The hyperbolic functions calculator above evaluates sinh, cosh, tanh, coth, sech and csch at any real number, together with the inverse functions arsinh, arcosh and artanh. It also shows the exponential decomposition each value is built from and runs the fundamental identity cosh²x − sinh²x = 1 as a live check, so you can see the structure rather than just the digits. Every result comes from the exponential definitions directly, which keeps the arithmetic honest at both small and large arguments.
Arb Digital built this page because hyperbolic functions get taught briefly, used rarely, and then turn up unannounced in engineering, physics and machine learning. The shape of a hanging cable, the Lorentz transformations of special relativity, the response of a transmission line, and the activation function that dominated a decade of neural networks are all hyperbolic. The names look like trigonometry and the identities look almost like trigonometry, which is exactly what makes them easy to get subtly wrong.
What This Hyperbolic Functions Calculator Does
It takes a single real argument and returns all six hyperbolic functions plus one selected inverse. The reciprocal functions coth, sech and csch are reported in the summary line under the headline number, since they are simply one divided by tanh, cosh and sinh respectively and the tool guards each division rather than printing an infinity. The identity panel confirms that cosh²x − sinh²x equals one, which is the hyperbolic analogue of sin²θ + cos²θ = 1 and the single most useful sanity check in the subject.
This is a different tool from our trigonometric functions calculator, which handles the circular functions sine, cosine and tangent on an angle measured in degrees or radians. The two families share notation and share many identity patterns, but they are not variants of one another: circular functions parameterise a circle x² + y² = 1 and are periodic, while hyperbolic functions parameterise the right branch of the hyperbola x² − y² = 1 and are not periodic over the reals at all. If you need raw exponential or logarithmic evaluation rather than the hyperbolic packaging, the exponent calculator and logarithm calculator cover those directly.
How to Use It
- Type a real number for x. There is no unit selector because there is no angle involved. A value of 1 means the number one, not one radian.
- Choose which inverse to display. arsinh accepts anything, arcosh needs x at least 1, and artanh needs x strictly between −1 and 1.
- Check the identity panel. If cosh²x − sinh²x is not showing 1, either the argument is enormous enough to hit floating-point limits or something upstream is wrong.
- Read the exponential bars. They show how much of the answer comes from e^x versus e^−x, which explains the behaviour at both ends of the range.
- Push x to extremes deliberately. Try 0.001 and then 20 to see the small-argument and large-argument regimes described below.
The Definitions and How They're Calculated
Everything follows from two exponential definitions: sinh x = (e^x − e^−x) ÷ 2 and cosh x = (e^x + e^−x) ÷ 2. Tangent follows as their ratio, tanh x = sinh x ÷ cosh x = (e^x − e^−x) ÷ (e^x + e^−x), and the remaining three are reciprocals: coth = 1/tanh, sech = 1/cosh, csch = 1/sinh. The NIST Digital Library of Mathematical Functions section on hyperbolic definitions gives the same six along with the relationships to their circular counterparts.
Take x = 1. Then e^1 = 2.718282 and e^−1 = 0.367879. Their difference over two is 1.175201, which is sinh 1; their sum over two is 1.543081, which is cosh 1; and the ratio is 0.761594, which is tanh 1. Squaring the last two and subtracting gives 2.381098 − 1.381098 = 1 exactly, as the identity demands. The reciprocals are coth 1 = 1.313035, sech 1 = 0.648054 and csch 1 = 0.850918. Those are the numbers on screen when the page loads.
The inverses have closed forms in logarithms rather than needing numerical inversion: arsinh x = ln(x + √(x² + 1)), arcosh x = ln(x + √(x² − 1)) for x ≥ 1, and artanh x = ½·ln((1 + x) ÷ (1 − x)) for |x| < 1. The DLMF section on inverse hyperbolic functions sets out these logarithmic forms and the branch cuts that matter once you move into the complex plane. Note the naming convention: the prefix is "ar" for area, not "arc", because the hyperbolic argument corresponds to twice the area of a hyperbolic sector rather than to an arc length.
The Identities, and Where They Differ From Trigonometry
The hyperbolic identities mirror the circular ones almost exactly, with a pattern of sign flips known as Osborn's rule: take a trigonometric identity, swap each function for its hyperbolic counterpart, and reverse the sign of every term containing a product of two sines. So sin²θ + cos²θ = 1 becomes cosh²x − sinh²x = 1, and the addition formula sin(A+B) = sin A cos B + cos A sin B carries over unchanged as sinh(x+y) = sinh x cosh y + cosh x sinh y, while cos(A+B) = cos A cos B − sin A sin B becomes cosh(x+y) = cosh x cosh y + sinh x sinh y with the sign flipped.
The double-angle results follow the same pattern: sinh 2x = 2 sinh x cosh x, and cosh 2x = cosh²x + sinh²x = 2cosh²x − 1 = 1 + 2sinh²x. That last chain is the hyperbolic mirror of the relationships our half angle calculator handles on the circular side. The structural differences worth remembering are that cosh x is never less than 1, tanh x is confined strictly between −1 and 1, and none of these functions repeats — there is no period, no unit circle, and no quadrant sign rules to worry about.
