y = \sinh x

Graph of the Hyperbolic Sine $y = \sinh x$

The hyperbolic sine function $y = \sinh x$ is defined in terms of the exponential function¹.

\sinh x = \frac{e^x - e^{-x}}{2}

It is the hyperbolic counterpart of the ordinary sine $\sin x$ .

Domain and range

The domain is all real numbers
The range is all real numbers
It increases monotonically
It is an odd function

Being built from the difference of $e^x$ and $e^{-x}$ , it has a value at every real number. Unlike the sine, its values are not bounded, and that is the sharpest difference between the two.

Symmetry

Since $\sinh(-x) = \dfrac{e^{-x} - e^{x}}{2} = -\sinh x$ , the function is odd and its graph is symmetric about the origin.

Monotonicity

The derivative is $\dfrac{d}{dx}\sinh x = \cosh x$ . Since $\cosh x$ is always at least $1$ , and therefore positive, $\sinh x$ is strictly increasing on the whole real line and has no extrema.

The second derivative is $\sinh x$ itself, so it changes sign only at the origin. The single inflection point is $(0, 0)$ , where the curve turns from concave down to concave up.

Relation to the exponential

\begin{align*} \cosh x + \sinh x &= e^x \\ \cosh x - \sinh x &= e^{-x} \end{align*}

One may read $\cosh$ and $\sinh$ as the even and the odd parts into which the exponential function splits.

Near the origin and far out

Range	Approximation	Behavior
near the origin	$\sinh x \approx x$	tangent to the line $y = x$
$x \to +\infty$	$\sinh x \approx \dfrac{e^x}{2}$	grows exponentially
$x \to -\infty$	$\sinh x \approx -\dfrac{e^{-x}}{2}$	falls exponentially

There are no horizontal asymptotes. The Taylor series has only odd powers.

\sinh x = x + \frac{x^3}{3!} + \frac{x^5}{5!} + \cdots

The coefficients have the same magnitudes as those of $\sin x$ ; only the alternation of signs is missing.

Notable values

$x$	$\sinh x$
$0$	$0$
$1$	$\approx 1.1752$
$2$	$\approx 3.6269$
$3$	$\approx 10.0179$

Relation to the hyperbola

With the hyperbolic cosine it satisfies the identity $\cosh^2 x - \sinh^2 x = 1$ , from which the point $(\cosh t, \sinh t)$ always lies on the hyperbola $x^2 - y^2 = 1$ .

Item	Trigonometric	Hyperbolic
Identity	$\cos^2 t + \sin^2 t = 1$	$\cosh^2 t - \sinh^2 t = 1$
Curve the point lies on	the unit circle	the unit hyperbola
Derivative on the sine side	$\cos t$	$\cosh t$
Derivative on the cosine side	$-\sin t$	$\sinh t$
Period	$2\pi$	none

This corresponds to the fact that $(\cos t, \sin t)$ lies on the unit circle, and it is the origin of the name hyperbolic functions.

Applications

The catenary, the shape a chain or a cable takes under its own weight
The equations of a vibrating string and of heat conduction
The addition of velocities in special relativity, through rapidity²

Hyperbolic functions, Wikipedia
Rapidity, Wikipedia

Graph of the Hyperbolic Sine y=sinh⁡xy = \sinh xy=sinhx