Hyperbolic functions and their inverses

What you’ll learn

• Use exponential definitions and hyperbolic identities.
• Solve hyperbolic equations and recognise inverse domains.

Understand the idea

Hyperbolic sine and cosine are built from exponentials. The function $\sinh x$ is odd and increasing, whereas $\cosh x$ is even with minimum value $1$. Define $\tanh x=\sinh x/\cosh x$ and the reciprocal functions $\operatorname{sech}x$, $\operatorname{cosech}x$ and $\coth x$ by taking reciprocals of $\cosh x$, $\sinh x$ and $\tanh x$.

To solve an equation involving hyperbolic functions, exponential definitions often turn it into a quadratic in $e^x$, which must be positive. The inverse $\operatorname{arsinh}$ is defined for all real inputs; $\operatorname{arcosh}$ uses inputs at least $1$ and returns a non-negative value; $\operatorname{artanh}$ requires $|x|<1$.

Key formulae

$$\sinh x=\frac{e^x-e^{-x}}2,\qquad\cosh x=\frac{e^x+e^{-x}}2$$

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

$$\operatorname{arsinh}x=\ln(x+\sqrt{x^2+1})$$

$$\operatorname{arcosh}x=\ln(x+\sqrt{x^2-1}),\quad\operatorname{artanh}x=\frac12\ln\frac{1+x}{1-x}$$

Worked example

Solve $\cosh x=5/4$ for real $x$.

1. Write $(e^x+e^{-x})/2=5/4$.
2. Let $u=e^x>0$; multiplying through gives $2u^2-5u+2=0$.
3. Factor as $(2u-1)(u-2)=0$, giving $u=1/2$ or $2$.

Answer: $x=\pm\ln2$.