Věta. sinh⁡\sinh je prostá funkce.
Důkaz. ea−e−a2=eb−e−b2\frac{e^a - e^{-a}}{2} = \frac{e^b - e^{-b}}{2} ea−e−a=eb−e−be^a - e^{-a} = e^b - e^{-b} eaeb(ea−e−a)=eaeb(eb−e−b)e^a e^b (e^a - e^{-a}) = e^a e^b (e^b - e^{-b}) eb(e2a−1)=ea(e2b−1)e^b (e^{2a} - 1) = e^a (e^{2b} - 1) ebe2a−eb=eae2b−eae^b e^{2a} - e^b = e^a e^{2b} - e^a eaeb(ea−eb)+(ea−eb)=0e^a e^b (e^a - e^b) + (e^a - e^b) = 0 (ea−eb)(eaeb+1)⏟>1=0(e^a - e^b) \underbrace{(e^a e^b + 1)}_{> 1} = 0 ea−eb=0e^a - e^b = 0 a=ba = b
Věta. sinh⁡−1(x)=argsinh⁡(x)=ln⁡(x+x2+1)\sinh^{-1}(x) = \operatorname{argsinh}(x) = \ln\left(x + \sqrt{x^2 + 1}\right).
Důkaz. x=ey−e−y2x = \frac{e^y - e^{-y}}{2} 2x=ey−e−y2x = e^y - e^{-y} 2x=ey−1ey2x = e^y - \frac1{e^y} 2x=e2y−1ey2x = \frac{e^{2y} - 1}{e^y} (ey)2−2xey−1=0(e^y)^2 - 2xe^y - 1 = 0 ey=2x±4x2+42e^y = \frac{2x \pm \sqrt{4x^2 + 4}}{2} ey=x±x2+1e^y = x \pm \sqrt{x^2 + 1} ey=x+x2+1∵ey>0e^y = x + \sqrt{x^2 + 1} \because e^y > 0 y=ln⁡(x+x2+1)y = \ln\left(x + \sqrt{x^2 + 1}\right)
Věta. cosh⁡\cosh zúžená na R0+\mathbb R^+_0 je prostá funkce.
Důkaz. ea+e−a2=eb+e−b2\frac{e^a + e^{-a}}{2} = \frac{e^b + e^{-b}}{2} ea+e−a=eb+e−be^a + e^{-a} = e^b + e^{-b} eaeb(ea+e−a)=eaeb(eb+e−b)e^a e^b (e^a + e^{-a}) = e^a e^b (e^b + e^{-b}) eb(e2a+1)=ea(e2b+1)e^b (e^{2a} + 1) = e^a (e^{2b} + 1) ebe2a+eb=eae2b+eae^b e^{2a} + e^b = e^a e^{2b} + e^a eaeb(ea−eb)−(ea−eb)=0e^a e^b (e^a - e^b) - (e^a - e^b) = 0 (ea−eb)(eaeb−1)=0(e^a - e^b) (e^a e^b - 1) = 0 ea−eb=0∨eaeb=1e^a - e^b = 0 \vee e^a e^b = 1 a=b∨a=−ba = b \vee a = -b
Věta. Pro cosh⁡\cosh zúženou na R0+\mathbb R^+_0: cosh⁡−1(x)=argcosh⁡(x)=ln⁡(x+x2−1)\cosh^{-1}(x) = \operatorname{argcosh}(x) = \ln\left(x + \sqrt{x^2 - 1}\right).
Důkaz. x=ey+e−y2x = \frac{e^y + e^{-y}}{2} 2x=ey+e−y2x = e^y + e^{-y} 2x=ey+1ey2x = e^y + \frac1{e^y} 2x=e2y+1ey2x = \frac{e^{2y} + 1}{e^y} (ey)2−2xey+1=0(e^y)^2 - 2xe^y + 1 = 0 ey=2x±4x2−42e^y = \frac{2x \pm \sqrt{4x^2 - 4}}{2} ey=x±x2−1e^y = x \pm \sqrt{x^2 - 1} ey=x+x2−1∵ey≥1e^y = x + \sqrt{x^2 - 1} \because e^y \ge 1 y=ln⁡(x+x2−1)y = \ln\left(x + \sqrt{x^2 - 1}\right)
Věta. sinh⁡(2x)=2sinh⁡(x)cosh⁡(x)\sinh(2x) = 2 \sinh(x) \cosh(x)
Důkaz. e2x−e−2x2=(ex)2−(e−x)22=2ea−e−a2ea−e−a2=2sinh⁡(x)cosh⁡(x)\frac{e^{2x} - e^{-2x}}{2} = \frac{(e^x)^2 - (e^{-x})^2}{2} = 2 \frac{e^a - e^{-a}}{2} \frac{e^a - e^{-a}}{2} = 2 \sinh(x) \cosh(x)
Věta. cosh⁡(2x)=cosh⁡(x)2+sinh⁡(x)2\cosh(2x) = \cosh(x)^2 + \sinh(x)^2
Věta. cosh⁡(x)2−sinh⁡(x)2=1\cosh(x)^2 - \sinh(x)^2 = 1