Hyperbolic functions — sinh, cosh, tanh and their reciprocals — are defined from the exponential function and mirror the properties of trig functions, but for the unit hyperbola instead of the unit circle. They arise everywhere from hanging cables to special relativity.
All six functions are defined in terms of eˣ and e⁻ˣ. The three primary functions are sinh, cosh, and tanh; the other three are their reciprocals.
sinh x
pronounced "sinch" or "shine"
sinh x = (eˣ − e⁻ˣ) / 2
sinh(0) = 0
Odd: sinh(−x) = −sinh(x)
Unbounded S-curve through origin. Grows exponentially for large |x|.
cosh x
pronounced "cosh"
cosh x = (eˣ + e⁻ˣ) / 2
cosh(0) = 1
Even: cosh(−x) = cosh(x)
U-shaped catenary. Minimum value 1 at x = 0. Never negative.
tanh x
pronounced "tanch" or "than"
tanh x = sinh x / cosh x
tanh(0) = 0
Odd: tanh(−x) = −tanh(x)
Bounded S-curve (sigmoid). Horizontal asymptotes at y = ±1.
csch x
pronounced "coseech"
csch x = 1 / sinh x
csch(0) undefined
Odd: csch(−x) = −csch(x)
Two branches, asymptotes at x = 0 and y = 0. Like a hyperbolic cosecant.
sech x
pronounced "seech"
sech x = 1 / cosh x
sech(0) = 1
Even: sech(−x) = sech(x)
Bell-shaped curve. Maximum value 1 at x = 0. Approaches 0 as x → ±∞.
coth x
pronounced "coth"
coth x = cosh x / sinh x
coth(0) undefined
Odd: coth(−x) = −coth(x)
Two branches outside (−1, 1). Vertical asymptote x = 0, horizontal asymptotes y = ±1.
Hyperbolic identities closely parallel trig identities — but watch the signs. The fundamental identity has a minus where trig has a plus.
cosh²x − sinh²x = 1
Trig analog: cos²x + sin²x = 1
The fundamental hyperbolic identity. Notice the minus sign (not plus).
tanh²x + sech²x = 1
Trig analog: tan²x + sec²x = — (not analogous)
Divide the fundamental identity by cosh²x. Compare: 1 − tanh²x = sech²x.
coth²x − csch²x = 1
Trig analog: cot²x + csc²x = csc²x − 1... different form
Divide the fundamental identity by sinh²x.
sinh(2x) = 2 sinh x cosh x
Trig analog: sin(2x) = 2 sin x cos x
Double angle for sinh. Exactly the same form as the trig double angle.
cosh(2x) = cosh²x + sinh²x
Trig analog: cos(2x) = cos²x − sin²x
Double angle for cosh. Note the plus sign — not minus as in trig.
cosh(x+y) = cosh x cosh y + sinh x sinh y
Trig analog: cos(x+y) = cos x cos y − sin x sin y
Addition formula for cosh. Again: plus sign where trig has minus.
Osborn's Rule
A quick way to convert trig identities to hyperbolic ones: replace every trig function with its hyperbolic counterpart, and flip the sign of every product of two sinh terms. For example, cos(x+y) = cos x cos y − sin x sin y becomes cosh(x+y) = cosh x cosh y + sinh x sinh y (the − sin·sin product becomes + sinh·sinh).
The parallel structure is real — but the differences matter, especially the lack of periodicity and the sign flip in the derivative of cosh.
| Property | Trig Functions | Hyperbolic Functions |
|---|---|---|
| Definition basis | Unit circle x² + y² = 1 | Unit hyperbola x² − y² = 1 |
| Periodic? | Yes (period 2π for sin/cos) | No — no periodicity |
| Parity: cos/cosh | cos is even | cosh is even |
| Parity: sin/sinh | sin is odd | sinh is odd |
| Pythagorean identity | cos²x + sin²x = 1 | cosh²x − sinh²x = 1 |
| Derivative of cos/cosh | −sin x | +sinh x |
| Derivative of sin/sinh | cos x | cosh x |
| Boundedness of tanh/tan | tan is unbounded (has asymptotes) | tanh is bounded: (−1, 1) |
| cosh range | cos range: [−1, 1] | [1, ∞) — always ≥ 1 |
| Double angle (cos/cosh) | cos(2x) = cos²x − sin²x | cosh(2x) = cosh²x + sinh²x |
y = sinh x
Increasing through all reals. Odd function symmetric about origin. Grows like eˣ/2 for large x, like −e⁻ˣ/2 for large negative x.
y = cosh x
U-shaped catenary. Even function symmetric about y-axis. Minimum at (0,1). Never below 1. Resembles a parabola near origin but grows exponentially.
y = tanh x
Bounded S-curve (sigmoid). Odd function. Increases from −1 to 1. Steepest at origin (slope = sech²(0) = 1). Never reaches ±1.
