# Hyperbolic Functions

## Exponential functions related to the hyperbola

### Hyperbolic Functions

Hyperbolic functions are combinations of the exponentials ex and e-x.

sinh x = (ex - e-x) / 2                  (1)

cosh x = (ex + e-x) / 2                   (2)

tanh x = sinh x / cosh x = (ex - e-x) / (ex + e-x)                          (3)

### Values for Hyperbolic Functions

xsinh xcosh xtanh x
0 0.00000 1.0000 0.00000
0.1 0.10017 1.00500 0.09967
0.3 0.30452 1.04534 0.29131
0.4 0.41075 1.08107 0.37995
0.5 0.52110 1.12763 0.46212
0.6 0.63665 1.18547 0.53705
0.7 0.75858 1.25517 0.60437
0.8 0.88811 1.33743 0.66404
0.9 1.02652 1.43309 0.71630
1.0 1.17520 1.54308 0.76159
1.1 1.33565 1.66852 0.80050
1.2 1.50946 1.81066 0.83365
1.3 1.69838 1.97091 0.86172
1.4 1.90430 2.15090 0.88535
1.5 2.12928 2.35241 0.90515
1.6 2.37557 2.57746 0.92167
1.7 2.64563 2.82832 0.93541
1.8 2.94217 3.10747 0.94681
1.9 3.26816 3.41773 0.95624
2.0 3.62686 3.76220 0.96403
2.5 6.05020 6.13229 0.98661
3.0 10.0179 10.0677 0.99505
3.5 16.5426 16.5728 0.99818
4.0 27.2899 27.3082 0.99933
4.5 45.0030 45.0141 0.99975
5.0 74.2032 74.2099 0.99991

### Inverse Hyperbolic Functions

sinh-1 x = loge(x + (x2 + 1)1/2)                 (4)

cosh-1 x = loge(x ± (x2 + 1)1/2)   x ≥ 1                     (5)

tanh-1 x = 1/2 loge((1 + x) / (1 - x))   x2 ‹ 1                        (6)

## Related Topics

• Mathematics - Mathematical rules and laws - numbers, areas, volumes, exponents, trigonometric functions and more

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