Hyperbolic functions

Intuition [Beginner]

Sine and cosine describe rotation around a circle. If you walk around the unit circle at constant speed, your horizontal position is cosine and your vertical position is sine. The key identity is that the sum of squares equals one: this encodes the geometry of the circle.

There is a parallel set of functions built from $e^x$ instead of rotation. They describe points on a hyperbola (the curve $x^2-y^2=1$) instead of a circle. These are the hyperbolic functions: hyperbolic sine (sinh) and hyperbolic cosine (cosh). Their defining identity uses subtraction instead of addition: ${\cosh }^{2}t-{\sinh }^{2}t=1$.

Why does this concept exist? Because many physical and geometric phenomena — the shape of a hanging chain, the geometry of spacetime, solutions to differential equations — are described naturally by these functions, and they share a structural analogy with ordinary trigonometry that simplifies computation.

Visual [Beginner]

The picture shows two curves side by side. On the left, a unit circle with a point moving around it; its coordinates are $(\cos t,\sin t)$. On the right, the right branch of the hyperbola $x^2-y^2=1$ with a point sliding along it; its coordinates are $(\cosh t,\sinh t)$. Both points trace their respective curves as $t$ increases.

The circle uses addition of squares; the hyperbola uses subtraction. This is the single structural difference between circular and hyperbolic trigonometry.

Worked example [Beginner]

Compute $\sinh (1)$ and $\cosh (1)$ using the definitions, given $e=2.718$ and $1/e=0.368$.

Step 1. By definition, $\sinh (x)=(e^x-e^{-x})/2$ and $\cosh (x)=(e^x+e^{-x})/2$.

Step 2. For $x=1$: $e^1=2.718$ and $e^{-1}=0.368$. So $\sinh (1)=(2.718-0.368)/2=2.350/2=1.175$.

Step 3. Similarly, $\cosh (1)=(2.718+0.368)/2=3.086/2=1.543$.

What this tells us: $\cosh (1)$ is always greater than $1$ (the minimum of $\cosh$ is at $t=0$, where $\cosh (0)=1$), and $\sinh (1)$ is positive for positive inputs.

Check your understanding [Beginner]

Formal definition [Intermediate+]

Definition (Hyperbolic sine and cosine). Define

$$\sinh (x)=\frac{e^x-e^{-x}}{2},\cosh (x)=\frac{e^x+e^{-x}}{2}.$$

These are well-defined for all $x\in \mathbb{R}$. The remaining hyperbolic functions are defined by analogy with circular trigonometry:

$$\tanh (x)=\frac{\sinh (x)}{\cosh (x)},\mathrm{csch}(x)=\frac{1}{\sinh (x)},\mathrm{sech}(x)=\frac{1}{\cosh (x)},\coth (x)=\frac{\cosh (x)}{\sinh (x)}.$$

The derivatives follow directly from the exponential definitions 02.06.01:

$$(\sinh x{)}^{\prime }=\cosh x,(\cosh x{)}^{\prime }=\sinh x,(\tanh x{)}^{\prime }={\mathrm{sech}}^{2}x.$$

Note the sign difference with circular functions: $(\cos x{)}^{\prime }=-\sin x$, but $(\cosh x{)}^{\prime }=+\sinh x$.

Counterexamples to common slips

• $\cosh$ is not periodic. Unlike $\cos$, which repeats every $2\pi$, $\cosh (x)$ grows exponentially as $x\to \pm \infty$. The hyperbolic functions are not periodic.

• $\sinh$ is not bounded. The function $\sinh (x)=(e^x-e^{-x})/2$ satisfies $\sinh (x)\to \infty$ as $x\to \infty$ and $\sinh (x)\to -\infty$ as $x\to -\infty$.

• The identity uses subtraction, not addition. Confusing ${\cosh }^2-{\sinh }^2=1$ with ${\cos }^2+{\sin }^2=1$ is the most common error. The sign difference is the defining structural distinction.

Key theorem with proof [Intermediate+]

Theorem (Hyperbolic identity and addition formulas). For all $x,y\in \mathbb{R}$:

1. ${\cosh }^{2}x-{\sinh }^{2}x=1$.
2. $\cosh (x+y)=\cosh x\cosh y+\sinh x\sinh y$.
3. $\sinh (x+y)=\sinh x\cosh y+\cosh x\sinh y$.

