Whether you're a high school student or a proficient mathematician, this hyperbolic functions calculator will surely be useful. It computes the values of the six basic hyperbolic functions — sinh, cosh, tanh, coth, sech, and csch — all in the blink of an eye. You can also use it to calculate the inverse hyperbolic functions.
What are hyperbolic functions?
Hyperbolic functions are analogical to the trigonometric functions you probably already know, such as sine or cosine. Visit our sine calculator and cosine calculator to explore those topics further.
What's the difference, then? If you plot points with coordinates (cos x, sin x) in a Cartesian coordinate system, they form a circle. But if you plot points with coordinates (cosh x, sinh x), they trace out a hyperbola — which is exactly where the name hyperbolic comes from.
How to calculate sinh, cosh, and tanh
We can define all of these functions in terms of exponential functions. If you're unsure what these are, head over to our exponent calculator for a more detailed explanation.
The hyperbolic sine is defined as:
sinh x = ½ (eˣ − e⁻ˣ)
The formula for the hyperbolic cosine is quite similar:
cosh x = ½ (eˣ + e⁻ˣ)
You can calculate tanh x, coth x, sech x, and csch x (hyperbolic tangent, cotangent, secant, and cosecant) analogically to trigonometry:
| Function | Definition |
|---|---|
| sinh x | (eˣ − e⁻ˣ) / 2 |
| cosh x | (eˣ + e⁻ˣ) / 2 |
| tanh x | sinh x / cosh x = (eˣ − e⁻ˣ) / (eˣ + e⁻ˣ) |
| coth x | cosh x / sinh x = 1 / tanh x (x ≠ 0) |
| sech x | 1 / cosh x |
| csch x | 1 / sinh x (x ≠ 0) |
A key identity ties them together, mirroring the Pythagorean identity in trigonometry: cosh²x − sinh²x = 1.
Inverse hyperbolic functions
The inverse hyperbolic functions (also called area hyperbolic functions) reverse the operation: given a value y, they return the argument x. Because hyperbolic functions are built from exponentials, their inverses can be written with the natural logarithm:
| Inverse | Formula | Domain |
|---|---|---|
| arcsinh y | ln(y + √(y² + 1)) | all real y |
| arccosh y | ln(y + √(y² − 1)) | y ≥ 1 |
| arctanh y | ½·ln[(1 + y) / (1 − y)] | −1 < y < 1 |
| arccoth y | ½·ln[(y + 1) / (y − 1)] | |y| > 1 |
| arcsech y | ln[(1 + √(1 − y²)) / y] | 0 < y ≤ 1 |
| arccsch y | ln(1/y + √(1/y² + 1)) | y ≠ 0 |
Example: how to use the hyperbolic functions calculator
- Keep the mode on Hyperbolic functions and type a value of x — for example, 1.
- Click Calculate. The tool instantly returns sinh 1 ≈ 1.1752, cosh 1 ≈ 1.5431, tanh 1 ≈ 0.7616, and the remaining three reciprocal functions.
- To go the other way, switch the mode to Inverse hyperbolic, pick a function (say arcsinh), enter the value, and read off the argument x.
FAQs
What is the difference between hyperbolic and trigonometric functions?
Trigonometric functions parametrize a circle (x² + y² = 1), while hyperbolic functions parametrize a hyperbola (x² − y² = 1). Trig functions are periodic; hyperbolic functions are not. sinh and cosh grow without bound as x increases.
What are the values of sinh, cosh, and tanh at 0?
At x = 0: sinh 0 = 0, cosh 0 = 1, and tanh 0 = 0. Note that coth 0 and csch 0 are undefined because they divide by sinh 0 = 0.
Is cosh always positive?
Yes. Since cosh x = (eˣ + e⁻ˣ)/2 and both exponentials are positive, cosh x ≥ 1 for every real x, reaching its minimum of 1 at x = 0. This is why sech x = 1/cosh x is always defined and lies in (0, 1].
What is the range of tanh?
The hyperbolic tangent maps every real number into the open interval (−1, 1). As x → +∞, tanh x → 1, and as x → −∞, tanh x → −1. This S-shaped curve is widely used as an activation function in machine learning.
How do inverse hyperbolic functions relate to logarithms?
Because hyperbolic functions are combinations of eˣ, solving for x turns them into natural logarithms. For instance, arcsinh y = ln(y + √(y² + 1)). The calculator uses these closed forms and verifies each answer by feeding it back through the direct function.