Free · Algebra and geometric models

Hyperbolic Tangent and Its Six-Function Family

Calculate hyperbolic functions and principal inverse values at a nonzero real argument.

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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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Inputs and results stay in this browser tab. Bookify does not upload or store the values you enter.

Formula and method

For x=1, hyperbolic tangent is approximately 0.761594 and cosh is approximately 1.543081.

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

Worked example

Enter these known values and leave the other values blank.

x
1
tanh(x)
0.7615942
coth(x)
1.3130353
sinh(x)
1.1752012
cosh(x)
1.5430806
csch(x)
0.8509181
sech(x)
0.6480543

Assumptions and limitations

  • The complete six-function family uses nonzero real x, because coth and csch divide by sinh x.
  • Tanh has range strictly between −1 and 1. Cosh and sech are even; inverse cosh/sech selects a nonnegative argument, so sign information needs an odd function.
  • The graph crosses zero, but x=0 is excluded when requesting all six reciprocal-family results.
  • The supported argument magnitude is at most 10000. For large arguments, finite precision may round tanh or coth to ±1; those are forward approximations, while an exactly entered inverse endpoint cannot give a finite argument.

Common questions

Why is zero excluded?

Tanh, sinh, cosh and sech are defined at zero, but coth and csch are not. This model returns the complete six-function family at nonzero arguments.

Which inverse sign is selected?

Cosh and sech do not distinguish x from −x. Their principal inverse returns a nonnegative x; an odd function preserves the sign.

References

The captured tanh inverse logs a negative square root. It is replaced by atanh(tanhx). Signed positive-log forms recover sinh and csch inverses without negative log arguments or subtractive cancellation for large negative arguments. The captured graph is replaced with an original textual account of odd symmetry, intercept, slope, monotonicity, range and asymptotes. Real inverse domains require a complete finite hyperbolic family. The argument range matches the existing local hyperbolic runtime bound. Finite known arguments can produce rounded tanh/coth endpoints; those forward approximations remain valid while standalone inverse endpoints remain invalid.

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