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Article 11 — Appendix A.31

tanh or th — hyperbolic tangent function

Category. Mathematics.

Abstract. Hyperbolic tangent: definition, graph, properties and identities.

References. This article is a part of scientific calculator Li-L, scientific calculator Li-X, scientific calculator Li-Lc and scientific calculator Li-Xc products.

1. Definition

Hyperbolic tangent is defined as

tanhx ≡ (ex − ex) /(ex + ex)

2. Graph

Hyperbolic tangent is antisymmetric function defined everywhere on real axis. Its graph is depicted below — fig. 1.

Fig. 1. Graph of the hyperbolic tangent function y = tanh x Fig. 1. Graph of the hyperbolic tangent function y = tanhx.

Function codomain is limited to the range (−1, 1).

3. Identities

Base:

tanh2x + sech2x = 1

By definition:

tanhx ≡ sinhx /coshx ≡ 1 /cothx

Property of antisymmetry:

tanh−x = −tanhx

Half-argument:

tanh(x/2) = (coshx − 1) /sinhx
tanh(x/2) = sinhx /(1 + coshx)
tanhx = 2 tanh(x/2) /[1 + tanh2(x/2)]

Double argument:

tanh(2x) = 2 tanhx /(tanh2x + 1)

Triple argument:

tanh(3x) = (tanh3x + 3 tanhx) /(3 tanh2x + 1)

Quadruple argument:

tanh(4x) = (4 tanh3x + 4 tanhx) /(tanh4x + 6 tanh2x + 1)

Power reduction:

tanh2x = (cosh(2x) − 1) /(cosh(2x) + 1)
tanh3x = (sinh(3x) − 3 sinhx) /(cosh(3x) + 3 coshx)
tanh4x = (cosh(4x) − 4 cosh(2x) + 3) /(cosh(4x) + 4 cosh(2x) + 3)
tanh5x = (sinh(5x) − 5 sinh(3x) + 10 sinhx) /(cosh(5x) + 5 cosh(3x) + 10 coshx)

Sum and difference of arguments:

tanh(x + y) = (tanhx + tanhy) /(1 + tanhx tanhy)
tanh(xy) = (tanhx − tanhy) /(1 − tanhx tanhy)

Product:

tanhx tanhy = [cosh(x + y) − cosh(xy)] /[cosh(x + y) + cosh(xy)]

Sum:

tanhx + tanhy = sinh(x + y) /(coshx coshy)
tanhx − tanhy = sinh(xy) /(coshx coshy)

4. Support

Hyperbolic tangent function tanh or th is supported in:

Hyperbolic tangent function of the complex argument tanh or th is supported in:

5. How to use

To calculate hyperbolic tangent of the number:

tanh(-1);

To calculate hyperbolic tangent of the current result:

tanh(Rslt);

To calculate hyperbolic tangent of the angle x in memory:

tanh(Mem[x]);