= Hyperbolic sine
{title2=$\sinh$}
The <hyperbolic sine> is the <odd function>
$$
\sinh x=\frac{e^x-e^{-x}}2.
$$
Its <derivative> is $\cosh x\geq1$, so it increases strictly from $-\infty$ to $\infty$ and is a <bijection> of $\mathbb R$ onto itself. For example, $\sinh0=0$, $\sinh1\approx1.175201$, and $\sinh(-1)\approx-1.175201$. It is the vertical coordinate on the unit <hyperbola> parametrized by $(\cosh x,\sinh x)$.
The <Taylor series>
$$
\sinh x=x+\frac{x^3}{3!}+\frac{x^5}{5!}+\cdots
$$
shows that $\sinh x\approx x$ near zero. Its <tangent line> there is $y=x$; unlike the ordinary <sine>, the <hyperbolic sine> grows without bound. Its <antiderivative> is $\cosh x+C$. In the <catenary> $y=a\cosh(x/a)$, the <hyperbolic sine> gives both the slope, through the <derivative> $y'=\sinh(x/a)$, and the <arc length> $s(x)=a\sinh(x/a)$ from the lowest point for $x\geq0$.
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