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Article 11 — Appendix A.27

sinh or sh — hyperbolic sine function

Category. Mathematics.

Abstract. Hyperbolic sine: 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 sine is defined as

sinhx ≡ (ex − e−x) /2

2. Graph

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

Fig. 1. Graph of the hyperbolic sine function y = sinh x Fig. 1. Graph of the hyperbolic sine function y = sinhx.

Function codomain is entire real axis.

3. Identities

Base:

cosh2x − sinh2x = 1

Connection to exponential function:

sinhx + coshx = ex
coshx − sinhx = e−x

By definition:

sinhx ≡ 1 /cschx

Property of antisymmetry:

sinh−x = −sinhx

Half-argument:

sinh(x/2) = √[(coshx − 1) /2]
sinhx = 2 tanh(x/2) /[1 − tanh2(x/2)]

Double argument:

sinh(2x) = 2 sinhx coshx

Triple argument:

sinh(3x) = 4 sinh3x + 3 sinhx

Quadruple argument:

sinh(4x) = 4 sinh3x coshx + 4 sinhx cosh3x

Power reduction:

sinh2x = (cosh(2x) − 1) /2
sinh3x = (sinh(3x) − 3 sinhx) /4
sinh4x = (cosh(4x) − 4 cosh(2x) + 3) /8
sinh5x = (sinh(5x) − 5 sinh(3x) + 10 sinhx) /16

Sum and difference of arguments:

sinh(x + y) = sinhx coshy + coshx sinhy
sinh(x − y) = sinhx coshy − coshx sinhy

Product-to-sum:

sinhx sinhy = [cosh(x + y) − cosh(x − y)] /2
sinhx coshy = [sinh(x + y) + sinh(x − y)] /2

Sum-to-product:

sinhx + sinhy = 2 sinh[(x + y) /2] cosh[(x − y) /2]
sinhx − sinhy = 2 sinh[(x − y) /2] cosh[(x + y) /2]

4. Support

Hyperbolic sine function sinh or sh is supported in:

Hyperbolic sine function of the complex argument sinh or sh is supported in:

5. How to use

To calculate hyperbolic sine of the number:

sinh(-1);

To calculate hyperbolic sine of the current result:

sinh(Rslt);

To calculate hyperbolic sine of the number x in memory:

sinh(Mem[x]);