Hyperbolic functions

The formula

sinh x = (eˣ − e⁻ˣ)/2
cosh x = (eˣ + e⁻ˣ)/2
tanh x = sinh x / cosh x

Reading: sinh is the odd part of eˣ, cosh is the even part, and adding them back gives cosh x + sinh x = eˣ exactly ^[1]^[2]. tanh is their ratio — it starts out as the 45° line near zero and flattens to ±1.

cosh²x − sinh²x = 1

Reading: the Pythagorean identity with a minus sign. Circular functions live on the unit circle x² + y² = 1; hyperbolic functions live on the unit hyperbola x² − y² = 1. That one sign is the entire difference between the two families ^[2].

d/dx (sinh x) = cosh x        d/dx (cosh x) = sinh x

Reading: no sign flip on the second derivative — differentiate twice and you get the same function back with a plus sign. Sine and cosine pick up a minus. This is why y'' = +k²·y (tension members, lossy lines, fin conduction) is solved by sinh and cosh, while y'' = −k²·y (vibration, buckling) is solved by sin and cos.

asinh x = ln(x + √(x² + 1))
acosh x = ln(x + √(x² − 1))          x ≥ 1
atanh x = (1/2)·ln((1 + x)/(1 − x))   |x| < 1

Reading: the inverses are just logarithms in disguise ^[7]. Anywhere you need to solve for the argument — back out a span from a measured sag, or a length from an attenuation ratio — you are one log away, and every scientific library ships them directly.

y(x) = a·cosh(x/a),   a = H/w

Reading: the catenary. A cable hanging under its own weight takes this shape, where H is the horizontal tension and w is the weight per unit length ^[4]. Half-span arc length is s = a·sinh(x/a), and the sag from the supports is a·(cosh(L/2a) − 1).

Where you meet it

The antenna range, doing a sag calc on a feeder run. A coax or messenger cable spanning between poles hangs in a catenary, and the sag-tension trade is a cosh problem. The parabola shortcut sag ≈ w·L²/(8H) is the first two terms of the cosh Taylor series — for a 300 m span with a = H/w = 1000 m, the exact sag is 11.271 m and the parabola says 11.250 m, low by 0.19%. Tight spans, fine. Slack spans, the error grows fast.
The RF bench, working a lossy line. The input impedance of a transmission line of length ℓ is Zin = Z0·(ZL + Z0·tanh(γℓ)) / (Z0 + ZL·tanh(γℓ)). On a lossless line γ is purely imaginary and tanh collapses to j·tan — the Smith-chart formula you already know. With real loss, tanh(γℓ) walks toward 1 as ℓ grows, which is the math saying a long lossy cable looks like Z0 no matter what you hang on the far end. That is also why a bad load can hide behind 30 m of coax during a VSWR check.
The stress review, beam-column with axial tension. Put a tie-rod or a pressurized skin panel under axial tension P plus a lateral load and the governing equation is EI·y'''' − P·y'' = q. The homogeneous solution is A·cosh(kx) + B·sinh(kx) + C·x + D with k = √(P/EI) — four constants for the four boundary conditions. Flip the axial load to compression and the same equation turns into sines and cosines — and eventually Euler buckling. Same beam, opposite sign, different function family.
The thermal bench, sizing a heat-sink fin. A straight fin with an insulated tip has efficiency η = tanh(m·L)/(m·L) with m = √(h·P/(k·A)). At mL = 2 that is 0.482 — the outer half of a long fin is barely working, which is the number that ends most "just make the fin longer" conversations.

How it works

Think of sinh and cosh as the two ways to average a growing exponential with a decaying one. Near zero they behave like their circular cousins: sinh x ≈ x, cosh x ≈ 1 + x²/2, tanh x ≈ x. Far from zero the decaying half dies and both sinh and cosh become eˣ/2 — at x = 3 they already agree with it to within 0.25%. There is no periodicity on the real line. cosh is even and never drops below 1; sinh and tanh are odd; tanh saturates, hitting 0.7616 at x = 1, 0.9951 at x = 3, and 0.99991 at x = 5, which is why it gets borrowed as a soft limiter everywhere from wave mechanics to neural nets.

