Hyperbolic Sine CalculatorFree online math calculator — instant results, no sign-up
Calculate the hyperbolic sine of a real number.
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About the Hyperbolic Sine Calculator
The Hyperbolic Sine Calculator evaluates sinh(x) for a real number using the difference between two exponential terms. With the default input of 1, it returns 1.1752011936. Unlike ordinary sine, hyperbolic sine is not periodic and is not restricted to values between −1 and 1.
- Real number
- The dimensionless real value x at which to evaluate hyperbolic sine; the default is 1, and no degree conversion is required.
Formula and method
sinh(x) = (e^x − e^(−x)) / 2; for x = 1, sinh(1) = (e^1 − e^(−1)) / 2 ≈ 1.1752011936.
- xThe dimensionless real-number input.
- eEuler’s number, the base of the natural exponential function.
- sinh(x)The dimensionless hyperbolic sine of x, shown as the result.
Evaluate e raised to x and e raised to −x, subtract the second value from the first, and divide the difference by 2. For the default input x = 1, this gives sinh(1) ≈ 1.1752011936 when rounded to ten decimal places. Numerical implementations may use an equivalent built-in sinh function to improve accuracy and efficiency.
Worked example
These are the calculator's starting values and the results it actually produces for them.
Inputs
- Real number1
Results
- Result1.1752011936
How to use the Hyperbolic Sine Calculator
- 1Enter your value in the Real number field, or leave the default value of 1.
- 2Use the value directly as a dimensionless input; do not convert it from degrees.
- 3Run the calculation to evaluate sinh(x) = (e^x − e^(−x)) / 2.
- 4Read the Result row; for the default input of 1, it shows 1.1752011936.
Reading your result
The Result row gives sinh(x), not ordinary sine or inverse hyperbolic sine. For the default input of 1, Result = 1.1752011936 is a rounded, dimensionless value. The result has the same sign as the input, and a magnitude greater than 1 is valid.
Limitations
This calculator accepts real inputs, not complex numbers. Hyperbolic sine is mathematically defined for every real number, but finite-precision arithmetic limits the accuracy and range of a numerical result. Very large input magnitudes can cause overflow, and directly subtracting the exponential terms near zero can lose precision. Displayed digits are rounded rather than an exact symbolic value.
Common questions
Is sinh(x) the same as sin(x)?+
No. Hyperbolic sine uses real exponentials, whereas ordinary sine describes a circular trigonometric relationship. Hyperbolic sine is unbounded and nonperiodic, so sinh(1) = 1.1752011936 to ten decimal places is valid even though it exceeds 1.
Should I enter degrees or radians?+
Enter a dimensionless real number directly. This calculation has no degree-based angle mode and does not apply a degree-to-radian conversion.
How does hyperbolic sine handle negative inputs?+
Hyperbolic sine is an odd function: sinh(−x) = −sinh(x). Reversing the input’s sign reverses the result’s sign without changing its magnitude.
Does this calculator find the inverse hyperbolic sine?+
No. It calculates sinh(x). To recover the input from a hyperbolic sine value, use the inverse function asinh, also called arsinh; using a rounded result may not recover the input exactly.
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.