= Hyperbolic function
{title2=$\sinh,\cosh,\tanh$}
{wiki}
<Hyperbolic functions> combine the <exponential function> to describe a <hyperbola> much as <trigonometric functions> describe a <circle>. Their two basic members are the <hyperbolic cosine> and <hyperbolic sine>, defined for every <real number> $t$ by
$$
\cosh t=\frac{e^t+e^{-t}}2,\qquad \sinh t=\frac{e^t-e^{-t}}2.
$$
Their <derivatives> exchange the two <functions>:
$$
\frac{d}{dt}\cosh t=\sinh t,\qquad \frac{d}{dt}\sinh t=\cosh t.
$$
These formulas follow by applying <differentiation> to the <exponential function> definitions. https://openstax.org/books/calculus-volume-2/pages/2-9-calculus-of-the-hyperbolic-functions[Gilbert Strang and Edwin Herman, OpenStax Calculus Volume 2, §2.9: hyperbolic definitions and derivatives]{ref}
Adding and subtracting gives $\cosh t+\sinh t=e^t$ and $\cosh t-\sinh t=e^{-t}$. Multiplying these equalities proves the <hyperbolic Pythagorean identity>:
$$
\boxed{\cosh^2t-\sinh^2t=1.}
$$
Consequently the <parametric equation> $(x,y)=(\cosh t,\sinh t)$ traverses the entire right branch of the unit <hyperbola> $x^2-y^2=1$, where $x\geq1$. For example, $t=0$ gives $(1,0)$, while $t=\pm1$ gives approximately $(1.543,\pm1.175)$. Compare $(\cos\theta,\sin\theta)$ on the <unit circle>, where the squared coordinates add to one. The minus sign changes the geometry from a closed <circle> to an unbounded <hyperbola>.
\Image[/hyperbolic-functions-and-unit-hyperbola.png]
{title=Hyperbolic cosine and sine become the coordinates of the right branch of a unit hyperbola}
{description=The same inputs $t=-1,0,1$ are marked on both <function> graphs and the unit <hyperbola>. The shaded region has <area> $1/2$ and corresponds to <hyperbolic angle> $t=1$.}
The other members are ratios and reciprocals: <hyperbolic tangent> is $\tanh t=\sinh t/\cosh t$, <hyperbolic cotangent> is $\coth t=\cosh t/\sinh t$, and <hyperbolic secant> is $\operatorname{sech}t=1/\cosh t$; denominators must be nonzero. The <hyperbolic sine> and <hyperbolic cosine> both solve the <second-order linear differential equation> $f''=f$. In fact $f(t)=A\cosh t+B\sinh t$ has $f(0)=A$ and $f'(0)=B$, making this <basis> convenient for an <initial value problem>. Their geometry also makes <hyperbolic substitution> useful for <integrals> containing $\sqrt{1+u^2}$, and the <hyperbolic cosine> gives the physical <catenary> of a hanging chain.
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