Cyklometrická funkce

Cyklometrické funkce jsou inverzní zobrazení ke goniometrickým funkcím.

Mezi cyklometrické funkce patří:

• Arkus sinus – ${\displaystyle \arcsin }$
• Arkus kosinus – ${\displaystyle \arccos }$
• Arkus tangens – ${\displaystyle \mathrm {arctg} }$
• Arkus kotangens – ${\displaystyle \mathrm {arccotg} }$
• Arkus sekans – ${\displaystyle \operatorname {arcsec} }$
• Arkus kosekans – ${\displaystyle \operatorname {arccsc} }$

Aby mohla k libovolné funkci existovat inverzní funkce, daná funkce musí být prostá, to znamená, že různým dvěma prvkům musí přiřazovat dvě různé hodnoty. Protože jsou ale goniometrické funkce periodické, tzn. nejsou prosté, musíme nejprve ošetřit jejich definiční obor a také definiční obory goniometrických funkcí. To znamená, že vybereme jen tu podmnožinu definičního oboru dané geometrické funkce, na které je prostá.

${\displaystyle \arcsin(\sin x)=x}$, pokud platí ${\displaystyle \ |x|\leq {\frac {\pi }{2}}}$

${\displaystyle \sin(\arcsin x)=x}$, pokud platí ${\displaystyle \ |x|\leq 1}$

${\displaystyle \arccos(\cos x)=x}$, pokud platí ${\displaystyle \ 0\leq x\leq \pi }$

${\displaystyle \cos(\arccos x)=x}$, pokud platí ${\displaystyle \ |x|\leq 1}$

${\displaystyle \operatorname {arctg} (\operatorname {tg} x)=x}$, pokud platí ${\displaystyle \ |x|<{\frac {\pi }{2}}}$

${\displaystyle \operatorname {tg} (\operatorname {arctg} x)=x}$

${\displaystyle \operatorname {arccotg} (\operatorname {cotg} x)=x}$, pokud platí ${\displaystyle \ 0<x<\pi }$

${\displaystyle \operatorname {cotg} (\operatorname {arcotg} x)=x}$

${\displaystyle {\begin{aligned}\arcsin x&={\frac {\pi }{2}}-\arccos x=\operatorname {arctg} \left({\frac {x}{\sqrt {1-x^{2}}}}\right)={\frac {\pi }{2}}-\operatorname {arccotg} \left({\frac {x}{\sqrt {1-x^{2}}}}\right)\\[12pt]\arccos x&={\frac {\pi }{2}}-\arcsin x={\frac {\pi }{2}}-\operatorname {arctg} \left({\frac {x}{\sqrt {1-x^{2}}}}\right)=\operatorname {arccotg} \left({\frac {x}{\sqrt {1-x^{2}}}}\right)\\[12pt]\operatorname {arctg} x&=\arcsin \left({\frac {x}{\sqrt {1+x^{2}}}}\right)={\frac {\pi }{2}}-\arccos \left({\frac {x}{\sqrt {1+x^{2}}}}\right)={\frac {\pi }{2}}-\operatorname {arccotg} x\\[12pt]\operatorname {arccotg} x&={\frac {\pi }{2}}-\arcsin \left({\frac {x}{\sqrt {1+x^{2}}}}\right)=\arccos \left({\frac {x}{\sqrt {1+x^{2}}}}\right)={\frac {\pi }{2}}-\operatorname {arctg} x\end{aligned}}}$

Dále platí:

${\displaystyle \operatorname {arccotg} x={\begin{cases}\operatorname {arctg} \displaystyle {\frac {1}{x}}\,,&{\text{pokud platí }}x>0,\\[12pt]\pi +\operatorname {arctg} \displaystyle {\frac {1}{x}}\,,&{\text{pokud platí }}x<0.\end{cases}}}$

${\displaystyle {\begin{aligned}\arcsin(-x)&=-\arcsin x\\\arccos(-x)&=\pi -\arccos x\\\operatorname {arctg} (-x)&=-\operatorname {arctg} x\\\operatorname {arccotg} (-x)&=\pi -\operatorname {arccotg} x\end{aligned}}}$

