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The Art of Interface |
Article 11 — Appendix A.14cosh or ch hyperbolic cosine functionCategory. 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. DefinitionHyperbolic cosine is defined as coshx ≡ (ex + e−x) /22. GraphHyperbolic cosine is symmetric function defined everywhere on real axis. Its chain-line graph is depicted below — fig. 1. ![]() Function codomain is range [1, +∞). 3. IdentitiesBase: cosh2x − sinh2x = 1Connection to exponential function: sinhx + coshx = excoshx − sinhx = e−x By definition: coshx ≡ 1 /sechxProperty of symmetry: cosh−x = coshxHalf-argument: cosh(x/2) = √[(coshx + 1) /2]coshx = [1 + tanh2(x/2)] /[1 − tanh2(x/2)] Doulbe argument: cosh(2x) = sinh2x + cosh2xcosh(2x) = 2 cosh2x − 1 cosh(2x) = 2 sinh2x + 1 Triple argument: cosh(3x) = 4 cosh3x − 3 coshxQuadruple argument: cosh(4x) = 8 cosh4x − 8 cosh2x + 4 = 8 cosh2x sinh2x + 1 = 8 sinh4x + 8 sinh2x + 1Power reduction: cosh2x = (cosh(2x) + 1) /2cosh3x = (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 sinhycosh(x − y) = coshx coshy − sinhx sinhy Product-to-sum: coshx coshy = [cosh(x + y) + cosh(x − y)] /2sinhx coshy = [sinh(x + y) + sinh(x − y)] /2 Sum-to-product: coshx + coshy = 2 cosh[(x + y) /2] cosh[(x − y) /2]coshx − coshy = 2 sinh[(x + y) /2] sinh[(x − y) /2] 4. SupportHyperbolic cosine function cosh or ch is supported in: Hyperbolic cosine function of the complex argument cosh or ch is supported in:
5. How to useTo calculate hyperbolic cosine of the number:
To calculate hyperbolic cosine of the current result:
To calculate hyperbolic cosine of the number x in memory:
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