Hyperbolic Functions: Definitions

The hyperbolic functions sinh, cosh, and tanh are analogues of the trigonometric functions sin, cos, and tan, but they are defined using the exponential function rather than the unit circle. They arise naturally in many areas of mathematics and physics.

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1 Definitions

sinh x = (e^x - e^(-x)) / 2 (pronounced "shine" or "sinch")

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

tanh x = sinh x / cosh x = (e^x - e^(-x)) / (e^x + e^(-x)) (pronounced "tanch" or "than")

The reciprocal functions are:

cosech x = 1 / sinh x (x is not 0)

sech x = 1 / cosh x

coth x = cosh x / sinh x (x is not 0)

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2 Key Values

x	sinh x	cosh x	tanh x
0	0	1	0
1	(e - e^(-1))/2 approx 1.175	(e + e^(-1))/2 approx 1.543	approx 0.762
-1	approx -1.175	approx 1.543	approx -0.762

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3 Graphs

y = sinh x