Hyperbolic Functions Calculator
Calculate hyperbolic and inverse hyperbolic functions: sinh, cosh, tanh, and more
Calculate Hyperbolic Functions
Domain: All real numbers
Function Values
Formulas Used:
sinh(x) = (e^x - e^(-x)) / 2
cosh(x) = (e^x + e^(-x)) / 2
tanh(x) = sinh(x) / cosh(x)
Hyperbolic Function Formulas
Forward Functions
Inverse Functions
Common Values
At x = 0
sinh(0) = 0
cosh(0) = 1
tanh(0) = 0
At x = 1
sinh(1) ≈ 1.175
cosh(1) ≈ 1.543
tanh(1) ≈ 0.762
At x = ln(2)
sinh(ln(2)) = 0.75
cosh(ln(2)) = 1.25
tanh(ln(2)) = 0.6
Function Properties
Parity
sinh, tanh: odd functions
cosh, sech: even functions
Range
sinh: (-∞, ∞)
cosh: [1, ∞)
tanh: (-1, 1)
Identities
cosh²(x) - sinh²(x) = 1
1 - tanh²(x) = sech²(x)
Quick Tips
Hyperbolic functions relate to hyperbolas like trig functions to circles
Unlike trig functions, hyperbolic functions are not periodic
cosh(x) ≥ 1 for all real x
tanh(x) approaches ±1 as x approaches ±∞
Understanding Hyperbolic Functions
What are Hyperbolic Functions?
Hyperbolic functions are analogues of trigonometric functions but are based on hyperbolas rather than circles. They are defined using exponential functions and have many similar properties to trigonometric functions.
Key Differences from Trigonometric Functions
- •Not periodic (don't repeat values)
- •Based on exponential functions rather than circular motion
- •Points (cosh x, sinh x) form a hyperbola, not a circle
Applications
Physics
Special relativity, wave equations, heat transfer
Engineering
Catenary curves (hanging cables), structural analysis
Mathematics
Complex analysis, differential equations, calculus
Computer Science
Machine learning, neural networks (activation functions)