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The Art of Interface |
Article 11 — Appendix A.27sinh or sh hyperbolic sine functionCategory. 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. DefinitionHyperbolic sine is defined as sinhx ≡ (ex − e−x) /22. GraphHyperbolic sine is antisymmetric function defined everywhere on real axis. Its graph is depicted below — fig. 1. ![]() Function codomain is entire real axis. 3. IdentitiesBase: cosh2x − sinh2x = 1Connection to exponential function: sinhx + coshx = excoshx − sinhx = e−x By definition: sinhx ≡ 1 /cschxProperty of antisymmetry: sinh−x = −sinhxHalf-argument: sinh(x/2) = √[(coshx − 1) /2]sinhx = 2 tanh(x/2) /[1 − tanh2(x/2)] Double argument: sinh(2x) = 2 sinhx coshxTriple argument: sinh(3x) = 4 sinh3x + 3 sinhxQuadruple argument: sinh(4x) = 4 sinh3x coshx + 4 sinhx cosh3xPower reduction: sinh2x = (cosh(2x) − 1) /2sinh3x = (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 sinhysinh(x − y) = sinhx coshy − coshx sinhy Product-to-sum: sinhx sinhy = [cosh(x + y) − cosh(x − y)] /2sinhx 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. SupportHyperbolic sine function sinh or sh is supported in: Hyperbolic sine function of the complex argument sinh or sh is supported in:
5. How to useTo calculate hyperbolic sine of the number:
To calculate hyperbolic sine of the current result:
To calculate hyperbolic sine of the number x in memory:
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