Hyperbolic Cosine CalculatorFree online math calculator — instant results, no sign-up
Calculate the hyperbolic cosine of a real number.
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About the Hyperbolic Cosine Calculator
The Hyperbolic Cosine Calculator evaluates cosh(x) for a real number using the average of two exponential terms, e^x and e^(-x). With the default input of 1, it returns Result = 1.5430806348. Unlike ordinary cosine, hyperbolic cosine is not periodic and is always at least 1 for real inputs.
- Real number
- The dimensionless real value x at which to evaluate hyperbolic cosine; the default is 1, and no degree conversion is applied.
Formula and method
Result = cosh(x) = (e^x + e^(-x)) / 2. For x = 1: cosh(1) = (e^1 + e^(-1)) / 2 ≈ 1.5430806348.
- xThe dimensionless real number entered into the calculator.
- eEuler's number, the base of the natural exponential function.
- coshThe hyperbolic cosine function.
- ResultThe dimensionless value of cosh(x).
Evaluate e raised to the input x and e raised to its negative, add the two values, and divide the sum by 2. For the default x = 1, this gives (e^1 + e^(-1)) / 2, which rounds to 1.5430806348 at ten decimal places.
Worked example
These are the calculator's starting values and the results it actually produces for them.
Inputs
- Real number1
Results
- Result1.5430806348
How to use the Hyperbolic Cosine Calculator
- 1Enter a value in the Real number field, or keep the default value of 1.
- 2Treat the input as dimensionless; do not convert it from degrees.
- 3Run the calculation to evaluate cosh(x) = (e^x + e^(-x)) / 2.
- 4Read the Result row: with the default input of 1, the displayed result is 1.5430806348.
Reading your result
The Result row reports the hyperbolic cosine of the entered real number, with no units. For the default input of 1, Result = 1.5430806348 is a rounded decimal approximation. A valid real-input result is at least 1; it equals 1 only when the input is zero and increases as the input's absolute value increases.
Limitations
This tool is intended for real inputs, not complex numbers or angle-unit conversions. Displayed decimals are rounded, and finite numerical precision can hide small differences near zero. Very large input magnitudes can exceed the numerical range of the implementation because hyperbolic cosine grows exponentially.
Common questions
Is hyperbolic cosine the same as ordinary cosine?+
No. Hyperbolic cosine uses (e^x + e^(-x)) / 2, is not periodic, and has a real-input range of [1, ∞). Ordinary cosine is periodic and stays between -1 and 1.
Does a negative input produce a negative hyperbolic cosine?+
No. Hyperbolic cosine is even: cosh(-x) = cosh(x). Changing the sign of the input swaps the two exponential terms but leaves their average unchanged.
Should I enter degrees or radians for hyperbolic cosine?+
Enter a dimensionless real value directly. This calculator evaluates the exponential definition of cosh and does not perform a degree-to-radian conversion.
Why does the default input of 1 give 1.5430806348?+
Substituting x = 1 gives cosh(1) = (e^1 + e^(-1)) / 2. Averaging these exponential terms and rounding to ten decimal places yields 1.5430806348.
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.