Proof of (1).

$${\cosh }^{2}x-{\sinh }^{2}x={\left(\frac{e^x+e^{-x}}{2}\right)}^2-{\left(\frac{e^x-e^{-x}}{2}\right)}^2$$

$$=\frac{e^{2x}+2+e^{-2x}}{4}-\frac{e^{2x}-2+e^{-2x}}{4}=\frac{4}{4}=1.$$

The cross terms $+2$ and $-2$ cancel, leaving $1$.

$\square$

Proof of (2).

$$\cosh (x+y)=\frac{e^{x+y}+e^{-(x+y)}}{2}=\frac{e^x{e}^y+e^{-x}{e}^{-y}}{2}.$$

Now expand the right side:

$$\cosh x\cosh y+\sinh x\sinh y=\frac{(e^x+e^{-x})(e^y+e^{-y})}{4}+\frac{(e^x-e^{-x})(e^y-e^{-y})}{4}.$$

Multiplying out the numerators:

$$\frac{e^{x+y}+e^{x-y}+e^{-x+y}+e^{-x-y}}{4}+\frac{e^{x+y}-e^{x-y}-e^{-x+y}+e^{-x-y}}{4}$$

$$=\frac{2e^{x+y}+2e^{-(x+y)}}{4}=\frac{e^{x+y}+e^{-(x+y)}}{2}=\cosh (x+y).$$

$\square$

Proof of (3). The argument is identical to (2) but with signs adjusted: $\sinh (x+y)=(e^{x+y}-e^{-(x+y)})/2$, and expanding $\sinh x\cosh y+\cosh x\sinh y$ gives the same cancellation pattern with signs flipped on the $e^{x-y}$ and $e^{-x+y}$ terms. $\square$

Bridge. This theorem is the foundational reason that hyperbolic functions mirror circular functions with signs flipped, and this is exactly the bridge from the exponential definition to a complete trigonometric calculus. The central insight is that the sign change in the identity ($-$ instead of $+$) propagates through every formula: addition formulas, double-angle formulas, and derivatives. The result builds toward the Gudermannian function 02.06.04 connecting circular and hyperbolic angles, and appears again in the ODE solution techniques 02.08.02 where the characteristic equation with complex roots produces sines and cosines while repeated real roots produce hyperbolic functions. Putting these together with the exponential definition 02.06.01, the bridge is that $\sinh$ and $\cosh$ are the "linear pieces" of $e^x$ and $e^{-x}$, splitting the exponential into its odd and even parts.

Exercises [Intermediate+]

Advanced results [Master]

Theorem 1 (The catenary). The curve formed by a uniform flexible chain hanging under gravity between two fixed supports is $y=a\cosh (x/a)$, where $a$ is a positive constant determined by the tension and linear mass density. The proof uses the calculus of variations: minimising the potential energy functional subject to a fixed-length constraint yields the ODE $y^{\prime \prime }=(1/a)\sqrt{1+(y^{\prime }{)}^2}$, whose solution is $y=a\cosh (x/a)+C$.

Theorem 2 (Inverse hyperbolic functions as logarithms).

$$\mathrm{arcsinh}(x)=\ln \left(x+\sqrt{x^2+1}\right),\mathrm{arccosh}(x)=\ln \left(x+\sqrt{x^2-1}\right)(x\ge 1).$$

The proof solves $y=\sinh (t)$ for $t$: set $e^t=u$, so $y=(u-1/u)/2$, giving $u^2-2yu-1=0$, hence $u=y+\sqrt{y^2+1}$ (taking the positive root), and $t=\ln u$.

Theorem 3 (Osborn's rule). Every trigonometric identity converts to a valid hyperbolic identity by replacing $\sin \to \sinh$, $\cos \to \cosh$, $\tan \to \tanh$, and negating one factor of $\sinh$ for each occurrence of a product of two sine-like terms. The proof follows from the identities $\cos (ix)=\cosh (x)$ and $\sin (ix)=i\sinh (x)$, verified by comparing Taylor series.

