Last updated: July 31, 2026
Hyperbolic Functions Calculator
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Formula
sinh(x)=(e^x−e^−x)/2, cosh(x)=(e^x+e^−x)/2, tanh(x)=sinh(x)/cosh(x)
Where:
- x=Real input value
- e=Euler's number
Worked Examples
x = 1
Standard positive input.
- 1Compute sinh(1) ≈ 1.175201.
- 2Compute cosh(1) ≈ 1.543081.
- 3Divide to get tanh(1) ≈ 0.761594.
- 4Take reciprocals for coth, sech, and csch where defined.
x = 0
Important undefined reciprocal case.
- 1sinh(0) = 0.
- 2cosh(0) = 1.
- 3tanh(0) = 0.
- 4coth and csch are undefined because they divide by zero.
x = -2
Check odd/even symmetry.
- 1sinh is odd, so sinh(-2) = -sinh(2).
- 2cosh is even, so cosh(-2) = cosh(2).
- 3Reciprocal functions follow from those base values.
Introduction
Hyperbolic functions appear in calculus, differential equations, complex analysis, catenary curves, and special relativity. This calculator evaluates the six standard hyperbolic functions directly and flags undefined reciprocal values instead of throwing errors.
Core Definitions
Hyperbolic functions are built from exponentials rather than circular geometry.
sinh uses (e^x − e^-x)/2.
cosh uses (e^x + e^-x)/2.
tanh is sinh divided by cosh.
sech and csch are reciprocals.
coth is the reciprocal of tanh.
Symmetry Properties
Several quick checks come from odd and even behavior.
sinh is odd.
cosh is even.
tanh is odd.
coth is odd when defined.
sech is even.
Undefined Cases
Reciprocal functions cannot divide by zero, so x = 0 needs special handling.
sinh(0) = 0.
tanh(0) = 0.
csch(0) is undefined.
coth(0) is undefined.
The calculator returns the string "undefined" for those outputs.
How the Calculator Computes
The implementation uses JavaScript's built-in hyperbolic functions and rounds to six decimals.
Math.sinh computes sinh(x).
Math.cosh computes cosh(x).
Math.tanh computes tanh(x).
Reciprocals are derived from those values.
Non-finite x is rejected before any math runs.
Input Rules
Any finite real number can be evaluated directly.
- 1
Positive inputs are valid.
- 2
Negative inputs are valid.
- 3
Zero is valid.
- 4
Infinity is rejected.
- 5
NaN is rejected.
Where Hyperbolic Functions Appear
These functions describe many natural growth and geometry patterns.
Catenary curves
Differential equations
Complex analysis identities
Signal and systems models
Relativity formulas
How to Read the Outputs
Look at the base functions first, then interpret reciprocal outputs.
Start with sinh and cosh.
Use tanh for bounded ratios.
Check whether coth is defined.
Check whether csch is defined.
Use sech to see reciprocal damping.
FAQs
What does this calculator return?
It returns sinh, cosh, tanh, coth, sech, and csch for a single real input x.
Why are coth and csch undefined at x = 0?
Because tanh
- 1= 0 and sinh
- 2= 0, so their reciprocals would divide by zero
Can I enter negative values of x?
Yes. All finite real numbers, including negatives, are allowed.
How are values rounded?
Numeric outputs are rounded to six decimal places.
Does the calculator use radians or degrees?
Hyperbolic functions take a plain real number x, not an angle measure, so there is no degree/radian mode.
What is sech(x)?
sech(x) is the reciprocal of cosh(x), so sech(x) = 1 / cosh(x).
What is the difference between trig and hyperbolic functions?
Trigonometric functions are linked to circles, while hyperbolic functions are built from exponentials and hyperbolas.
How can I verify x = 1 manually?
Compute e and 1/e, plug them into the sinh and cosh definitions, then derive the remaining outputs from ratios and reciprocals.