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

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

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

Domain and range

Because $\sinh x$ is built from the exponentials $e^x$ and $e^{-x}$, it is defined for every real number $x$, so its domain is all of $(-\infty, \infty)$. It also attains every real value, so its range is $(-\infty, \infty)$ as well.

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 $\dfrac{d}{dx}\sinh x = \cosh x$. Because $\cosh x \geq 1 > 0$ everywhere, $\sinh x$ is strictly increasing on the whole real line.

Limits and asymptotic behavior

Near the origin $\sinh x \approx x$, so the curve hugs the line $y = x$. For large $x$ the term $e^{-x}$ becomes negligible and $\sinh x \approx \dfrac{e^x}{2}$, giving exponential growth. As $x \to -\infty$ the term $-\dfrac{e^{-x}}{2}$ dominates and the function decreases rapidly to $-\infty$. There are no horizontal asymptotes.

Notable points

The curve passes through the origin with $\sinh 0 = 0$. This point is also an inflection point, where the graph changes from concave down to concave up.

Relationships with other functions

With the hyperbolic cosine it satisfies the identity $\cosh^2 x - \sinh^2 x = 1$. Consequently the point $(\cosh t, \sinh t)$ always lies on the hyperbola $x^2 - y^2 = 1$, just as $(\cos t, \sin t)$ lies on the unit circle. This is the origin of the name "hyperbolic functions."

Applications

$\sinh x$ appears in the description of the catenary (the shape of a hanging chain or cable), in equations for vibrating strings and heat conduction, and in the addition of velocities (rapidity) in special relativity.