Hyperbolic Functions

Exponential functions related to the hyperbola.

Hyperbolic Functions

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

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

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

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

Values for Hyperbolic Functions

Hyperbolic Functions
x	e^x	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) = log_e(x + (x² + 1)^1/2) (4)

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

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