Free · Hyperbolic functions

Hyperbolic Cosine and Related Functions

Evaluate six real hyperbolic functions and recover a real argument using documented inverse branches and domain checks.

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Calculator inputs

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How to use this calculator

  1. Enter the known values in the units shown. Results update as you type.
  2. Where results are editable, change one to solve backwards. Lock a value to hold it fixed.
  3. Use the worked example to check the method. Reset restores the starting fields.

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Formula and method

A hyperbolic cosine of 1.25 gives the nonnegative argument ln(2), with sinh = 0.75, tanh = 0.6 and sech = 0.8.

cosh(x) = (exp(x)+exp(−x))/2; sinh(x) = (exp(x)−exp(−x))/2; tanh = sinh/cosh; coth = 1/tanh; sech = 1/cosh; csch = 1/sinh.

Worked example

Enter these known values and leave the other values blank.

cosh(x)
1.25
x
0.6931472
sinh(x)
0.75
tanh(x)
0.6
coth(x)
1.6666667
sech(x)
0.8
csch(x)
1.3333333

Assumptions and limitations

  • Hyperbolic functions use a dimensionless real argument, not a degree angle. Cosh and sech are even; their inverses select a nonnegative argument.
  • Sinh, tanh, coth and csch retain signed real inverse branches. Finite inputs require cosh ≥ 1, −1 < tanh < 1, 0 < sech ≤ 1, |coth| > 1, and csch ≠ 0.
  • Coth and csch have a pole at x = 0. Their unavailable values do not change cosh(0) = 1, sinh(0) = 0, tanh(0) = 0 or sech(0) = 1. Clear an unknown before using an inverse.
  • The evaluator has finite numerical resolution. At large absolute arguments, tanh and coth can round to plus or minus one; resulting endpoint warnings indicate a numerical resolution limit, not a physical or mathematical pole at that argument.

Common questions

Does hyperbolic cosine use degrees?

Hyperbolic functions use a dimensionless real argument, not a degree angle. Cosh and sech are even; their inverses select a nonnegative argument.

Does an inverse always have a unique answer?

Cosh and sech are even, so a nonzero argument and its negative share the same value. This calculator selects the nonnegative principal inverse.

Why can a large argument show an endpoint warning?

Finite numerical precision can round tanh or coth to plus or minus one when the true value is extremely close. This is a numerical resolution limit; it does not mean the real function has a pole there.

References

Use stable real inverse hyperbolic identities, including signed sinh/csch and factored half-angle cosh/sech branches near zero; retain nonnegative principal cosh/sech inverse branches. At exactly zero, represent only the two undefined reciprocal forward values as absent; preserve their inverse controls and every defined function. Enforce strict finite-real inverse ranges at poles and endpoints. Zero is valid for x, sinh and tanh, but not for csch or sech. Replace external curve image with an original accessible description of symmetry, minimum and inverse ambiguity.

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