sinh or sh — hyperbolic sine function

1. Definition

Hyperbolic sine is defined as

sinhx ≡ (e^x − e^−x) /2

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 = sinhx.

Function codomain is entire real axis.

Base:

cosh²x − sinh²x = 1

sinhx + coshx = e^x
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]

Hyperbolic sine derivative:

sinh′x = coshx

Indefinite integral of the hyperbolic sine:

∫ sinhx dx = coshx + C

where C is an arbitrary constant.

To calculate hyperbolic sine of the number:

sinh(−1);

To get hyperbolic sine of the complex number:

sinh(−1+i);

To get hyperbolic sine of the current result:

sinh(rslt);

To get hyperbolic sine of the number z in calculator memory:

sinh(mem[z]);

Hyperbolic sine of the real argument is supported in free version of the Librow calculator.

Hyperbolic sine of the complex argument is supported in professional version of the Librow calculator.