Theorem 4 (The Gudermannian function). Define $\mathrm{gd}(x)={\int }_0^x\mathrm{sech}(t)dt$. Then $\mathrm{gd}$ is a strictly increasing bijection $\mathbb{R}\to (-\pi /2,\pi /2)$ satisfying

$$\sinh (x)=\tan (\mathrm{gd}(x)),\cosh (x)=\sec (\mathrm{gd}(x)),\tanh (x)=\sin (\mathrm{gd}(x)).$$

This function connects circular and hyperbolic angles and appears in the theory of map projections (the Mercator projection uses ${\mathrm{gd}}^{-1}$).

Theorem 5 (Hyperbolic functions and the Lorentz group). The matrix

$$\left(\begin{array}{cc}\cosh \varphi & \sinh \varphi \\ \sinh \varphi & \cosh \varphi \end{array}\right)$$

is an element of ${\mathrm{SO}}^{+}(1,1)$, the identity component of the Lorentz group in $1+1$ dimensions. The rapidity $\varphi$ plays the role that the angle $\theta$ plays for rotations, and velocity addition corresponds to addition of rapidities via the addition formula for $\tanh$.

Theorem 6 (Taylor series).

$$\cosh (x)=\sum _{n=0}^{\infty }\frac{x^{2n}}{(2n)!},\sinh (x)=\sum _{n=0}^{\infty }\frac{x^{2n+1}}{(2n+1)!}.$$

These are the even and odd parts of the exponential series $e^x=\sum x^n/n!$, convergent for all $x\in \mathbb{R}$.

Synthesis. The foundational reason hyperbolic functions mirror circular functions is that both arise from the exponential: $\cosh$ and $\sinh$ are the even and odd parts of $e^x$ 02.06.01, while $\cos$ and $\sin$ are the even and odd parts of $e^{ix}$. The central insight is that replacing $i$ by $1$ converts every circular identity into a hyperbolic one (Osborn's rule), and this is exactly the bridge between Euclidean geometry (${\cos }^2+{\sin }^2=1$) and Minkowskian geometry (${\cosh }^2-{\sinh }^2=1$). Putting these together with the Gudermannian function, which identifies the circular and hyperbolic angle parameters, the pattern generalises to the full Lorentz group where rapidity plays the role of angle. The bridge is that the catenary 02.06.04 (a purely geometric object) is governed by $\cosh$, and this same function appears again in ODEs 02.08.02 whenever the characteristic equation has real roots — builds toward the connection between geometry and differential equations.

Full proof set [Master]

Proposition 1 (The identity ${\cosh }^{2}x-{\sinh }^{2}x=1$). Restated from Key theorem.

Proof. Writing $\cosh x=(e^x+e^{-x})/2$ and $\sinh x=(e^x-e^{-x})/2$:

$${\cosh }^{2}x-{\sinh }^{2}x=\frac{(e^x+e^{-x}{)}^2}{4}-\frac{(e^x-e^{-x}{)}^2}{4}=\frac{4e^x\cdot e^{-x}}{4}=\frac{4}{4}=1,$$

since $(e^x+e^{-x}{)}^2-(e^x-e^{-x}{)}^2=4e^x{e}^{-x}=4$. $\square$

Proposition 2 (Inverse hyperbolic sine as a logarithm). $\mathrm{arcsinh}(y)=\ln (y+\sqrt{y^2+1})$ for all $y\in \mathbb{R}$.

Proof. Set $x=\mathrm{arcsinh}(y)$, so $y=\sinh (x)=(e^x-e^{-x})/2$. Let $u=e^x$, so $y=(u-1/u)/2$, giving $u^2-2yu-1=0$. By the quadratic formula, $u=y\pm \sqrt{y^2+1}$. Since $u=e^x>0$ and $y-\sqrt{y^2+1}<0$ (as $\sqrt{y^2+1}>|y|$), we must take $u=y+\sqrt{y^2+1}$. Thus $x=\ln (y+\sqrt{y^2+1})$. $\square$

Proposition 3 (The Gudermannian satisfies $\tanh (x)=\sin (\mathrm{gd}(x))$).