${\displaystyle \arcsin x\,+\,\arcsin y={\begin{cases}\arcsin \left(x{\sqrt {1-y^{2}}}+y{\sqrt {1-x^{2}}}\right),&{\text{pokud platí }}xy\leq 0{\text{ nebo }}x^{2}+y^{2}\leq 1,\\[12pt]\pi -\arcsin \left(x{\sqrt {1-y^{2}}}+y{\sqrt {1-x^{2}}}\right),&{\text{pokud platí }}x>0,y>0,x^{2}+y^{2}>1,\\[12pt]-\pi -\arcsin \left(x{\sqrt {1-y^{2}}}+y{\sqrt {1-x^{2}}}\right),&{\text{pokud platí }}x<0,y<0,x^{2}+y^{2}>1.\end{cases}}}$

${\displaystyle \arcsin x\,-\,\arcsin y={\begin{cases}\arcsin \left(x{\sqrt {1-y^{2}}}-y{\sqrt {1-x^{2}}}\right),&{\text{pokud platí }}xy\geq 0{\text{ nebo }}x^{2}+y^{2}\leq 1,\\[12pt]\pi -\arcsin \left(x{\sqrt {1-y^{2}}}-y{\sqrt {1-x^{2}}}\right),&{\text{pokud platí }}x>0,y<0,x^{2}+y^{2}>1,\\[12pt]-\pi -\arcsin \left(x{\sqrt {1-y^{2}}}+y{\sqrt {1-x^{2}}}\right),&{\text{pokud platí }}x<0,y>0,x^{2}+y^{2}>1.\end{cases}}}$

${\displaystyle \arccos x\,+\,\arccos y={\begin{cases}\arccos \left(xy-{\sqrt {1-x^{2}}}\cdot {\sqrt {1-y^{2}}}\right),&{\text{pokud platí }}x+y\geq 0,\\[12pt]2\pi -\arccos \left(xy-{\sqrt {1-x^{2}}}\cdot {\sqrt {1-y^{2}}}\right),&{\text{pokud platí }}x+y<0.\end{cases}}}$

${\displaystyle \arccos x\,-\,\arccos y={\begin{cases}-\arccos \left(xy+{\sqrt {1-x^{2}}}\cdot {\sqrt {1-y^{2}}}\right),&{\text{pokud platí }}x\geq y,\\[12pt]\arccos \left(xy+{\sqrt {1-x^{2}}}\cdot {\sqrt {1-y^{2}}}\right),&{\text{pokud platí }}x<y.\end{cases}}}$

${\displaystyle \operatorname {arctg} x\,+\,\operatorname {arctg} y={\begin{cases}\operatorname {arctg} \left(\displaystyle {\frac {x+y}{1-xy}}\right),&{\text{pokud platí }}xy<1,\\[12pt]\pi +\operatorname {arctg} \left(\displaystyle {\frac {x+y}{1-xy}}\right),&{\text{pokud platí }}xy>1,x>0\\[12pt]-\pi +\operatorname {arctg} \left(\displaystyle {\frac {x+y}{1-xy}}\right),&{\text{pokud platí }}xy>1,x<0.\end{cases}}}$

${\displaystyle \operatorname {arctg} x\,-\,\operatorname {arctg} y={\begin{cases}\operatorname {arctg} \left(\displaystyle {\frac {x-y}{1+xy}}\right),&{\text{pokud platí }}xy>-1,\\[12pt]\pi +\operatorname {arctg} \left(\displaystyle {\frac {x-y}{1+xy}}\right),&{\text{pokud platí }}xy<-1,x>0\\[12pt]-\pi +\operatorname {arctg} \left(\displaystyle {\frac {x-y}{1+xy}}\right),&{\text{pokud platí }}xy<-1,x<0.\end{cases}}}$

${\displaystyle \operatorname {arccotg} x\,+\,\operatorname {arccotg} y={\begin{cases}\operatorname {arccotg} \left(\displaystyle {\frac {xy-1}{x+y}}\right),&{\text{pokud platí }}x>-y,\\[10pt]\pi +\operatorname {arccotg} \left(\displaystyle {\frac {xy-1}{x+y}}\right),&{\text{pokud platí }}x<-y.\end{cases}}}$

