Graph of the Hyperbolic Cosine $y = \cosh x$

The hyperbolic cosine function $y = \cosh x$ is defined using the exponential function as

It is the hyperbolic analogue of the ordinary cosine $\cos x$.

Domain and range

Defined as a sum of exponentials, it exists for every real $x$, so its domain is $(-\infty, \infty)$. By the arithmetic-geometric mean inequality, $\dfrac{e^x + e^{-x}}{2} \geq \sqrt{e^x \cdot e^{-x}} = 1$, so its range is $[1, \infty)$; the value of $\cosh x$ is never less than $1$.

Symmetry

Since $\cosh(-x) = \dfrac{e^{-x} + e^{x}}{2} = \cosh x$, the function is even, and its graph is symmetric about the $y$-axis.

Monotonicity and minimum

The derivative is $\dfrac{d}{dx}\cosh x = \sinh x$, which is negative for $x < 0$ and positive for $x > 0$. Hence $\cosh x$ decreases on $(-\infty, 0)$ and increases on $(0, \infty)$, attaining its minimum at the point $(0, 1)$.

Limits and asymptotic behavior

For large $x$, $\cosh x \approx \dfrac{e^x}{2}$, and as $x \to -\infty$, $\cosh x \approx \dfrac{e^{-x}}{2}$. On both sides it grows exponentially like $\dfrac{e^{|x|}}{2}$ and diverges to $+\infty$. There are no horizontal asymptotes.

Notable points

The curve passes through its lowest point $(0, 1)$, with $\cosh 0 = 1$. The entire graph is concave up, forming a smooth valley shape.

Relationships with other functions

With the hyperbolic sine it satisfies $\cosh^2 x - \sinh^2 x = 1$, so $(\cosh t, \sinh t)$ lies on the right branch of the hyperbola $x^2 - y^2 = 1$. It also satisfies $\cosh x + \sinh x = e^x$ and $\cosh x - \sinh x = e^{-x}$.

Applications

The most famous application of $\cosh x$ is the catenary: a chain or cable hanging under its own weight from two fixed points takes the shape $y = a\cosh\dfrac{x}{a}$. Although it resembles a parabola, it is mathematically distinct. The curve also arises in the design of arches and power transmission lines.