What is the hyperbolic cosine (cosh)?

cosh(x) is defined as (e^x + e^-x) / 2, where e is Euler's number (~2.71828). It is analogous to the regular cosine function but based on the hyperbola rather than the circle.

What is the minimum value of cosh(x)?

cosh(0) = 1 is the minimum value. For all other real x, cosh(x) > 1 because the function is always greater than or equal to 1.

How does cosh differ from cos?

cos(x) is bounded between -1 and 1 and is periodic. cosh(x) is always >= 1 and grows without bound as |x| increases. They are related through complex numbers: cosh(x) = cos(ix).

Why would I use degrees instead of radians?

Radians are the standard unit for hyperbolic functions in mathematics. However, if your source data uses degrees (e.g., from an angle measurement), this calculator converts them to radians automatically before computing cosh.

Can cosh return a negative value?

No. cosh(x) is always >= 1 for all real numbers x. It equals exactly 1 only when x = 0, and increases toward infinity as x moves away from zero in either direction.

What is the inverse of cosh?

The inverse hyperbolic cosine is arccosh (or cosh^-1), defined for x >= 1. It equals ln(x + sqrt(x^2 - 1)).

Hyperbolic Cosine Calculator

Enter a number in radians or degrees to instantly compute its hyperbolic cosine (cosh).

Input

Value of x

Batch Mode

Enter one value per line (uses selected unit above).

x	cosh(x)

Quick Reference

x (rad)	cosh(x)

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Summary

The hyperbolic cosine (cosh) is a fundamental mathematical function defined as cosh(x) = (e^x + e^-x) / 2. It appears in catenary curves (hanging chains), special relativity, and differential equations. This calculator accepts input in radians or degrees and returns the exact cosh value with full precision.

How it works

Enter the value of x in the input field.
Select whether your input is in radians or degrees.
The calculator instantly computes cosh(x) using the formula (e^x + e^-x) / 2.
View the result along with the intermediate e^x and e^-x values.
Use the batch mode to calculate cosh for a list of values at once.

Use cases

Solving catenary equations for hanging cables and suspension bridges.
Computing hyperbolic identities in advanced mathematics coursework.
Verifying cosh values when studying special relativity or Lorentz factors.
Checking results from differential equations involving hyperbolic functions.
Engineering calculations for arch structures and flexible beams.
Quick reference for students learning about hyperbolic vs. trigonometric functions.

Frequently Asked Questions

Last updated: 2026-05-29 · Reviewed by Nham Vu