A few anchor values worth having in your head for sanity-checking a spreadsheet:

sinh(1) = 1.1752    cosh(1) = 1.5431    tanh(1) = 0.7616
sinh(0) = 0         cosh(0) = 1         tanh(0) = 0
cosh(x) ≥ 1 always;  sinh and tanh carry the sign of x

The identity family is trig with deliberate sign damage. Every circular identity has a hyperbolic twin, and Osborn's rule of thumb is that a product of two sinhs flips sign relative to the sine version: cosh(a+b) = cosh a·cosh b + sinh a·sinh b. The bridge between the families is exact: sinh(jx) = j·sin x and cosh(jx) = cos x. That single substitution is what turns the lossy-line tanh into the lossless-line tan, and it is why a solver that handles one family handles both if it accepts complex arguments.

The mistakes that actually bite. One: the sign of k². Setting up a beam-column or a lossy line and grabbing sin/cos when the physics wants sinh/cosh produces a solution that satisfies the boundary conditions and is still wrong everywhere in between — oscillatory where the real answer grows monotonically. Two: fitting a parabola to a deep catenary. Utility spans with small sag-to-span ratios forgive it; a slack mooring line, a long guy wire, or an aerostat tether does not, and the arc length (which sets how much cable you buy) diverges from the parabola faster than the sag does. Three: overflow. In double precision, cosh and sinh overflow just past x = 710, and long before that they amplify any error in the argument by a factor of eˣ/2 — a lossy-line calc with a big γℓ product will happily return infinities. Four: cancellation at the other end. Computing sinh from (eˣ − e⁻ˣ)/2 at small x subtracts two nearly equal numbers; at x = 10⁻⁹ the naive form carries a relative error near 3×10⁻⁸ while the library sinh is exact. Use the library function, or expm1, and never roll your own from two exp calls.

tanh has a second career as the universal interpolator between two regimes. Water-wave phase speed obeys c² = (g·λ/2π)·tanh(2π·d/λ): in deep water the tanh saturates at 1 and speed depends only on wavelength; in shallow water tanh of a small argument is the argument itself and the formula collapses to c = √(g·d). One function, both limits, and a smooth handoff in between — the same trick the fin-efficiency and lossy-line formulas are pulling. And atanh runs the trick backwards: its addition rule atanh a + atanh b = atanh((a+b)/(1+ab)) turns awkward cascade combinations into plain sums, which is exactly the relativistic velocity-addition law and the reason rapidity, not velocity, is the quantity that adds.

One more physical reading worth keeping: in a catenary, the tension at any point equals w·y — weight per unit length times the height above the curve's directrix. The parameter a is not an abstraction; it is the height at which the cable's own weight over that length equals its horizontal tension. Stiff, tight cable: big a, flat curve. Slack chain: small a, deep bight. An anchor rode works precisely because a heavy chain buys a small a, keeping the pull at the anchor horizontal ^[4].

History

The curve came long before the functions. Galileo discussed the hanging chain in Two New Sciences (1638) and treated it as close to a parabola — a claim Joachim Jungius took apart, with the disproof published after his death in 1669 ^[4]^[5]. In 1690 Jakob Bernoulli put the problem to the mathematical world as an open challenge, and in June 1691 Acta Eruditorum carried three independent correct solutions: Leibniz, Christiaan Huygens, and Jakob's younger brother Johann ^[4]^[5]. Huygens had already given the curve its name, coining catenaria — from the Latin catena, chain — in a 1690 letter to Leibniz ^[5]^[6]. Family legend adds that Johann needled Jakob about failing to solve his own challenge for the rest of their lives.

Here is the odd part: all three 1691 solutions were written without hyperbolic functions, because the functions did not exist yet as named objects. Vincenzo Riccati — son of Jacopo, of Riccati-equation fame — formally introduced them in 1757 in his Opusculorum, writing Sh. and Ch., working out the addition formulas and derivatives, and using them to solve cubics ^[2]^[3]. Johann Heinrich Lambert systematized the functions in the 1760s, tied them cleanly to the exponential, and used notation that became the modern sinh and cosh; he is often credited as the originator, but Riccati beat him to print by more than a decade ^[2]^[3]. So the catenary was solved sixty-six years before anyone could write its equation the way we do now — and the shape stood for a long time after that too: the Gateway Arch in St. Louis is an inverted weighted catenary, deliberately flattened by making the legs heavier than the crown ^[4].

Related tools

/tools/beam-deflection
/tools/euler-buckling
/tools/coax-loss
/tools/vswr-return-loss

Sources

Written by HE in our own words from the cited sources — engineering judgment included, your stamp still required. All entries →