Proof. Let $\theta =\mathrm{gd}(x)$, so $d\theta /dx=\mathrm{sech}(x)$. Then $\frac{d}{dx}\sin (\theta )=\cos (\theta )\cdot \mathrm{sech}(x)$. Using $\cos (\theta )=1/\cosh (x)$ (which follows from $\cosh (x)=\sec (\mathrm{gd}(x))$), we get $\frac{d}{dx}\sin (\theta )=\mathrm{sech}(x)/\cosh (x)=1/{\cosh }^2(x)={\mathrm{sech}}^2(x)=(\tanh x{)}^{\prime }$. Since $\sin (\mathrm{gd}(0))=\sin (0)=0=\tanh (0)$, and both functions have the same derivative, they are equal: $\sin (\mathrm{gd}(x))=\tanh (x)$. $\square$

Connections [Master]

• Logarithm as an integral 02.06.01. The exponential function $e^x$ is the inverse of the natural logarithm defined as an integral, and $\sinh$, $\cosh$ are built from $e^x$ and $e^{-x}$. Every property of the hyperbolic functions traces back to the integral definition of $\ln$ through the chain: integral defines $\ln$, inverse gives $\exp$, splitting $\exp$ into even and odd parts gives $\cosh$ and $\sinh$.

• Complex numbers and Euler's formula 02.09.01. The identities $\cos (ix)=\cosh (x)$ and $\sin (ix)=i\sinh (x)$ follow from comparing Taylor series. This is the algebraic mechanism behind Osborn's rule: substituting $ix$ for $x$ in the circular addition formulas yields the hyperbolic addition formulas with appropriate sign changes. Euler's formula $e^{ix}=\cos x+i\sin x$ becomes $e^x=\cosh x+\sinh x$ when $i$ is removed.

• Second-order linear ODEs 02.08.02. The ODE $y^{\prime \prime }-y=0$ has characteristic equation $r^2-1=0$ with roots $r=\pm 1$, and its general solution is $y=A{e}^x+B{e}^{-x}=C\cosh (x)+D\sinh (x)$. The hyperbolic functions are the natural basis for solutions to constant-coefficient ODEs with real characteristic roots, just as sines and cosines serve for complex roots.

Historical & philosophical context [Master]

Riccati 1757, in his Opusculorum ad res physicas et mathematicas pertinentium ^{[Riccati1757]}, introduced the hyperbolic sine and cosine as formal objects satisfying the relation ${\cosh }^2-{\sinh }^2=1$, recognising the structural analogy with circular trigonometry. Lambert 1768, in the Memoires de l'Academie royale des Sciences de Berlin ^{[Lambert1768]}, developed the systematic calculus of hyperbolic functions, derived their addition formulas, and applied them to the computation of logarithms of complex numbers. Lambert's work on the hyperbolic functions was part of his broader investigation into the nature of $\pi$ and $e$, including his proof that $e$ is irrational.

Euler had already encountered these functions in his 1748 Introductio ^[Euler1748], where the exponential expansions made the even/odd decomposition natural, but the explicit naming and systematic development is due to Riccati and Lambert. The connection to the catenary was known much earlier: Galileo conjectured that the hanging chain was a parabola, but Huygens, Leibniz, and Johann Bernoulli independently showed (1691) that it is the curve $y=a\cosh (x/a)$. The name "catenary" (from Latin catena, chain) is due to Huygens.

Bibliography [Master]

@article{Lambert1768,
  author = {Lambert, Johann Heinrich},
  title = {M\'emoire sur quelques propri\'et\'es remarquables des quantit\'es transcendentes circulaires et logarithmiques},
  journal = {M\'emoires de l'Acad\'emie royale des Sciences de Berlin},
  year = {1768},
  pages = {265--322},
}

@book{Riccati1757,
  author = {Riccati, Vincenzo},
  title = {Opusculorum ad res physicas et mathematicas pertinentium},
  publisher = {Ex Typographia Remondini},
  year = {1757},
}

@book{Euler1748,
  author = {Euler, Leonhard},
  title = {Introductio in Analysin Infinitorum},
  publisher = {Marcus-Michaelis Bousquet},
  year = {1748},
}

@book{Apostol1967,
  author = {Apostol, Tom M.},
  title = {Calculus, Volume 1},
  publisher = {John Wiley \& Sons},
  year = {1967},
}