${\displaystyle \arcsin x+\arccos x={\frac {\pi }{2}},}$ pokud platí ${\displaystyle \ |x|\leq 1}$

${\displaystyle \operatorname {arctg} x+\operatorname {arccotg} x={\frac {\pi }{2}}}$

Cyklometrické funkce se dají také vyjádřit použitím logaritmů a komplexních čísel:

${\displaystyle {\begin{aligned}\arcsin x&{}=-\mathrm {i} \ln \left(\mathrm {i} x+{\sqrt {1-x^{2}}}\right)&{}\\[10pt]\arccos x&{}={\frac {\pi }{2}}\,+\mathrm {i} \ln \left(\mathrm {i} x+{\sqrt {1-x^{2}}}\right)={\frac {\pi }{2}}-\arcsin x&{}\\[10pt]\operatorname {arctg} x&{}={\frac {\mathrm {i} }{2}}\left(\ln \left(1-\mathrm {i} x\right)-\ln \left(1+\mathrm {i} x\right)\right)=\operatorname {arccotg} {\frac {1}{x}}\\[10pt]\operatorname {arccotg} x&{}={\frac {\mathrm {i} }{2}}\left(\ln \left(x-\mathrm {i} \right)-\ln \left(x+\mathrm {i} \right)\right)=\operatorname {arctg} {\frac {1}{x}}\end{aligned}}}$

Vztahy goniometrických a cyklometrických funkcí je možné jednoduše odvodit z pravoúhlého trojúhelníka ze znalosti Pythagorovy věty.

${\displaystyle \theta }$	${\displaystyle \sin \theta }$	${\displaystyle \cos \theta }$	${\displaystyle \operatorname {tg} \theta }$	Diagram
${\displaystyle \arcsin x}$	${\displaystyle \sin(\arcsin x)=x}$	${\displaystyle \cos(\arcsin x)={\sqrt {1-x^{2}}}}$	${\displaystyle \operatorname {tg} (\arcsin x)={\frac {x}{\sqrt {1-x^{2}}}}}$
${\displaystyle \arccos x}$	${\displaystyle \sin(\arccos x)={\sqrt {1-x^{2}}}}$	${\displaystyle \cos(\arccos x)=x}$	${\displaystyle \operatorname {tg} (\arccos x)={\frac {\sqrt {1-x^{2}}}{x}}}$
${\displaystyle \operatorname {arctg} x}$	${\displaystyle \sin(\operatorname {arctg} x)={\frac {x}{\sqrt {1+x^{2}}}}}$	${\displaystyle \cos(\operatorname {arctg} x)={\frac {1}{\sqrt {1+x^{2}}}}}$	${\displaystyle \operatorname {tg} (\operatorname {arctg} x)=x}$
${\displaystyle \operatorname {arccotg} x}$	${\displaystyle \sin(\operatorname {arccotg} x)={\frac {1}{\sqrt {1+x^{2}}}}}$	${\displaystyle \cos(\operatorname {arccotg} x)={\frac {x}{\sqrt {1+x^{2}}}}}$	${\displaystyle \operatorname {tg} (\operatorname {arccotg} x)={\frac {1}{x}}}$
${\displaystyle \operatorname {arcsec} x}$	${\displaystyle \sin(\operatorname {arcsec} x)={\frac {\sqrt {x^{2}-1}}{x}}}$	${\displaystyle \cos(\operatorname {arcsec} x)={\frac {1}{x}}}$	${\displaystyle \operatorname {tg} (\operatorname {arcsec} x)={\sqrt {x^{2}-1}}}$
${\displaystyle \operatorname {arccsc} x}$	${\displaystyle \sin(\operatorname {arccsc} x)={\frac {1}{x}}}$	${\displaystyle \cos(\operatorname {arccsc} x)={\frac {\sqrt {x^{2}-1}}{x}}}$	${\displaystyle \operatorname {tg} (\operatorname {arccsc} x)={\frac {1}{\sqrt {x^{2}-1}}}}$

