Free · Geometry and function relationships

Hyperbolic Sine and Related Functions

Hyperbolic sine and cosine use exponential growth rather than periodic angles. The reciprocal functions csch and coth are undefined at zero; inverse branches use real principal values.

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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, sinhx = 1.175201193643801456882381850595600815155717981334095870229565413, coshx = 1.543080634815243778477905620757061682601529112365863704737402215, tanhx = 0.7615941559557648881194582826047935904127685972579365515968105001.

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

Worked example

Enter these known values and leave the other values blank.

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

Assumptions and limitations

  • Hyperbolic sine and cosine use exponential growth rather than periodic angles. The reciprocal functions csch and coth are undefined at zero; inverse branches use real principal values.
  • Clear the unknown field before solving. Locks preserve known quantities; an inverse may have multiple solutions and this model selects its principal branch.
  • Numerical rounding does not describe measurement uncertainty. Extremely large arguments or near-singular inverses can exceed the available precision; use moderate values for interpretable finite results.

Common questions

How do I enter a consistent set of values?

Hyperbolic sine and cosine use exponential growth rather than periodic angles. The reciprocal functions csch and coth are undefined at zero; inverse branches use real principal values.

Can I calculate a missing input?

Clear that field and provide enough independent known quantities. Output-only fields and nonunique inverse branches may need a different mode.

References

Finite real inverse hyperbolic values require tanh in (−1,1), positive sech, |coth|>1 and nonzero csch. Source bounds include infinite endpoints and omit reciprocal-domain checks.

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