Why a Hanging Chain Is a Cosh and Not a Parabola
Suspend a uniform flexible chain from two points and it settles into a curve called a catenary, whose equation is y = a·cosh(x/a). The reason is a force balance: at every point the horizontal component of tension is constant, while the vertical component must support the weight of the chain hanging below that point. Because the weight is proportional to arc length rather than to horizontal distance, the differential equation that falls out has cosh as its solution, and the parameter a is the horizontal tension divided by the weight per unit length.
A parabola looks extremely similar and is often mistaken for it. The distinction is physical rather than aesthetic: a parabola is what you get when the load is distributed evenly along the horizontal, which is why the main cables of a loaded suspension bridge really are close to parabolic while an unloaded chain, a power line between pylons, or a free-hanging rope is a catenary. Increase the tension and a grows, cosh(x/a) flattens toward its own quadratic approximation, and the two curves converge — which is precisely why a taut cable is hard to tell apart from a parabola by eye and why sag calculations on long spans need the hyperbolic form.
Rapidity: the Hyperbolic Angle in Special Relativity
Velocities in special relativity do not add. Two speeds of 0.75c do not combine to 1.5c; the relativistic composition law gives 0.96c, and nothing can exceed c. This awkwardness disappears if you stop working with velocity and use rapidity instead, defined by φ = artanh(v/c). Rapidities add in the ordinary way, so a Lorentz boost along a single axis is a hyperbolic rotation by an angle φ, and composing two boosts just means adding their rapidities.
The rest of the structure follows immediately. The Lorentz factor γ is cosh φ, the quantity γv/c is sinh φ, and the transformation matrix has exactly the form of a rotation with the trigonometric functions replaced by hyperbolic ones — which is the sense in which spacetime geometry is hyperbolic rather than circular. The speed limit becomes a statement about range: tanh is asymptotic to 1, so an infinite rapidity still yields v strictly below c. Try artanh with x = 0.75 in the tool to get the rapidity of a 0.75c boost, then double it and take tanh to recover the composed velocity of 0.96c.
tanh in Machine Learning and Signal Processing
Beyond physics, tanh earns its keep as a squashing function. It maps the whole real line into the open interval (−1, 1), it is smooth and monotonic everywhere, it is centred on zero, and its derivative is the clean expression 1 − tanh²x. That combination made it the standard hidden-layer activation in neural networks before rectified linear units took over, and it is still the default in recurrent architectures where a bounded, zero-centred output keeps the state from drifting.
Its weakness is visible in the tool. Set x to 5 and tanh returns 0.999909; the derivative 1 − tanh²x is then about 0.00018. A gradient that small effectively stops learning, which is the vanishing-gradient problem in one line of arithmetic. The same saturation behaviour is exploited deliberately elsewhere: soft clipping in audio processing uses tanh precisely because it compresses loud signals smoothly instead of chopping their peaks off, producing gentle harmonic distortion rather than harsh clipping artefacts. The property that ruins deep gradients is the property that makes a pleasant limiter.
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Explore Content Marketing Talk To Our TeamCommon Mistakes to Avoid
- Treating x as an angle — hyperbolic functions take a plain real number, and switching a calculator to degree mode does nothing useful to them.
- Copying the trigonometric sign pattern — cosh²x − sinh²x = 1 has a minus where the circular identity has a plus, and cosh(x+y) has a plus where the circular version has a minus.
- Expecting periodicity — there is no repeating cycle over the real numbers, so there is no reference angle, no coterminal argument and no quadrant rule.
- Ignoring the inverse domains — arcosh needs an argument of at least 1 and artanh needs one strictly inside −1 to 1, and outside those ranges no real value exists.
- Reading a hanging cable as a parabola — the two curves are close under high tension but the catenary is the correct model for a free-hanging uniform chain, and sag estimates diverge on long spans.
Related Free Tools From Arb Digital
Evaluate the circular family with the trigonometric functions calculator, work with powers using the exponent calculator, invert an exponential with the logarithm calculator, rewrite a compound log with the logarithm expansion calculator. Everything we publish is listed on the free online tools hub.
Frequently Asked Questions
They are the six functions sinh, cosh, tanh, coth, sech and csch, defined from the exponential function. sinh x is (e to the x minus e to the minus x) over two, and cosh x is the same expression with a plus sign.
Circular functions parameterise the circle and repeat every full turn. Hyperbolic functions parameterise the right branch of a hyperbola, take a plain real number rather than an angle, and never repeat over the real numbers.
cosh squared x minus sinh squared x equals one. It is the hyperbolic counterpart of sin squared plus cos squared equals one, with a minus sign in place of the plus.
Because the prefix stands for area, not arc. The hyperbolic argument corresponds to twice the area of a sector of the hyperbola, whereas a circular argument corresponds to an arc length.
arsinh accepts every real number. arcosh requires an argument of one or greater. artanh requires an argument strictly between minus one and one, because tanh never actually reaches either endpoint.
Because its weight is distributed along its own length rather than along the horizontal. The resulting force balance produces the catenary y equals a times cosh of x over a, where a is horizontal tension divided by weight per unit length.
It is used as a bounded activation function in neural networks, as a soft clipping curve in audio processing, and in special relativity where the rapidity of a boost is artanh of velocity over the speed of light.
This page explains mathematical functions for educational purposes. Results are computed in double-precision floating point, so extremely large arguments will saturate and any engineering calculation should be checked against the tolerances of your own method.
