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• A-Level Further Maths

# Hyperbolic Functions

Hyperbolic functions are similar to trigonometric functions, but are defined in terms of exponentials. There are three fundamental hyperbolic functions: sinh, cosh and tanh:

Similarly, the reciprocal of each function exists:

### Hyperbolic Graphs

For any value of x, sinh(-x) = -sinh(x)
• y = sinh(x) has no asymptotes

For any value of x, cosh(-x) = -cosh(x)
• y = cosh(x) never goes below y=1

• y = tanh(x) has asymptotes at y = ±1, and always stays between these

## Inverse Hyperbolic Functions

Just like sin, cos and tan, the hyperbolic functions have inverses, arcsinh, arcosh and artanh:

The graphs of these are their respective reflections in the line y=x: