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Article 11 — Appendix A.5

arcosh or arch — arc-hyperbolic cosine function

Category. Mathematics.

Abstract. Arc-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

Arc-hyperbolic cosine is inverse of hyperbolic cosine function. With the help of natural logarithm it can be represented as:

arcoshx ≡ ln[x + √(x2 − 1)]

2. Graph

Arc-hyperbolic cosine is monotone function defined in the range [1, +∞). Its graph is depicted below — fig. 1.

Fig. 1. Graph of the arc-hyperbolic cosine function y = arcosh x Fig. 1. Graph of the arc-hyperbolic cosine function y = arcoshx.

Function codomain is non-negative part of real axis: [0, +∞).

3. Identities

Reciprocal argument:

arcosh(1/x) = arsechx

Sum and difference:

arcoshx + arcoshy = arcosh{xy + √[(x2 − 1)(y2 − 1)]}
arcoshx − arcoshy = arcosh{xy − √[(x2 − 1)(y2 − 1)]}
arsinhx + arcoshy = arsinh{xy + √[(x2 + 1)(y2 − 1)]} = arcosh[y√(x2 + 1) + x√(y2 − 1)]

4. Support

Arc-hyperbolic cosine function arcosh or arch is supported in:

Arc-hyperbolic cosine function of the complex argument arcosh or arch is supported in:

5. How to use

To calculate arc-hyperbolic cosine of the number:

arcosh(2);

To calculate arc-hyperbolic cosine of the current result:

arcosh(Rslt);

To calculate arc-hyperbolic cosine of the number x in memory:

arcosh(Mem[x]);