The key difference from trig: d/dx[cosh x] = +sinh x, not −sinh x. All other derivatives follow the same sign pattern as their trig counterparts.
| f(x) | f ′(x) | Trig analog | Note |
|---|---|---|---|
| sinh x | cosh x | sin x → cos x | Same form as trig. |
| cosh x | sinh x | cos x → −sin x | Positive! Not negative like trig. |
| tanh x | sech²x | tan x → sec²x | Same structure as trig. |
| csch x | −csch x coth x | csc x → −csc x cot x | Same form as trig. |
| sech x | −sech x tanh x | sec x → sec x tan x | Minus sign; trig has plus. |
| coth x | −csch²x | cot x → −csc²x | Same form as trig. |
Watch out on exams
The most common error is writing d/dx[cosh x] = −sinh x by analogy with cosine. It is +sinh x. The sign difference traces back to the minus in the fundamental identity cosh²x − sinh²x = 1.
Each inverse hyperbolic function can be written exactly as a natural logarithm. This is possible because sinh and cosh are defined with exponentials — solving, for instance, y = sinh x for x means solving an equation in eˣ, which gives a quadratic in eˣ.
arcsinh(x)
also: sinh⁻¹(x)
ln(x + √(x² + 1))
Defined for all real x. The +1 under the radical means the expression inside ln is always positive.
arccosh(x)
also: cosh⁻¹(x)
ln(x + √(x² − 1))
Requires x ≥ 1. At x = 1: arccosh(1) = ln(1 + 0) = 0. Only the principal (positive) branch.
arctanh(x)
also: tanh⁻¹(x)
(1/2) ln((1 + x)/(1 − x))
Requires |x| < 1. Vertical asymptotes at x = ±1. Useful in integration of 1/(1 − x²).
arccsch(x)
also: csch⁻¹(x)
ln(1/x + √(1/x² + 1))
Can be written as arcsinh(1/x). Defined for all nonzero real x.
arcsech(x)
also: sech⁻¹(x)
ln(1/x + √(1/x² − 1))
Can be written as arccosh(1/x). Requires 0 < x ≤ 1.
arccoth(x)
also: coth⁻¹(x)
(1/2) ln((x + 1)/(x − 1))
Requires |x| > 1. Related to arctanh by arccoth(x) = arctanh(1/x).
Why the logarithm forms matter in calculus
The logarithmic forms let you evaluate and differentiate inverse hyperbolic functions without a special table. For example, d/dx[arcsinh x] = d/dx[ln(x + √(x²+1))] = 1/√(x²+1). Similarly, d/dx[arctanh x] = 1/(1−x²). These are standard integration results: arcsinh x arises from ∫ 1/√(x²+1) dx, and arctanh x from ∫ 1/(1−x²) dx.
A flexible, inextensible chain hanging under gravity assumes the shape y = a·cosh(x/a). This appears in suspension bridges, power lines, spider webs, and the iconic Gateway Arch in St. Louis (an inverted weighted catenary).
The Lorentz boost (changing between inertial reference frames) can be parametrized with rapidity φ so that v/c = tanh φ, γ = cosh φ, and γv/c = sinh φ. This makes velocity composition simply additive in rapidity.
Hyperbolic functions appear as solutions to linear ODEs and PDEs with constant coefficients. For example, sinh and cosh solve y″ = y (compare: sin and cos solve y″ = −y). This makes them natural in boundary value problems.
∫ 1/√(x²+1) dx = arcsinh(x) + C; ∫ 1/√(x²−1) dx = arccosh(x) + C; ∫ 1/(1−x²) dx = arctanh(x) + C for |x| < 1. These parallel the trig substitution results and are often simpler to write.
tanh(x) is a classic activation function in neural networks. Its output is bounded in (−1, 1), it is smooth and differentiable everywhere, and its derivative sech²(x) is easy to compute. It is related to the sigmoid by: tanh(x) = 2·sigmoid(2x) − 1.
