If we follow the $x$ and $y$ coordinates of a point $P$ that is on the unit circle, we will see that the $x=\cos(\theta)$ and $y=\sin(\theta)$, where $\theta$ is the angle of the point. This is called circular trigonometry.

We can also have hyperbolic trigonometry, if we replace the circle with a hyperbola. The $x$-coordinate of a point $Q$ that traces out the shape of a hyperbola will have the formula $\cosh$ and its $y$ coordinate will be $\sinh$.

We can define the hyperbolic functions in terms of the exponential function: $$\cosh x = \frac{e^{x} + e^{-x}}{2}, \qquad \sinh x = \frac{e^{x} - e^{-x}}{2}.$$

• $\cosh$ is an even function, while $\sinh$ is odd.

In fact you can think of $\cosh x$ as the “even part of $e^x$,

  and $\sinh x$ as the odd part of $e^x$ since
  \[
   e^x = \cosh x + \sinh x.
  \]

The equivalent of the circular-trigonometric identity $\cos^2 \theta + \sin^2 \theta = 1$, is the following: $$\cosh^2 x - \sinh^2 x = 1.$$

A long wire suspended between two lamp posts will sag in the middle producing a curved shape. The exact equation that describes the shape of the cable is the hyperbolic cosine function: $\cosh x$.