Hyperbolic Tangent CalculatorFree online math calculator — instant results, no sign-up
Calculate the hyperbolic tangent of a real number.
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About the Hyperbolic Tangent Calculator
The Hyperbolic Tangent Calculator computes tanh(x) for a real-number input using the ratio of hyperbolic sine to hyperbolic cosine. With the default input of 1, it returns 0.761594156. Unlike ordinary tangent, hyperbolic tangent is defined for every real input and has values strictly between −1 and 1.
- Real number
- The dimensionless real value x at which to evaluate hyperbolic tangent; the default is 1, and no degree or radian setting is needed.
Formula and method
y = tanh(x) = sinh(x) / cosh(x) = (e^x − e^(−x)) / (e^x + e^(−x)); for x = 1, y = (e^1 − e^(−1)) / (e^1 + e^(−1)) ≈ 0.761594156.
- xThe dimensionless real-number input.
- yThe dimensionless hyperbolic tangent result.
- eEuler’s number, the base of the natural exponential function.
- tanhThe hyperbolic tangent function.
- sinhThe hyperbolic sine function, defined by sinh(x) = (e^x − e^(−x)) / 2.
- coshThe hyperbolic cosine function, defined by cosh(x) = (e^x + e^(−x)) / 2.
Evaluate hyperbolic tangent at the supplied real number. Mathematically, this means subtracting e^(−x) from e^x and dividing by their sum. For the default x = 1, this ratio rounds to 0.761594156 at nine decimal places. Numerically stable implementations may evaluate an equivalent expression rather than form both exponentials directly.
Worked example
These are the calculator's starting values and the results it actually produces for them.
Inputs
- Real number1
Results
- Result0.761594156
How to use the Hyperbolic Tangent Calculator
- 1Enter a finite real value in the Real number field, or keep the default value of 1.
- 2Treat the input as dimensionless; do not convert it between degrees and radians.
- 3Run the calculation to evaluate tanh(x).
- 4Read the Result row: with the default input of 1, it shows 0.761594156.
- 5Interpret the sign and magnitude as the hyperbolic tangent value, not as an angle.
Reading your result
The Result row reports tanh(x), a dimensionless number. For x = 1, Result = 0.761594156 is the rounded hyperbolic tangent value, not a percentage or an angle. The output has the same sign as the input, is zero when the input is zero, and approaches 1 or −1 as the input becomes increasingly positive or negative.
Limitations
This tool is for real inputs, not complex numbers. Displayed results are rounded, so they are not exact symbolic values. For finite real inputs, the mathematical result never equals −1 or 1, although rounding or finite-precision arithmetic can display these endpoints for large input magnitudes. Direct evaluation of the exponential formula can overflow for extreme inputs unless the implementation uses a numerically stable method.
Common questions
Does the Hyperbolic Tangent Calculator use degrees or radians?+
Neither angle mode is required. It evaluates tanh(x) for a dimensionless real input, rather than the ordinary tangent of an angle.
Why does the default input of 1 give 0.761594156?+
Substituting x = 1 into (e^x − e^(−x)) / (e^x + e^(−x)) gives tanh(1), which rounds to 0.761594156 at nine decimal places.
Can hyperbolic tangent exceed 1 or be undefined for a real input?+
No. For every finite real input, cosh(x) is positive, so the ratio is defined and lies strictly between −1 and 1. A displayed endpoint can result from rounding.
Is tanh the same as inverse tangent?+
No. tanh is hyperbolic tangent, defined as sinh(x) / cosh(x). Inverse ordinary tangent is arctan, while inverse hyperbolic tangent is artanh.
Does the calculator save my information?+
Calculations run in your browser. While signed in, inputs and results are automatically recorded in account history; saving a result creates a separate saved record.
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Results are estimates for informational purposes and are not financial, medical or legal advice.