Hyperbolic Functions
Exponential functions related to the hyperbola.
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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
| x | ex | e-x | sinh x | cosh x | tanh 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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