Rozvoj cyklometrických funkcí lze psát jako:

${\displaystyle {\begin{aligned}\arcsin z&=z\,+\,\left({\frac {1}{2}}\right){\frac {z^{3}}{3}}\,+\,\left({\frac {1\cdot 3}{2\cdot 4}}\right){\frac {z^{5}}{5}}\,+\,\left({\frac {1\cdot 3\cdot 5}{2\cdot 4\cdot 6}}\right){\frac {z^{7}}{7}}\,+\,\dots \ =\sum _{n=0}^{\infty }{\frac {{\binom {2n}{n}}z^{2n+1}}{4^{n}(2n+1)}}\,,\qquad {\text{je-li }}|z|\leq 1\\[10pt]\arccos z&={\frac {\pi }{2}}\,-\,\arcsin z={\frac {\pi }{2}}\,-\,\left(z+\left({\frac {1}{2}}\right){\frac {z^{3}}{3}}+\left({\frac {1\cdot 3}{2\cdot 4}}\right){\frac {z^{5}}{5}}+\dots \ \right)={\frac {\pi }{2}}\,-\,\sum _{n=0}^{\infty }{\frac {{\binom {2n}{n}}z^{2n+1}}{4^{n}(2n+1)}}\,,\qquad {\text{je-li }}|z|\leq 1\\[10pt]\operatorname {arctg} z&=z-{\frac {z^{3}}{3}}+{\frac {z^{5}}{5}}-{\frac {z^{7}}{7}}+\dots \ =\sum _{n=0}^{\infty }{\frac {(-1)^{n}z^{2n+1}}{2n+1}}\,,\qquad {\text{je-li }}|z|\leq 1,\ z\neq \pm \mathrm {i} \\[10pt]\operatorname {arccotg} z&={\frac {\pi }{2}}-\operatorname {arctg} z\ ={\frac {\pi }{2}}\,-\,\left(z-{\frac {z^{3}}{3}}+{\frac {z^{5}}{5}}-{\frac {z^{7}}{7}}+\dots \ \right)={\frac {\pi }{2}}\,-\,\sum _{n=0}^{\infty }{\frac {(-1)^{n}z^{2n+1}}{2n+1}}\,,\qquad {\text{je-li }}|z|\leq 1,\ z\neq \pm \mathrm {i} \\[10pt]\operatorname {arcsec} z&=\arccos {(1/z)}={\frac {\pi }{2}}\,-\,\left(z^{-1}+\left({\frac {1}{2}}\right){\frac {z^{-3}}{3}}+\left({\frac {1\cdot 3}{2\cdot 4}}\right){\frac {z^{-5}}{5}}+\dots \ \right)={\frac {\pi }{2}}\,-\,\sum _{n=0}^{\infty }{\frac {{\binom {2n}{n}}z^{-(2n+1)}}{4^{n}(2n+1)}}\,,\qquad {\text{je-li }}|z|\geq 1\\[10pt]\operatorname {arccsc} z&=\arcsin {(1/z)}=z^{-1}\,+\,\left({\frac {1}{2}}\right){\frac {z^{-3}}{3}}\,+\,\left({\frac {1\cdot 3}{2\cdot 4}}\right){\frac {z^{-5}}{5}}\,+\,\dots \ =\sum _{n=0}^{\infty }{\frac {{\binom {2n}{n}}z^{-(2n+1)}}{4^{n}(2n+1)}}\,,\qquad {\text{je-li }}|z|\geq 1\end{aligned}}}$

V tomto článku byl použit překlad textu z článku Cyklometrická funkcia na slovenské Wikipedii.

• REKTORYS, Karel, a spol. Přehled užité matematiky I.. 7. vyd. Praha: Prometheus, 2000. ISBN 978-80-7196-179-6. OCLC 85008168 
• HANS-JOCHEN, Bartsch. Matematické vzorce. 2., revidované vyd. Praha: SNTL, 1987. 832 s. OCLC 39432784 

• Obrázky, zvuky či videa k tématu cyklometrická funkce na Wikimedia Commons