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Real Hyperbolic Functions & Principal Inverses

Evaluate sinh, cosh, tanh and their reciprocals, or recover a real argument from an admissible function value.

  • Formula & worked example
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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.

Private by default

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

Formula and method

At x = ln 2, the exponential pair is 2 and 1/2. Therefore sinh x = 3/4, cosh x = 5/4, tanh x = 3/5 and sech x = 4/5.

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.

Real argument x
0.6931471805599453094172321214581765680755001343602552541206800095
Hyperbolic sine sinh x
0.75
Hyperbolic cosine cosh x
1.25
Hyperbolic tangent tanh x
0.6
Hyperbolic cotangent coth x
1.6666667
Hyperbolic secant sech x
0.8
Hyperbolic cosecant csch x
1.3333333

Assumptions and limitations

  • Supported arguments range from −10000 through 10000 to keep high-precision calculation responsive. Larger magnitudes are rejected before function evaluation.
  • The argument is a dimensionless real number; these functions do not use a degree/radian selector. The reciprocal functions are coth = 1/tanh, sech = 1/cosh and csch = 1/sinh.
  • For finite real inverses, cosh is at least 1, tanh lies strictly between −1 and 1, absolute coth exceeds 1, sech lies above 0 and at most 1, and csch is nonzero.
  • Cosh and sech are even. Their inverse alone cannot recover the sign of x; the calculator selects the nonnegative principal argument. The other listed inverses preserve the appropriate sign.
  • At x = 0, sinh and tanh are 0 while cosh and sech are 1. Coth and csch are undefined and remain blank.
  • A displayed value rounded to 1 for tanh or 0 for sech is not a valid finite inverse input. Use sufficient unrounded precision when calculating backwards.

Common questions

Can a cosh value tell whether x was negative?

No. cosh x equals cosh(−x). This inverse uses the nonnegative principal value; a known sign must come from other information.

Why can I not enter tanh x = 1?

For finite real x, tanh x stays strictly below 1 in magnitude. The endpoint is approached only as x tends to infinity.

References

Bookify corrects the real inverse branches, avoids cancellation in signed inverse sinh/csch, and leaves singular reciprocal outputs blank at zero. Bookify validates the finite real inverse ranges and bounds hyperbolic arguments to minus 10000 through 10000 to keep calculations responsive.

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