Compute sinh(x), cosh(x), tanh(x), coth(x), sech(x), csch and their inverses. Enter a number or a function value to solve for x.
This smart calculator computes hyperbolic functions such as sinh(x), cosh(x), tanh(x), coth(x), sech(x), csch(x) as well as their inverse hyperbolic functions instantly with accurate numeric results.
Easily select any function from the dropdown menu.
As soon as you enter a value, the calculator automatically computes the result without needing a submit button.
Fast, smooth, and beginner-friendly.
The calculator uses exponential definitions such as:
Inverse functions are calculated using logarithmic formulas.
The calculator automatically detects undefined or invalid values.
1️⃣ Select a hyperbolic or inverse hyperbolic function from the dropdown menu.
2️⃣ Enter a numeric value in the input field.
3️⃣ The calculator applies the correct mathematical formula:
4️⃣ If the input falls outside the valid domain, the calculator shows “invalid input” or “undefined”.
5️⃣ The result is displayed instantly in the format:
function(value) = result
Hyperbolic functions are analogs of trigonometric functions but for a hyperbola rather than a circle. They are widely used in calculus, physics, engineering, and complex analysis. This calculator computes both the basic hyperbolic functions sinh(x), cosh(x), tanh(x), coth(x), sech(x), csch(x) and their inverse counterparts arsinh(x), arcosh(x), artanh(x), arcoth(x), arsech(x), arcsch(x).
Each function can be expressed using exponential formulas. These formulas allow precise numeric computation and help handle special cases for undefined or invalid inputs.
sinh(x) – Hyperbolic sine: sinh(x) = (eˣ − e⁻ˣ) ÷ 2
cosh(x) – Hyperbolic cosine: cosh(x) = (eˣ + e⁻ˣ) ÷ 2
tanh(x) – Hyperbolic tangent: tanh(x) = sinh(x) ÷ cosh(x)
coth(x) – Hyperbolic cotangent: coth(x) = cosh(x) ÷ sinh(x)
sech(x) – Hyperbolic secant: sech(x) = 1 ÷ cosh(x)
csch(x) – Hyperbolic cosecant: csch(x) = 1 ÷ sinh(x)
arsinh(x): arsinh(x) = ln(x + √(x² + 1))
arcosh(x): arcosh(x) = ln(x + √(x² − 1)), x ≥ 1
artanh(x): artanh(x) = ½ ln((1 + x)/(1 − x)), |x| < 1
arcoth(x): arcoth(x) = ½ ln((x + 1)/(x − 1)), |x| > 1
arsech(x): arsech(x) = ln((1 + √(1 − x²))/x), 0 < x ≤ 1
arcsch(x): arcsch(x) = ln((1/x) + √(1 + 1/x²)), x ≠ 0
Using these formulas, the calculator evaluates inputs instantly. It checks for special cases such as division by zero or invalid domains, ensuring accurate results. For instance, coth(0) and csch(0) are undefined because they involve division by zero, while arcosh(x) is only valid for x ≥ 1.
These hyperbolic functions are essential in solving differential equations, modeling exponential growth, describing catenary curves, and in many physics and engineering problems. Mastering their formulas helps in both computational and theoretical understanding.
The inverse hyperbolic formulas also allow solving for x when given a function value. For example, if sinh(x) = 2, using arsinh formula:
x = arsinh(2) = ln(2 + √(2² + 1)) ≈ 1.4436
This systematic approach ensures clarity, accuracy, and quick calculations, making this calculator a powerful tool for students, engineers, and professionals dealing with hyperbolic functions.
If you select:
sinh(x)
And enter:
x = 1
The calculator computes:
sinh(1) ≈ 1.1752
Using the formula (e¹ − e⁻¹) / 2.
Clear answers about computing hyperbolic and inverse hyperbolic functions quickly and accurately.
It calculates the values of hyperbolic functions such as sinh(x), cosh(x), tanh(x), coth(x), sech(x), csch(x) and their inverse functions including arsinh(x), arcosh(x), artanh(x), arcoth(x), arsech(x), and arcsch(x).
It uses the exponential definitions: sinh(x) = (eˣ − e⁻ˣ) / 2 and cosh(x) = (eˣ + e⁻ˣ) / 2. Other functions like tanh(x) are derived from these formulas.
Inverse hyperbolic functions are computed using logarithmic formulas. For example, arsinh(x) = ln(x + √(x² + 1)) and artanh(x) = ½ ln((1 + x) / (1 − x)).
Yes. It automatically detects values outside the valid mathematical domain. For example, arcosh(x) is only defined for x ≥ 1, and artanh(x) is only valid when −1 < x < 1. If an invalid value is entered, the calculator displays “invalid input” or “undefined.”
No. The calculator works in real time. As soon as you enter a value or change the selected function, the result updates automatically.
Some functions are undefined at 0. For example, coth(0) and csch(0) are undefined because they involve division by zero. The calculator correctly displays “undefined” in such cases.
Yes. It is ideal for algebra, calculus, and engineering students who need quick and accurate hyperbolic function calculations for homework, quizzes, and exam preparation.
Yes. The calculator is fully responsive and works smoothly on smartphones, tablets, laptops, and desktop computers.
Yes. It is completely free to use with no sign-up or hidden fees required.
You can click the “Clear” button to remove the current input or use the “Reload Calculator” button to reset the function selection and start fresh.
Explore more algebra calculators, tools, and guides to simplify polynomial, equation, and expression problems quickly and accurately.