Definitions of Hyperbolic Functions

We build the hyperbolic functions directly from the exponential function $e^x$, in the same way circular (trigonometric) functions can be pictured on the unit circle. For any real number $x$, we define

The remaining four hyperbolic functions are built from these two exactly as the circular ratios are built from sine and cosine:

Since $e^x$ is defined and positive for every real $x$, $\cosh x$ is a sum of two positive quantities and can never vanish, so $\tanh x$ and $\operatorname{sech} x$ are defined for all real $x$. But $\sinh x = 0$ only at $x = 0$, so $\coth x$ and $\operatorname{csch} x$ exclude $x = 0$ from their domain.

A quick check of parity is useful before we go further. Replacing $x$ by $-x$ swaps $e^x$ and $e^{-x}$ inside the definitions, which immediately gives $\sinh(-x) = -\sinh x$ (an odd function) and $\cosh(-x) = \cosh x$ (an even function). Consequently $\tanh x$, $\coth x$, and $\operatorname{csch} x$ are odd, while $\operatorname{sech} x$ is even — a mirror image of how sine and cosine hand down their parity to tangent and secant in circular trigonometry.

A short worked check: at $x = 0$, $e^0 = 1$, so $\sinh 0 = \frac{1-1}{2} = 0$ and $\cosh 0 = \frac{1+1}{2} = 1$, giving $\tanh 0 = 0$ and $\operatorname{sech} 0 = 1$, consistent with the even/odd behaviour just noted. These six definitions are the raw material for every identity, graph, and equation in this chapter.