Trigonometrie plana si sferica

Trigonometrie plana si sferica

Functii hiperbolice

Functiile hiperbolice sunt: sinus hiperbolic (sh), cosinus hiperbolic (ch), tangenta hiperbolica (th) si cotangenta hiperbolica (cth)

Formule pentru functiile hiperbolice:

@d\operatorname{ch}^2x-\operatorname{sh}^2x=1@d

@d\operatorname{ch}(x\pm y)=\operatorname{ch} x\operatorname{ch} y\pm\operatorname{sh} x\operatorname{sh} y@d

@d\operatorname{sh}(x\pm y)=\operatorname{sh} x\operatorname{ch} y\pm\operatorname{ch} x\operatorname{sh} y@d

@d\operatorname{th}(x\pm y)=\displaystyle\frac{\operatorname{th} x\pm\operatorname{th} y}{1\pm\operatorname{th} x\operatorname{th} y}@d

@d\operatorname{cth}(x\pm y)=\displaystyle\frac{1\pm\operatorname{cth} x\operatorname{cth} y}{\operatorname{cth} x\pm\operatorname{cth} y}@d

@d\operatorname{ch}2x=\operatorname{ch}^2x+\operatorname{sh}^2x@d

@d\operatorname{sh}2x=2\operatorname{sh} x\operatorname{ch} x@d

@d\operatorname{th}2x=\displaystyle\frac{2\operatorname{th} x}{1+\operatorname{th}^2x}@d

@d\operatorname{cth}2x=\displaystyle\frac{1+\operatorname{cth}^2x}{2\operatorname{cth} x}@d

@d\operatorname{sh} x\pm\operatorname{sh} y=\displaystyle2\operatorname{sh}\frac{x\pm y}{2}\operatorname{ch}\frac{x\mp y}{2}@d

@d\operatorname{ch} x+\operatorname{ch} y=\displaystyle2\operatorname{ch}\frac{x+y}{2}\operatorname{ch}\frac{x-y}{2}@d

@d\operatorname{ch} x-\operatorname{ch} y=\displaystyle2\operatorname{sh}\frac{x+y}{2}\operatorname{sh}\frac{x-y}{2}@d

@d\operatorname{th} x\pm\operatorname{th} y=\displaystyle\frac{\operatorname{sh}(x\pm y)}{\operatorname{ch} x\operatorname{ch} y}@d

Functiile hiperbolice si inversele lor sunt derivabile pe domeniile lor de definitie si derivatele lor sunt:

@d(\operatorname{sh} x)'=\operatorname{ch} x;\ (\operatorname{ch} x)'=\operatorname{sh} x@d

@d(\operatorname{th} x)'=\displaystyle\frac{1}{\operatorname{ch}^2x};\ (\operatorname{cth} x)'=\displaystyle\frac{1}{\operatorname{sh}^2x}@d

@d(\operatorname{argsh}x)'=\displaystyle\frac{1}{\sqrt{x^2+1}}@d

@d(\operatorname{argch_+}x)'=\displaystyle\frac{1}{\sqrt{x^2-1}},\ x>1@d

@d(\operatorname{argth}x)'=\displaystyle\frac{1}{1-x^2},\ |x|<1@d

Dezvoltarile in serii de puteri ale functiilor hiperbolice sunt:

@d\operatorname{sh} x=\displaystyle\frac{x}{1!}+\frac{x^3}{3!}+\dots+\frac{x^{2n+1}}{(2n+1)!}+\dots,\forall x\in\mathbb{R}@d

@d\operatorname{ch} x=\displaystyle1+\frac{x^2}{2!}+\frac{x^4}{4!}+\dots+\frac{x^{2n}}{(2n)!}+\dots,\forall x\in\mathbb{R}@d

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