$, range of $\sinh x\in \mathbb{R}$.
• $\cosh x$ has a domain of $x\in \mathbb{R}$, range of $\cosh x\ge 1$.
• $\tanh x$ has a domain of $x\in \mathbb{R}$, range of $-1<\tanh x<1$.

Inverse hyperbolic functions

The inverse hyperbolic functions are $\text{DeclareMathOperator}arsinharsinh\text{arsinh}x$, $\text{DeclareMathOperator}arcosharcosh\text{arcosh}x$, and $\text{DeclareMathOperator}artanhartanh\text{artanh}x$, also denoted ${\sinh }^{-1}x,{\cosh }^{-1}x$ and ${\tanh }^{-1}x$ respectively.

The domains and ranges can be found:

• $\text{DeclareMathOperator}arsinharsinh\text{arsinh}x$ has a domain of $x\in \mathbb{R}$, range of $\text{DeclareMathOperator}arsinharsinh\text{arsinh}x\in \mathbb{R}$.
• $\text{DeclareMathOperator}arcosharcosh\text{arcosh}x$ has a domain of $x\ge 1$, range of $\text{DeclareMathOperator}arcosharcosh\text{arcosh}x\ge 0$.
• $\text{DeclareMathOperator}artanhartanh\text{artanh}x$ has a domain of $-1<x<1$, range of $\text{DeclareMathOperator}artanhartanh\text{artanh}x\in \mathbb{R}$.

The inverse hyperbolic functions have logarithmic forms that can be proved by using the exponential definition of the hyperbolic functions and solving the hidden quadratic. These are:

• $\text{DeclareMathOperator}arsinharsinh\text{arsinh}x=\ln (x+\sqrt{x^2+1})$
• $\text{DeclareMathOperator}arcosharcosh\text{arcosh}x=\ln (x+\sqrt{x^2-1})$
• $\text{DeclareMathOperator}artanhartanh\text{artanh}x=\frac{1}{2}\ln (\frac{1+x}{1-x})$ These are given in the formula book. Proofs are given below.

Hyperbolic identities

The most important hyperbolic identity is:

$${\cosh }^{2}x-{\sinh }^{2}x=1$$

When proving identities, it is often useful to return to the exponential definitions of $\cosh$, $\sinh$, and $\tanh$. Outside of the syllabus, there is Osborn’s Rule for converting normal trigonometric identities into hyperbolic identities.

Calculus of hyperbolic functions

The derivatives of hyperbolic functions are:

• $\frac{\text{d}y}{\text{d}x}(\sinh x)=\cosh x$
• $\frac{\text{d}y}{\text{d}x}(\cosh x)=\sinh x$
• $\frac{\text{d}y}{\text{d}x}(\tanh x)=\frac{1}{{\cosh }^{2}x}$ Only the last one is given. These can be found by returning to the exponential definitions.

The integrals of hyperbolic functions are:

• $\int \sinh x\text{d}x=\cosh x+C$
• $\int \cosh x\text{d}x=\sinh x+C$
• $\int \tanh x\text{d}x=\ln \cosh x+C$

Derivation of logarithm definitions of inverse hyperbolic functions

The following proofs are given to derive the logarithm definitions of inverse hyperbolic functions. Broadly, all follow a similar idea of completing the square on $e^x$ (this can even be used to find logarithmic definitions of inverse trigonometric functions, which has come up before on past papers).

Proof for ${\sinh }^{-1}x$

$$\begin{array}{rl}y=\sinh x& =\frac{e^x-e^{-x}}{2}\\ 2y& =e^x-e^{-x}\\ 2y{e}^x& =e^{2x}-1\\ 0& =(e^x{)}^2-2y(e^x)-1\\ & \\ e^x& =\frac{2y\pm \sqrt{(2y{)}^2-4(-1)}}{2}\\ & =\frac{2y\pm \sqrt{4y^2+4}}{2}\\ & =y\pm \sqrt{y^2+1}\\ & =y+\sqrt{y^2+1}(e^x>0)\\ x& =\ln (y+\sqrt{y^2+1})\end{array}$$

Proof for ${\cosh }^{-1}x$

$$\begin{array}{rl}y=\cosh x& =\frac{e^x+e^{-x}}{2}\\ 2y& =e^x+e^{-x}\\ 2y{e}^x& =e^{2x}+1\\ 0& =(e^x{)}^2-2y(e^x)+1\\ & \\ e^x& =\frac{2y\pm \sqrt{(2y{)}^2-4(1)}}{2}\\ & =\frac{2y\pm \sqrt{4y^2-4}}{2}\\ & =y\pm \sqrt{y^2-1}\\ & =y+\sqrt{y^2-1}(e^x>0)\\ x& =\ln (y+\sqrt{y^2-1})\end{array}$$