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

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

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

Domain and range

• The domain is all real numbers
• The range is $y \geq 1$
• The minimum $1$ is attained at $x = 0$
• It is an even function

The relation between the arithmetic and the geometric mean shows that the value never falls below $1$.

Equality holds only when $e^x = e^{-x}$, that is at $x = 0$.

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 $x > 0$. So $\cosh x$ decreases to the left of the origin and increases to the right, and the minimum point is $(0, 1)$.

The second derivative is $\cosh x$ itself, always positive, so the whole graph is concave up with no inflection point. It draws a smooth valley, like a stretched U.

Notable values

On both sides it grows exponentially like $\dfrac{e^{|x|}}{2}$, and there are no horizontal asymptotes. The Taylor series has only even powers.

Relation to the exponential

The identity $\cosh^2 x - \sinh^2 x = 1$ also holds, so the point $(\cosh t, \sinh t)$ lies on the right half of the hyperbola $x^2 - y^2 = 1$.

The catenary

The most famous application is the catenary². A chain or a cable hanging under its own weight from two fixed points takes the following shape.

It closely resembles a parabola but is mathematically a different curve. The two expansions part company from the third term, so the gap widens the further one goes from the origin.

Applications

• The shape of a hanging chain or power line
• The design of arches in architecture
• The equations of a vibrating string and of heat conduction

1. Hyperbolic functions, Wikipedia
2. Catenary, Wikipedia