Cosh Calculator
Calculate hyperbolic cosine (cosh) and its inverse (arcosh) with step-by-step solutions
Calculate Hyperbolic Cosine
Results
Manual Calculation Verification
Additional Information
Step-by-Step Solution
Common Values and Examples
| x | cosh(x) | sinh(x) | tanh(x) |
|---|---|---|---|
| 0 | 1.000 | 0.000 | 0.000 |
| 0.5 | 1.128 | 0.521 | 0.462 |
| 1 | 1.543 | 1.175 | 0.762 |
| 1.5 | 2.352 | 2.129 | 0.905 |
| 2 | 3.762 | 3.627 | 0.964 |
| 3 | 10.068 | 10.018 | 0.995 |
Cosh Properties
Definition
cosh(x) = (e^x + e^(-x)) / 2
Domain
(-∞, ∞) for cosh(x)
Range
[1, ∞) for cosh(x)
Even Function
cosh(-x) = cosh(x)
Derivative
d/dx[cosh(x)] = sinh(x)
Quick Reference
cosh(0) = 1 (minimum value)
cosh(x) ≥ 1 for all real x
cosh²(x) - sinh²(x) = 1
Describes catenary curves
arcosh(x) domain: [1, ∞)
Understanding the Hyperbolic Cosine Function
What is cosh(x)?
The hyperbolic cosine function, denoted as cosh(x), is defined as the average of the exponential function e^x and its reciprocal e^(-x). Despite its name, it's not directly related to the circular cosine function but shares some similar properties.
Key Properties
- •Even function: cosh(-x) = cosh(x)
- •Always positive: cosh(x) ≥ 1 for all real x
- •Not periodic: Unlike cos(x), cosh(x) doesn't repeat
- •Increasing: For x > 0, cosh(x) increases rapidly
Mathematical Definition
cosh(x) = (e^x + e^(-x)) / 2
Inverse Function
arcosh(x) = ln(x + √(x² - 1))
Domain: x ≥ 1
Hyperbolic Identity
cosh²(x) - sinh²(x) = 1
Similar to cos²(x) + sin²(x) = 1, but with a minus sign
Real-World Applications
Catenary Curves
The shape of a hanging chain or cable forms a catenary curve, described mathematically by the hyperbolic cosine function. This appears in suspension bridges and power lines.
Physics Applications
Hyperbolic functions appear in special relativity, describing the relationship between space and time coordinates, and in solutions to wave equations and heat transfer problems.
Engineering
Used in structural engineering for analyzing flexible cables and chains, and in electrical engineering for modeling transmission lines and signal propagation.
Architecture
Architects use catenary curves in designing structurally efficient arches and domes, as they naturally distribute weight and minimize material stress.