On a surface of constant negative curvature (hyperbolic geometry), the analog of circles uses cosh and sinh in the metric. The tractrix — the curve traced by a point on a taut string as the other end moves along a line — involves hyperbolic functions in its arc length formula.
Trigonometric functions are defined using the unit circle (x² + y² = 1), while hyperbolic functions are defined using the unit hyperbola (x² − y² = 1). Both share similar identities — compare cos²x + sin²x = 1 with cosh²x − sinh²x = 1 — but hyperbolic functions are not periodic and grow without bound. sinh is an odd function and cosh is an even function, just like sin and cos, but cosh never goes negative and sinh is unbounded.
A hanging flexible chain or cable under uniform gravity forms a curve called a catenary. This shape is described by the equation y = a·cosh(x/a), where a is a constant related to the tension and weight of the cable. The word 'catenary' comes from the Latin 'catena' meaning chain. The Gateway Arch in St. Louis is an upside-down catenary shape (technically a weighted catenary). Engineers and architects use cosh to model cables, power lines, and arched structures.
sinh(0) = (e⁰ − e⁻⁰)/2 = (1 − 1)/2 = 0. cosh(0) = (e⁰ + e⁻⁰)/2 = (1 + 1)/2 = 1. These are analogous to sin(0) = 0 and cos(0) = 1 in trigonometry. The point (cosh(0), sinh(0)) = (1, 0) lies on the unit hyperbola x² − y² = 1, just as (cos(0), sin(0)) = (1, 0) lies on the unit circle.
The derivative of tanh(x) is sech²(x). You can derive this using the quotient rule: tanh(x) = sinh(x)/cosh(x), so d/dx[tanh x] = (cosh x · cosh x − sinh x · sinh x)/cosh²x = (cosh²x − sinh²x)/cosh²x = 1/cosh²x = sech²x. This is analogous to d/dx[tan x] = sec²x in trigonometry.
Inverse hyperbolic functions can be expressed as logarithms: arcsinh(x) = ln(x + √(x² + 1)) for all real x; arccosh(x) = ln(x + √(x² − 1)) for x ≥ 1; arctanh(x) = (1/2)ln((1 + x)/(1 − x)) for |x| < 1. These logarithmic forms are useful in integration — for example, ∫ 1/√(x²+1) dx = arcsinh(x) + C = ln(x + √(x²+1)) + C.
No — regular tangent has vertical asymptotes and is unbounded, while tanh is bounded between −1 and 1. As x → +∞, tanh(x) → 1; as x → −∞, tanh(x) → −1. The graph of tanh is a smooth S-curve (sigmoid) with horizontal asymptotes at y = ±1. This bounded, S-shaped behavior makes tanh useful as an activation function in neural networks and as a model for saturation effects in physics.
In special relativity, the Lorentz transformation — which relates observations between two inertial frames moving at relative velocity v — can be written using hyperbolic functions. If we define rapidity φ by tanh(φ) = v/c, then the Lorentz factor is cosh(φ) and the velocity factor is sinh(φ). This hyperbolic parametrization makes the composition of velocities equivalent to adding rapidities, which is simpler than using the relativistic velocity addition formula directly.
This is one of the key differences from trig: d/dx[cos x] = −sin x, but d/dx[cosh x] = +sinh x. You can verify from the definitions: d/dx[cosh x] = d/dx[(eˣ + e⁻ˣ)/2] = (eˣ − e⁻ˣ)/2 = sinh x. The positive sign arises because both eˣ and e⁻ˣ contribute positively to the derivative of cosh, whereas in cosine the oscillating nature introduces the negative sign.
arcsin, arccos, arctan — domain restrictions, composition identities, right triangle method
eˣ, aˣ — the building blocks that define sinh and cosh
Pythagorean, double angle, sum/difference — compare with hyperbolic identities
Evaluating limits of hyperbolic functions, L'Hôpital's rule applications
Interactive problems covering definitions, identities, derivatives, and inverse hyperbolic functions — with step-by-step solutions. Free to try.
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