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# 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

Hyperbolic Functions
xexe-xsinh xcosh xtanh x
0.00 1 1 0 1 0
0.05 1.0513 0.9512 0.050021 1.00125 0.0499584
0.10 1.1052 0.9048 0.10017 1.00500 0.099668
0.15 1.1618 0.8607 0.15056 1.01127 0.14889
0.20 1.2214 0.8187 0.20134 1.02007 0.19738
0.25 1.2840 0.7788 0.25261 1.03141 0.24492
0.30 1.3499 0.7408 0.30452 1.04534 0.29131
0.35 1.4191 0.7047 0.35719 1.06188 0.33638
0.40 1.4918 0.6703 0.41075 1.08107 0.37995
0.45 1.5683 0.6376 0.46534 1.10297 0.42190
0.50 1.6487 0.6065 0.52110 1.12763 0.46212
0.55 1.7333 0.5769 0.57815 1.15510 0.50052
0.60 1.8221 0.5488 0.63665 1.18547 0.53705
0.65 1.9155 0.5220 0.69675 1.21879 0.57167
0.70 2.0138 0.4966 0.75858 1.25517 0.60437
0.75 2.1170 0.4724 0.82232 1.29468 0.63515
0.80 2.2255 0.4493 0.88811 1.33743 0.66404
0.85 2.3396 0.4274 0.95612 1.38353 0.69107
0.90 2.4596 0.4066 1.0265 1.43309 0.71630
0.95 2.5857 0.3867 1.0995 1.48623 0.73978
1.0 2.7183 0.3679 1.1752 1.54308 0.76159
1.1 3.0042 0.3329 1.3356 1.66852 0.80050
1.2 3.3201 0.3012 1.5095 1.81066 0.83365
1.3 3.6693 0.2725 1.6984 1.97091 0.86172
1.4 4.0552 0.2466 1.9043 2.15090 0.88535
1.5 4.4817 0.2231 2.1293 2.35241 0.90515
1.6 4.9530 0.2019 2.3756 2.57746 0.92167
1.7 5.4739 0.1827 2.6456 2.82832 0.93541
1.8 6.0496 0.1653 2.9422 3.10747 0.94681
1.9 6.6859 0.1496 3.2682 3.41773 0.95624
2.0 7.3891 0.1353 3.6269 3.76220 0.96403
2.5 12.182 0.08208 6.0502 6.13229 0.98661
3.0 20.086 0.04979 10.018 10.068 0.99505
3.5 33.115 0.03020 16.543 16.573 0.99818
4.0 54.598 0.01832 27.290 27.308 0.99933
4.5 90.017 0.01111 45.003 45.014 0.99975
5.0 148.41 0.006738 74.203 74.210 0.99991
5.5 244.69 0.004087 122.34 122.35 0.99997
6.0 403.43 0.002479 201.71 201.72 0.99999
6.5 665.14 0.001503 332.57 332.57 1.00000
7.0 1096.6 0.0009119 548.32 548.32 1.00000
7.5 1808.0 0.0005531 904.02 904.02 1.00000
8.0 2981.0 0.0003355 1490.5 1490.5 1.00000
8.5 4914.8 0.0002035 2457.4 2457.4 1.00000
9.0 8103.1 0.00012341 4051.5 4051.5 1.00000
9.5 13360 0.00007485 6679.9 6679.9 1.00000
10 22026 0.000045400 11013 11013 0.999999996

### 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)

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## Citation

• The Engineering ToolBox (2010). Hyperbolic Functions. [online] Available at: https://www.engineeringtoolbox.com/hyperbolic-functions-d_1663.html [Accessed Day Month Year].

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6.3.17

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