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Definition
The hyperbolic sine function is defined as the linear combination of two exponential functions $\frac{1}{2}e^{x} - \frac{1}{2}e^{-x}$ and is denoted as follows:
$$ \sinh x := \dfrac{e^{x} - e^{-x}}{2} $$
Similarly, the hyperbolic cosine function is defined as the linear combination of two exponential functions $\frac{1}{2}e^{x} + \frac{1}{2}e^{-x}$ and is denoted as follows:
$$ \cosh x := \dfrac{e^{x} + e^{-x}}{2} $$
Explanation
The names and notations of $\sinh$ and $\cosh$ are derived from “hyperbolic” + “sine (cosine).” From the definitions alone, it is hard to understand why they are named as such. Let’s find out why they are hyperbolic and why they are trigonometric functions.
Why Hyperbolic Functions?
To start with the conclusion, the name “hyperbolic functions” derives from the hyperbola. Represent the two points on the 2-dimensional plane $x$ and $y$ using $\cosh$ and $\sinh$.
$$ x(t) = \cosh t \\[1em] y(t) = \sinh t $$
Then the trajectory traced by the point $(x, y)$ forms a hyperbola. $$ \begin{align*} x^{2} - y^{2} = \cosh^{2} t - \sinh^{2} t &= \left( \dfrac{e^{x} + e^{-x}}{2} \right)^{2} - \left( \dfrac{e^{x} - e^{-x}}{2} \right)^{2}\\ &= \dfrac{e^{2x} + 2 + e^{-2x}}{4} - \dfrac{e^{2x} - 2 + e^{-2x}}{4} = 1 \end{align*} $$
It is similar to how a point on a circle is represented by the parametric equations $x(t) = \cos t$ and $y(t) = \sin t$.
Why Sine (Cosine) Functions?
To understand why they are named sine (cosine), we need to extend the domain to complex numbers. The following relations hold between hyperbolic functions and trigonometric functions.
$$ \sinh(ix) = i \sin x \\[1em] \cosh(ix) = \cos x $$
From the above equation, it is clear that hyperbolic functions and trigonometric functions are related. Due to this relationship, the two functions $\dfrac{e^{x} - e^{-x}}{2}$ and $\dfrac{e^{ix} + e^{-ix}}{2}$ are named after $\sin$ and $\cos$.