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Article 11 — Appendix A.14

cosh or ch — hyperbolic cosine function

Category. Mathematics.

Abstract. Hyperbolic cosine: 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 cosine is defined as

coshx ≡ (ex + ex) /2

2. Graph

Hyperbolic cosine is symmetric function defined everywhere on real axis. Its chain-line graph is depicted below — fig. 1.

Fig. 1. Graph of the hyperbolic cosine function y = cosh x Fig. 1. Graph of the hyperbolic cosine function y = coshx.

Function codomain is range [1, +∞).

3. Identities

Base:

cosh2x − sinh2x = 1

Connection to exponential function:

sinhx + coshx = ex
coshx − sinhx = ex

By definition:

coshx ≡ 1 /sechx

Property of symmetry:

cosh−x = coshx

Half-argument:

cosh(x/2) = √[(coshx + 1) /2]
coshx = [1 + tanh2(x/2)] /[1 − tanh2(x/2)]

Doulbe argument:

cosh(2x) = sinh2x + cosh2x
cosh(2x) = 2 cosh2x − 1
cosh(2x) = 2 sinh2x + 1

Triple argument:

cosh(3x) = 4 cosh3x − 3 coshx

Quadruple argument:

cosh(4x) = 8 cosh4x − 8 cosh2x + 4 = 8 cosh2x sinh2x + 1 = 8 sinh4x + 8 sinh2x + 1

Power reduction:

cosh2x = (cosh(2x) + 1) /2
cosh3x = (cosh(3x) + 3 coshx) /4
cosh4x = (cosh(4x) + 4 cosh(2x) + 3) /8
cosh5x = (cosh(5x) + 5 cosh(3x) + 10 coshx) /16

Sum and difference of arguments:

cosh(x + y) = coshx coshy + sinhx sinhy
cosh(xy) = coshx coshy − sinhx sinhy

Product-to-sum:

coshx coshy = [cosh(x + y) + cosh(xy)] /2
sinhx coshy = [sinh(x + y) + sinh(xy)] /2

Sum-to-product:

coshx + coshy = 2 cosh[(x + y) /2] cosh[(xy) /2]
coshx − coshy = 2 sinh[(x + y) /2] sinh[(xy) /2]

4. Support

Hyperbolic cosine function cosh or ch is supported in:

Hyperbolic cosine function of the complex argument cosh or ch is supported in:

5. How to use

To calculate hyperbolic cosine of the number:

cosh(-1);

To calculate hyperbolic cosine of the current result:

cosh(Rslt);

To calculate hyperbolic cosine of the number x in memory:

cosh(Mem[x]);